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

This library is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.

This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the COPYING file for more details.

Unit in the Last Place: our definition using fexp and its properties, successor and predecessor

Require Import Reals Psatz.
Require Import Raux Defs Round_pred Generic_fmt Float_prop.

Section Fcore_ulp.

Variable beta : radix.

Notation bpow e := (bpow beta e).

Variable fexp : Z -> Z.

Definition and basic properties about the minimal exponent, when it exists

Lemma Z_le_dec_aux: forall x y : Z, (x <= y)%Z \/ ~ (x <= y)%Z.beta:radix
fexp:Z -> Z
forall x y : Z, (x <= y)%Z \/ ~ (x <= y)%Z
Proof.beta:radix
fexp:Z -> Z
forall x y : Z, (x <= y)%Z \/ ~ (x <= y)%Z
intros.beta:radix
fexp:Z -> Z
x, y:Z
(x <= y)%Z \/ ~ (x <= y)%Z
destruct (Z_le_dec x y).beta:radix
fexp:Z -> Z
x, y:Z
l:(x <= y)%Z
(x <= y)%Z \/ ~ (x <= y)%Z
beta:radix
fexp:Z -> Z
x, y:Z
n:~ (x <= y)%Z
(x <= y)%Z \/ ~ (x <= y)%Z
now left.beta:radix
fexp:Z -> Z
x, y:Z
n:~ (x <= y)%Z
(x <= y)%Z \/ ~ (x <= y)%Z
now right.
Qed.

negligible_exp is either none (as in FLX) or Some n such that n <= fexp n.

Definition negligible_exp: option Z :=
  match (LPO_Z _ (fun z => Z_le_dec_aux z (fexp z))) with
   | inleft N => Some (proj1_sig N)
   | inright _ => None
 end.


Inductive negligible_exp_prop: option Z -> Prop :=
  | negligible_None: (forall n, (fexp n < n)%Z) -> negligible_exp_prop None
  | negligible_Some: forall n, (n <= fexp n)%Z -> negligible_exp_prop (Some n).


Lemma negligible_exp_spec: negligible_exp_prop negligible_exp.beta:radix
fexp:Z -> Z
negligible_exp_prop negligible_exp
Proof.beta:radix
fexp:Z -> Z
negligible_exp_prop negligible_exp
unfold negligible_exp; destruct LPO_Z as [(n,Hn)|Hn].beta:radix
fexp:Z -> Z
n:Z
Hn:(n <= fexp n)%Z
negligible_exp_prop
  (Some
     (proj1_sig
        (exist (fun n0 : Z => (n0 <= fexp n0)%Z) n Hn)))
beta:radix
fexp:Z -> Z
Hn:forall n : Z, ~ (n <= fexp n)%Z
negligible_exp_prop None
now apply negligible_Some.beta:radix
fexp:Z -> Z
Hn:forall n : Z, ~ (n <= fexp n)%Z
negligible_exp_prop None
apply negligible_None.beta:radix
fexp:Z -> Z
Hn:forall n : Z, ~ (n <= fexp n)%Z
forall n : Z, (fexp n < n)%Z
intros n; specialize (Hn n); omega.
Qed.

Lemma negligible_exp_spec': (negligible_exp = None /\ forall n, (fexp n < n)%Z)
           \/ exists n, (negligible_exp = Some n /\ (n <= fexp n)%Z).beta:radix
fexp:Z -> Z
negligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) \/
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z)
Proof.beta:radix
fexp:Z -> Z
negligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) \/
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z)
unfold negligible_exp; destruct LPO_Z as [(n,Hn)|Hn].beta:radix
fexp:Z -> Z
n:Z
Hn:(n <= fexp n)%Z
Some
  (proj1_sig
     (exist (fun n0 : Z => (n0 <= fexp n0)%Z) n Hn)) =
None /\ (forall n0 : Z, (fexp n0 < n0)%Z) \/
(exists n0 : Z,
   Some
     (proj1_sig
        (exist (fun n1 : Z => (n1 <= fexp n1)%Z) n Hn)) =
   Some n0 /\ (n0 <= fexp n0)%Z)
beta:radix
fexp:Z -> Z
Hn:forall n : Z, ~ (n <= fexp n)%Z
None = None /\ (forall n : Z, (fexp n < n)%Z) \/
(exists n : Z, None = Some n /\ (n <= fexp n)%Z)
right; simpl; exists n; now split.beta:radix
fexp:Z -> Z
Hn:forall n : Z, ~ (n <= fexp n)%Z
None = None /\ (forall n : Z, (fexp n < n)%Z) \/
(exists n : Z, None = Some n /\ (n <= fexp n)%Z)
left; split; trivial.beta:radix
fexp:Z -> Z
Hn:forall n : Z, ~ (n <= fexp n)%Z
forall n : Z, (fexp n < n)%Z
intros n; specialize (Hn n); omega.
Qed.

Context { valid_exp : Valid_exp fexp }.

Lemma fexp_negligible_exp_eq: forall n m, (n <= fexp n)%Z -> (m <= fexp m)%Z -> fexp n = fexp m.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall n m : Z,
(n <= fexp n)%Z -> (m <= fexp m)%Z -> fexp n = fexp m
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall n m : Z,
(n <= fexp n)%Z -> (m <= fexp m)%Z -> fexp n = fexp m
intros n m Hn Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n, m:Z
Hn:(n <= fexp n)%Z
Hm:(m <= fexp m)%Z
fexp n = fexp m
case (Zle_or_lt n m); intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n, m:Z
Hn:(n <= fexp n)%Z
Hm:(m <= fexp m)%Z
H:(n <= m)%Z
fexp n = fexp m
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n, m:Z
Hn:(n <= fexp n)%Z
Hm:(m <= fexp m)%Z
H:(m < n)%Z
fexp n = fexp m
apply valid_exp; omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n, m:Z
Hn:(n <= fexp n)%Z
Hm:(m <= fexp m)%Z
H:(m < n)%Z
fexp n = fexp m
apply sym_eq, valid_exp; omega.
Qed.

Definition and basic properties about the ulp

Now includes a nice ulp(0): ulp(0) is now 0 when there is no minimal exponent, such as in FLX, and beta^(fexp n) when there is a n such that n <= fexp n. For instance, the value of ulp(O) is then beta^emin in FIX and FLT. The main lemma to use is ulp_neq_0 that is equivalent to the previous "unfold ulp" provided the value is not zero.

Definition ulp x := match Req_bool x 0 with
  | true   => match negligible_exp with
            | Some n => bpow (fexp n)
            | None => 0%R
            end
  | false  => bpow (cexp beta fexp x)
 end.

Lemma ulp_neq_0 :
  forall x, x <> 0%R ->
  ulp x = bpow (cexp beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
x <> 0%R -> ulp x = bpow (cexp beta fexp x)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
x <> 0%R -> ulp x = bpow (cexp beta fexp x)
intros  x Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:x <> 0%R
ulp x = bpow (cexp beta fexp x)
unfold ulp; case (Req_bool_spec x); trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:x <> 0%R
x = 0%R ->
match negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end = bpow (cexp beta fexp x)
intros H; now contradict H.
Qed.

Notation F := (generic_format beta fexp).

Theorem ulp_opp :
  forall x, ulp (- x) = ulp x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, ulp (- x) = ulp x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, ulp (- x) = ulp x
intros x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
ulp (- x) = ulp x
unfold ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
(if Req_bool (- x) 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp (- x))) =
(if Req_bool x 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp x))
case Req_bool_spec; intros H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:(- x)%R = 0%R
match negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end =
(if Req_bool x 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:(- x)%R <> 0%R
bpow (cexp beta fexp (- x)) =
(if Req_bool x 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp x))
rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:(- x)%R = 0%R
x = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:(- x)%R <> 0%R
bpow (cexp beta fexp (- x)) =
(if Req_bool x 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp x))
rewrite <- (Ropp_involutive x), H1; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:(- x)%R <> 0%R
bpow (cexp beta fexp (- x)) =
(if Req_bool x 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp x))
rewrite Req_bool_false.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:(- x)%R <> 0%R
bpow (cexp beta fexp (- x)) = bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:(- x)%R <> 0%R
x <> 0%R
now rewrite cexp_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:(- x)%R <> 0%R
x <> 0%R
intros H2; apply H1; rewrite H2; ring.
Qed.

Theorem ulp_abs :
  forall x, ulp (Rabs x) = ulp x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, ulp (Rabs x) = ulp x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, ulp (Rabs x) = ulp x
intros x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
ulp (Rabs x) = ulp x
unfold ulp; case (Req_bool_spec x 0); intros H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:x = 0%R
(if Req_bool (Rabs x) 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp (Rabs x))) =
match negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:x <> 0%R
(if Req_bool (Rabs x) 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp (Rabs x))) =
bpow (cexp beta fexp x)
rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:x = 0%R
Rabs x = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:x <> 0%R
(if Req_bool (Rabs x) 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp (Rabs x))) =
bpow (cexp beta fexp x)
now rewrite H1, Rabs_R0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:x <> 0%R
(if Req_bool (Rabs x) 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp (Rabs x))) =
bpow (cexp beta fexp x)
rewrite Req_bool_false.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:x <> 0%R
bpow (cexp beta fexp (Rabs x)) =
bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:x <> 0%R
Rabs x <> 0%R
now rewrite cexp_abs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H1:x <> 0%R
Rabs x <> 0%R
now apply Rabs_no_R0.
Qed.

Theorem ulp_ge_0:
  forall x, (0 <= ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 <= ulp x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 <= ulp x)%R
intros x; unfold ulp; case Req_bool_spec; intros.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H:x = 0%R
(0 <=
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H:x <> 0%R
(0 <= bpow (cexp beta fexp x))%R
case negligible_exp; intros.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H:x = 0%R
z:Z
(0 <= bpow (fexp z))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H:x = 0%R
(0 <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H:x <> 0%R
(0 <= bpow (cexp beta fexp x))%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H:x = 0%R
(0 <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H:x <> 0%R
(0 <= bpow (cexp beta fexp x))%R
apply Rle_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
H:x <> 0%R
(0 <= bpow (cexp beta fexp x))%R
apply bpow_ge_0.
Qed.


Theorem ulp_le_id:
  forall x,
    (0 < x)%R ->
    F x ->
    (ulp x <= x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 < x)%R -> F x -> (ulp x <= x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 < x)%R -> F x -> (ulp x <= x)%R
intros x Zx Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(ulp x <= x)%R
rewrite <- (Rmult_1_l (ulp x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(1 * ulp x <= x)%R
pattern x at 2; rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(1 * ulp x <=
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(1 * bpow (cexp beta fexp x) <=
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
x <> 0%R
2: now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(1 * bpow (cexp beta fexp x) <=
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
unfold F2R; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(1 * bpow (cexp beta fexp x) <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x))%R
apply Rmult_le_compat_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(0 <= bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R
apply IZR_le, (Zlt_le_succ 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
apply gt_0_F2R with beta (cexp beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
now rewrite <- Fx.
Qed.

Theorem ulp_le_abs:
  forall x,
    (x <> 0)%R ->
    F x ->
    (ulp x <= Rabs x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, x <> 0%R -> F x -> (ulp x <= Rabs x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, x <> 0%R -> F x -> (ulp x <= Rabs x)%R
intros x Zx Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Fx:F x
(ulp x <= Rabs x)%R
rewrite <- ulp_abs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Fx:F x
(ulp (Rabs x) <= Rabs x)%R
apply ulp_le_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Fx:F x
(0 < Rabs x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Fx:F x
F (Rabs x)
now apply Rabs_pos_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Fx:F x
F (Rabs x)
now apply generic_format_abs.
Qed.

Theorem round_UP_DN_ulp :
  forall x, ~ F x ->
  round beta fexp Zceil x = (round beta fexp Zfloor x + ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
~ F x ->
round beta fexp Zceil x =
(round beta fexp Zfloor x + ulp x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
~ F x ->
round beta fexp Zceil x =
(round beta fexp Zfloor x + ulp x)%R
intros x Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
round beta fexp Zceil x =
(round beta fexp Zfloor x + ulp x)%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
round beta fexp Zceil x =
(round beta fexp Zfloor x + bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
unfold round.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
F2R
  {|
  Fnum := Zceil (scaled_mantissa beta fexp x);
  Fexp := cexp beta fexp x |} =
(F2R
   {|
   Fnum := Zfloor (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} +
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
F2R
  {|
  Fnum := Zceil (scaled_mantissa beta fexp x);
  Fexp := cexp beta fexp x |} =
(F2R
   {|
   Fnum := Zfloor (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} +
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
(IZR
   (Fnum
      {|
      Fnum := Zceil (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) *
 bpow
   (Fexp
      {|
      Fnum := Zceil (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}))%R =
(IZR
   (Fnum
      {|
      Fnum := Zfloor (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) *
 bpow
   (Fexp
      {|
      Fnum := Zfloor (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) +
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
(IZR (Zceil (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x))%R =
(IZR (Zfloor (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) + bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
rewrite Zceil_floor_neq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
(IZR (Zfloor (scaled_mantissa beta fexp x) + 1) *
 bpow (cexp beta fexp x))%R =
(IZR (Zfloor (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) + bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
IZR (Zfloor (scaled_mantissa beta fexp x)) <>
scaled_mantissa beta fexp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
rewrite plus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
((IZR (Zfloor (scaled_mantissa beta fexp x)) + 1) *
 bpow (cexp beta fexp x))%R =
(IZR (Zfloor (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) + bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
IZR (Zfloor (scaled_mantissa beta fexp x)) <>
scaled_mantissa beta fexp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
((IZR (Zfloor (scaled_mantissa beta fexp x)) + 1) *
 bpow (cexp beta fexp x))%R =
(IZR (Zfloor (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) + bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
IZR (Zfloor (scaled_mantissa beta fexp x)) <>
scaled_mantissa beta fexp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
IZR (Zfloor (scaled_mantissa beta fexp x)) <>
scaled_mantissa beta fexp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
H:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp x
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
apply Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
H:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp x
F x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
unfold generic_format, F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
H:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp x
x =
(IZR
   (Fnum
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) *
 bpow
   (Fexp
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
H:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp x
x =
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
rewrite <- H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
H:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp x
x =
(IZR
   (Ztrunc
      (IZR (Zfloor (scaled_mantissa beta fexp x)))) *
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
rewrite Ztrunc_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
H:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp x
x =
(IZR (Zfloor (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
rewrite H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
H:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp x
x =
(scaled_mantissa beta fexp x * bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
now rewrite scaled_mantissa_mult_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
x <> 0%R
intros V; apply Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
V:x = 0%R
F x
rewrite V.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
V:x = 0%R
F 0
apply generic_format_0.
Qed.

Theorem ulp_canonical :
  forall m e,
  m <> 0%Z ->
  canonical beta fexp (Float beta m e) ->
  ulp (F2R (Float beta m e)) = bpow e.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall m e : Z,
m <> 0%Z ->
canonical beta fexp {| Fnum := m; Fexp := e |} ->
ulp (F2R {| Fnum := m; Fexp := e |}) = bpow e
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall m e : Z,
m <> 0%Z ->
canonical beta fexp {| Fnum := m; Fexp := e |} ->
ulp (F2R {| Fnum := m; Fexp := e |}) = bpow e
intros m e Hm Hc.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
m, e:Z
Hm:m <> 0%Z
Hc:canonical beta fexp {| Fnum := m; Fexp := e |}
ulp (F2R {| Fnum := m; Fexp := e |}) = bpow e
rewrite ulp_neq_0 by now apply F2R_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
m, e:Z
Hm:m <> 0%Z
Hc:canonical beta fexp {| Fnum := m; Fexp := e |}
bpow (cexp beta fexp (F2R {| Fnum := m; Fexp := e |})) =
bpow e
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
m, e:Z
Hm:m <> 0%Z
Hc:canonical beta fexp {| Fnum := m; Fexp := e |}
cexp beta fexp (F2R {| Fnum := m; Fexp := e |}) = e
now apply sym_eq.
Qed.

Theorem ulp_bpow :
  forall e, ulp (bpow e) = bpow (fexp (e + 1)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall e : Z, ulp (bpow e) = bpow (fexp (e + 1))
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall e : Z, ulp (bpow e) = bpow (fexp (e + 1))
intros e.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
ulp (bpow e) = bpow (fexp (e + 1))
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow (cexp beta fexp (bpow e)) = bpow (fexp (e + 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow e <> 0%R
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
cexp beta fexp (bpow e) = fexp (e + 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow e <> 0%R
apply cexp_fexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(bpow (e + 1 - 1) <= Rabs (bpow e) < bpow (e + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow e <> 0%R
rewrite Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(bpow (e + 1 - 1) <= bpow e < bpow (e + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(0 <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow e <> 0%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(bpow (e + 1 - 1) <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(bpow e < bpow (e + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(0 <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow e <> 0%R
ring_simplify (e + 1 - 1)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(bpow e <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(bpow e < bpow (e + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(0 <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow e <> 0%R
apply Rle_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(bpow e < bpow (e + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(0 <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow e <> 0%R
apply bpow_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(e < e + 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(0 <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow e <> 0%R
apply Zlt_succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(0 <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow e <> 0%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
bpow e <> 0%R
apply Rgt_not_eq, Rlt_gt, bpow_gt_0.
Qed.

Lemma generic_format_ulp_0 :
  F (ulp 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
F (ulp 0)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
F (ulp 0)
unfold ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
F
  (if Req_bool 0 0
   then
    match negligible_exp with
    | Some n => bpow (fexp n)
    | None => 0%R
    end
   else bpow (cexp beta fexp 0))
rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
F
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end
case negligible_exp_spec.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(forall n : Z, (fexp n < n)%Z) -> F 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall n : Z, (n <= fexp n)%Z -> F (bpow (fexp n))
intros _; apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall n : Z, (n <= fexp n)%Z -> F (bpow (fexp n))
intros n H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n:Z
H1:(n <= fexp n)%Z
F (bpow (fexp n))
apply generic_format_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n:Z
H1:(n <= fexp n)%Z
(fexp (fexp n + 1) <= fexp n)%Z
now apply valid_exp.
Qed.

Lemma generic_format_bpow_ge_ulp_0 :
  forall e, (ulp 0 <= bpow e)%R ->
  F (bpow e).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall e : Z, (ulp 0 <= bpow e)%R -> F (bpow e)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall e : Z, (ulp 0 <= bpow e)%R -> F (bpow e)
intros e; unfold ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
((if Req_bool 0 0
  then
   match negligible_exp with
   | Some n => bpow (fexp n)
   | None => 0
   end
  else bpow (cexp beta fexp 0)) <= bpow e)%R ->
F (bpow e)
rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= bpow e)%R -> F (bpow e)
case negligible_exp_spec.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(forall n : Z, (fexp n < n)%Z) ->
(0 <= bpow e)%R -> F (bpow e)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
forall n : Z,
(n <= fexp n)%Z ->
(bpow (fexp n) <= bpow e)%R -> F (bpow e)
intros H1 _.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
H1:forall n : Z, (fexp n < n)%Z
F (bpow e)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
forall n : Z,
(n <= fexp n)%Z ->
(bpow (fexp n) <= bpow e)%R -> F (bpow e)
apply generic_format_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
H1:forall n : Z, (fexp n < n)%Z
(fexp (e + 1) <= e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
forall n : Z,
(n <= fexp n)%Z ->
(bpow (fexp n) <= bpow e)%R -> F (bpow e)
specialize (H1 (e+1)%Z); omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
forall n : Z,
(n <= fexp n)%Z ->
(bpow (fexp n) <= bpow e)%R -> F (bpow e)
intros n H1 H2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
F (bpow e)
apply generic_format_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
(fexp (e + 1) <= e)%Z
case (Zle_or_lt (e+1) (fexp (e+1))); intros H4.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(e + 1 <= fexp (e + 1))%Z
(fexp (e + 1) <= e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(fexp (e + 1) < e + 1)%Z
(fexp (e + 1) <= e)%Z
absurd (e+1 <= e)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(e + 1 <= fexp (e + 1))%Z
~ (e + 1 <= e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(e + 1 <= fexp (e + 1))%Z
(e + 1 <= e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(fexp (e + 1) < e + 1)%Z
(fexp (e + 1) <= e)%Z
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(e + 1 <= fexp (e + 1))%Z
(e + 1 <= e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(fexp (e + 1) < e + 1)%Z
(fexp (e + 1) <= e)%Z
apply Z.le_trans with (1:=H4).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(e + 1 <= fexp (e + 1))%Z
(fexp (e + 1) <= e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(fexp (e + 1) < e + 1)%Z
(fexp (e + 1) <= e)%Z
replace (fexp (e+1)) with (fexp n).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(e + 1 <= fexp (e + 1))%Z
(fexp n <= e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(e + 1 <= fexp (e + 1))%Z
fexp n = fexp (e + 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(fexp (e + 1) < e + 1)%Z
(fexp (e + 1) <= e)%Z
now apply le_bpow with beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(e + 1 <= fexp (e + 1))%Z
fexp n = fexp (e + 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(fexp (e + 1) < e + 1)%Z
(fexp (e + 1) <= e)%Z
now apply fexp_negligible_exp_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e, n:Z
H1:(n <= fexp n)%Z
H2:(bpow (fexp n) <= bpow e)%R
H4:(fexp (e + 1) < e + 1)%Z
(fexp (e + 1) <= e)%Z
omega.
Qed.

The three following properties are equivalent: Exp_not_FTZ ; forall x, F (ulp x) ; forall x, ulp 0 <= ulp x

Lemma generic_format_ulp :
  Exp_not_FTZ fexp ->
  forall x, F (ulp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Exp_not_FTZ fexp -> forall x : R, F (ulp x)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Exp_not_FTZ fexp -> forall x : R, F (ulp x)
unfold Exp_not_FTZ; intros H x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
F (ulp x)
case (Req_dec x 0); intros Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x = 0%R
F (ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
F (ulp x)
rewrite Hx; apply generic_format_ulp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
F (ulp x)
rewrite (ulp_neq_0 _ Hx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
F (bpow (cexp beta fexp x))
apply generic_format_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
(fexp (cexp beta fexp x + 1) <= cexp beta fexp x)%Z
apply H.
Qed.

Lemma not_FTZ_generic_format_ulp :
  (forall x,  F (ulp x)) ->
  Exp_not_FTZ fexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(forall x : R, F (ulp x)) -> Exp_not_FTZ fexp
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(forall x : R, F (ulp x)) -> Exp_not_FTZ fexp
intros H e.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall x : R, F (ulp x)
e:Z
(fexp (fexp e + 1) <= fexp e)%Z
specialize (H (bpow (e-1))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
H:F (ulp (bpow (e - 1)))
(fexp (fexp e + 1) <= fexp e)%Z
rewrite ulp_neq_0 in H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
H:F (bpow (cexp beta fexp (bpow (e - 1))))
(fexp (fexp e + 1) <= fexp e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
H:F (ulp (bpow (e - 1)))
bpow (e - 1) <> 0%R
2: apply Rgt_not_eq, bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
H:F (bpow (cexp beta fexp (bpow (e - 1))))
(fexp (fexp e + 1) <= fexp e)%Z
unfold cexp in H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
H:F (bpow (fexp (mag beta (bpow (e - 1)))))
(fexp (fexp e + 1) <= fexp e)%Z
rewrite mag_bpow in H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
H:F (bpow (fexp (e - 1 + 1)))
(fexp (fexp e + 1) <= fexp e)%Z
apply generic_format_bpow_inv' in H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
H:(fexp (fexp (e - 1 + 1) + 1) <= fexp (e - 1 + 1))%Z
(fexp (fexp e + 1) <= fexp e)%Z
now replace (e-1+1)%Z with e in H by ring.
Qed.


Lemma ulp_ge_ulp_0 :
  Exp_not_FTZ fexp ->
  forall x, (ulp 0 <= ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Exp_not_FTZ fexp -> forall x : R, (ulp 0 <= ulp x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Exp_not_FTZ fexp -> forall x : R, (ulp 0 <= ulp x)%R
unfold Exp_not_FTZ; intros H x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
(ulp 0 <= ulp x)%R
case (Req_dec x 0); intros Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x = 0%R
(ulp 0 <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
(ulp 0 <= ulp x)%R
rewrite Hx; now right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
(ulp 0 <= ulp x)%R
unfold ulp at 1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
((if Req_bool 0 0
  then
   match negligible_exp with
   | Some n => bpow (fexp n)
   | None => 0
   end
  else bpow (cexp beta fexp 0)) <= ulp x)%R
rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
(match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= ulp x)%R
case negligible_exp_spec'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
negligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) ->
(match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= ulp x)%R
intros (H1,H2); rewrite H1; apply ulp_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= ulp x)%R
intros (n,(H1,H2)); rewrite H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
(bpow (fexp n) <= ulp x)%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
(bpow (fexp n) <= bpow (cexp beta fexp x))%R
apply bpow_le; unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
(fexp n <= fexp (mag beta x))%Z
generalize (mag beta x); intros l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
l:mag_prop beta x
(fexp n <= fexp l)%Z
case (Zle_or_lt l (fexp l)); intros Hl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
l:mag_prop beta x
Hl:(l <= fexp l)%Z
(fexp n <= fexp l)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
l:mag_prop beta x
Hl:(fexp l < l)%Z
(fexp n <= fexp l)%Z
rewrite (fexp_negligible_exp_eq n l); trivial; apply Z.le_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
l:mag_prop beta x
Hl:(fexp l < l)%Z
(fexp n <= fexp l)%Z
case (Zle_or_lt (fexp n) (fexp l)); trivial; intros K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
l:mag_prop beta x
Hl:(fexp l < l)%Z
K:(fexp l < fexp n)%Z
(fexp n <= fexp l)%Z
absurd (fexp n <= fexp l)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
l:mag_prop beta x
Hl:(fexp l < l)%Z
K:(fexp l < fexp n)%Z
~ (fexp n <= fexp l)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
l:mag_prop beta x
Hl:(fexp l < l)%Z
K:(fexp l < fexp n)%Z
(fexp n <= fexp l)%Z
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
l:mag_prop beta x
Hl:(fexp l < l)%Z
K:(fexp l < fexp n)%Z
(fexp n <= fexp l)%Z
apply Z.le_trans with (2:= H _).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
l:mag_prop beta x
Hl:(fexp l < l)%Z
K:(fexp l < fexp n)%Z
(fexp n <= fexp (fexp l + 1))%Z
apply Zeq_le, sym_eq, valid_exp; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Z
x:R
Hx:x <> 0%R
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
l:mag_prop beta x
Hl:(fexp l < l)%Z
K:(fexp l < fexp n)%Z
(fexp l + 1 <= fexp n)%Z
omega.
Qed.

Lemma not_FTZ_ulp_ge_ulp_0:
  (forall x, (ulp 0 <= ulp x)%R) ->
  Exp_not_FTZ fexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(forall x : R, (ulp 0 <= ulp x)%R) -> Exp_not_FTZ fexp
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(forall x : R, (ulp 0 <= ulp x)%R) -> Exp_not_FTZ fexp
intros H e.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall x : R, (ulp 0 <= ulp x)%R
e:Z
(fexp (fexp e + 1) <= fexp e)%Z
apply generic_format_bpow_inv' with beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall x : R, (ulp 0 <= ulp x)%R
e:Z
F (bpow (fexp e))
apply generic_format_bpow_ge_ulp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall x : R, (ulp 0 <= ulp x)%R
e:Z
(ulp 0 <= bpow (fexp e))%R
replace e with ((e-1)+1)%Z by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall x : R, (ulp 0 <= ulp x)%R
e:Z
(ulp 0 <= bpow (fexp (e - 1 + 1)))%R
rewrite <- ulp_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:forall x : R, (ulp 0 <= ulp x)%R
e:Z
(ulp 0 <= ulp (bpow (e - 1)))%R
apply H.
Qed.

Lemma ulp_le_pos :
  forall { Hm : Monotone_exp fexp },
  forall x y: R,
  (0 <= x)%R -> (x <= y)%R ->
  (ulp x <= ulp y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall x y : R,
(0 <= x)%R -> (x <= y)%R -> (ulp x <= ulp y)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall x y : R,
(0 <= x)%R -> (x <= y)%R -> (ulp x <= ulp y)%R
intros Hm x y Hx Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 <= x)%R
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
destruct Hx as [Hx|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
(bpow (cexp beta fexp x) <= ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
(bpow (cexp beta fexp x) <= bpow (cexp beta fexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
apply bpow_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
(cexp beta fexp x <= cexp beta fexp y)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
apply Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
(mag beta x <= mag beta y)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
now apply mag_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
apply Rgt_not_eq, Rlt_gt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
(0 < y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
now apply Rlt_le_trans with (1:=Hx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:(0 < x)%R
Hxy:(x <= y)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
(ulp x <= ulp y)%R
rewrite <- Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
(ulp 0 <= ulp y)%R
apply ulp_ge_ulp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hx:0%R = x
Hxy:(x <= y)%R
Exp_not_FTZ fexp
apply monotone_exp_not_FTZ...
Qed.

Theorem ulp_le :
  forall { Hm : Monotone_exp fexp },
  forall x y: R,
  (Rabs x <= Rabs y)%R ->
  (ulp x <= ulp y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall x y : R,
(Rabs x <= Rabs y)%R -> (ulp x <= ulp y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall x y : R,
(Rabs x <= Rabs y)%R -> (ulp x <= ulp y)%R
intros Hm x y Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hxy:(Rabs x <= Rabs y)%R
(ulp x <= ulp y)%R
rewrite <- ulp_abs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hxy:(Rabs x <= Rabs y)%R
(ulp (Rabs x) <= ulp y)%R
rewrite <- (ulp_abs y).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hxy:(Rabs x <= Rabs y)%R
(ulp (Rabs x) <= ulp (Rabs y))%R
apply ulp_le_pos; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
x, y:R
Hxy:(Rabs x <= Rabs y)%R
(0 <= Rabs x)%R
apply Rabs_pos.
Qed.

Properties when there is no minimal exponent

Theorem eq_0_round_0_negligible_exp :
   negligible_exp = None -> forall rnd {Vr: Valid_rnd rnd} x,
     round beta fexp rnd x = 0%R -> x = 0%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
negligible_exp = None ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, round beta fexp rnd x = 0%R -> x = 0%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
negligible_exp = None ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, round beta fexp rnd x = 0%R -> x = 0%R
intros H rnd Vr x Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
x = 0%R
case (Req_dec x 0); try easy; intros Hx2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
x = 0%R
absurd (Rabs (round beta fexp rnd x) = 0%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
Rabs (round beta fexp rnd x) <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
Rabs (round beta fexp rnd x) = 0%R
2: rewrite Hx, Rabs_R0; easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
Rabs (round beta fexp rnd x) <> 0%R
apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(Rabs (round beta fexp rnd x) > 0)%R
apply Rlt_le_trans with (bpow (mag beta x - 1)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(0 < bpow (mag beta x - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(bpow (mag beta x - 1) <= Rabs (round beta fexp rnd x))%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(bpow (mag beta x - 1) <= Rabs (round beta fexp rnd x))%R
apply abs_round_ge_generic; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
F (bpow (mag beta x - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(bpow (mag beta x - 1) <= Rabs x)%R
apply generic_format_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(bpow (mag beta x - 1) <= Rabs x)%R
case negligible_exp_spec'; [intros (K1,K2)|idtac].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
K1:negligible_exp = None
K2:forall n : Z, (fexp n < n)%Z
(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(bpow (mag beta x - 1) <= Rabs x)%R
ring_simplify (mag beta x-1+1)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
K1:negligible_exp = None
K2:forall n : Z, (fexp n < n)%Z
(fexp (mag beta x) <= mag beta x - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(bpow (mag beta x - 1) <= Rabs x)%R
specialize (K2 (mag beta x)); now auto with zarith.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(bpow (mag beta x - 1) <= Rabs x)%R
intros (n,(Hn1,Hn2)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(bpow (mag beta x - 1) <= Rabs x)%R
rewrite Hn1 in H; discriminate.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rnd:R -> Z
Vr:Valid_rnd rnd
x:R
Hx:round beta fexp rnd x = 0%R
Hx2:x <> 0%R
(bpow (mag beta x - 1) <= Rabs x)%R
now apply bpow_mag_le.
Qed.

Definition and properties of pred and succ

Definition pred_pos x :=
  if Req_bool x (bpow (mag beta x - 1)) then
    (x - bpow (fexp (mag beta x - 1)))%R
  else
    (x - ulp x)%R.

Definition succ x :=
  if (Rle_bool 0 x) then
    (x+ulp x)%R
  else
    (- pred_pos (-x))%R.

Definition pred x := (- succ (-x))%R.

Theorem pred_eq_pos :
  forall x, (0 <= x)%R ->
  pred x = pred_pos x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 <= x)%R -> pred x = pred_pos x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 <= x)%R -> pred x = pred_pos x
intros x Hx; unfold pred, succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
(-
 (if Rle_bool 0 (- x)
  then - x + ulp (- x)
  else - pred_pos (- - x)))%R = pred_pos x
case Rle_bool_spec; intros Hx'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
(- (- x + ulp (- x)))%R = pred_pos x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(- x < 0)%R
(- - pred_pos (- - x))%R = pred_pos x
assert (K:(x = 0)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
x = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
K:x = 0%R
(- (- x + ulp (- x)))%R = pred_pos x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(- x < 0)%R
(- - pred_pos (- - x))%R = pred_pos x
apply Rle_antisym; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
(x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
K:x = 0%R
(- (- x + ulp (- x)))%R = pred_pos x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(- x < 0)%R
(- - pred_pos (- - x))%R = pred_pos x
apply Ropp_le_cancel.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
(- 0 <= - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
K:x = 0%R
(- (- x + ulp (- x)))%R = pred_pos x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(- x < 0)%R
(- - pred_pos (- - x))%R = pred_pos x
now rewrite Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
K:x = 0%R
(- (- x + ulp (- x)))%R = pred_pos x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(- x < 0)%R
(- - pred_pos (- - x))%R = pred_pos x
rewrite K; unfold pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
K:x = 0%R
(- (- 0 + ulp (- 0)))%R =
(if Req_bool 0 (bpow (mag beta 0 - 1))
 then (0 - bpow (fexp (mag beta 0 - 1)))%R
 else (0 - ulp 0)%R)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(- x < 0)%R
(- - pred_pos (- - x))%R = pred_pos x
rewrite Req_bool_false.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
K:x = 0%R
(- (- 0 + ulp (- 0)))%R = (0 - ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
K:x = 0%R
0%R <> bpow (mag beta 0 - 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(- x < 0)%R
(- - pred_pos (- - x))%R = pred_pos x
2: apply Rlt_not_eq, bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(0 <= - x)%R
K:x = 0%R
(- (- 0 + ulp (- 0)))%R = (0 - ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(- x < 0)%R
(- - pred_pos (- - x))%R = pred_pos x
rewrite Ropp_0; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
Hx':(- x < 0)%R
(- - pred_pos (- - x))%R = pred_pos x
now rewrite 2!Ropp_involutive.
Qed.

Theorem succ_eq_pos :
  forall x, (0 <= x)%R ->
  succ x = (x + ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 <= x)%R -> succ x = (x + ulp x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 <= x)%R -> succ x = (x + ulp x)%R
intros x Hx; unfold succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 <= x)%R
(if Rle_bool 0 x
 then (x + ulp x)%R
 else (- pred_pos (- x))%R) = (x + ulp x)%R
now rewrite Rle_bool_true.
Qed.

Theorem succ_opp :
  forall x, succ (-x) = (- pred x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, succ (- x) = (- pred x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, succ (- x) = (- pred x)%R
intros x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
succ (- x) = (- pred x)%R
now apply sym_eq, Ropp_involutive.
Qed.

Theorem pred_opp :
  forall x, pred (-x) = (- succ x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, pred (- x) = (- succ x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, pred (- x) = (- succ x)%R
intros x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
pred (- x) = (- succ x)%R
unfold pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
(- succ (- - x))%R = (- succ x)%R
now rewrite Ropp_involutive.
Qed.

Theorem pred_bpow :
  forall e, pred (bpow e) = (bpow e - bpow (fexp e))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall e : Z,
pred (bpow e) = (bpow e - bpow (fexp e))%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall e : Z,
pred (bpow e) = (bpow e - bpow (fexp e))%R
intros e.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
pred (bpow e) = (bpow e - bpow (fexp e))%R
rewrite pred_eq_pos by apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
pred_pos (bpow e) = (bpow e - bpow (fexp e))%R
unfold pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(if Req_bool (bpow e) (bpow (mag beta (bpow e) - 1))
 then (bpow e - bpow (fexp (mag beta (bpow e) - 1)))%R
 else (bpow e - ulp (bpow e))%R) =
(bpow e - bpow (fexp e))%R
rewrite mag_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(if Req_bool (bpow e) (bpow (e + 1 - 1))
 then (bpow e - bpow (fexp (e + 1 - 1)))%R
 else (bpow e - ulp (bpow e))%R) =
(bpow e - bpow (fexp e))%R
replace (e + 1 - 1)%Z with e by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
e:Z
(if Req_bool (bpow e) (bpow e)
 then (bpow e - bpow (fexp e))%R
 else (bpow e - ulp (bpow e))%R) =
(bpow e - bpow (fexp e))%R
now rewrite Req_bool_true.
Qed.

pred and succ are in the format

(* cannont be x <> ulp 0, due to the counter-example 1-bit FP format fexp: e -> e-1 *)
(* was pred_ge_bpow *)
Theorem id_m_ulp_ge_bpow :
  forall x e,  F x ->
  x <> ulp x ->
  (bpow e < x)%R ->
  (bpow e <= x - ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (x : R) (e : Z),
F x ->
x <> ulp x ->
(bpow e < x)%R -> (bpow e <= x - ulp x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (x : R) (e : Z),
F x ->
x <> ulp x ->
(bpow e < x)%R -> (bpow e <= x - ulp x)%R
intros x e Fx Hx' Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
(bpow e <= x - ulp x)%R
(* *)
assert (1 <= Ztrunc (scaled_mantissa beta fexp x))%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <= x - ulp x)%R
assert (0 <  Ztrunc (scaled_mantissa beta fexp x))%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <= x - ulp x)%R
apply gt_0_F2R with beta (cexp beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <= x - ulp x)%R
rewrite <- Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
(0 < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <= x - ulp x)%R
apply Rle_lt_trans with (2:=Hx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
(0 <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <= x - ulp x)%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <= x - ulp x)%R
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <= x - ulp x)%R
case (Zle_lt_or_eq _ _ H); intros Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <= x - ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
(* *)
pattern x at 1 ; rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <=
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} - ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <=
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} -
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <=
 IZR
   (Fnum
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) *
 bpow
   (Fexp
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) -
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) - bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
pattern (bpow (cexp beta fexp x)) at 2 ; rewrite <- Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) - 1 * bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
rewrite <- Rmult_minus_distr_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <=
 (IZR (Ztrunc (scaled_mantissa beta fexp x)) - 1) *
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
rewrite <- minus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <=
 IZR (Ztrunc (scaled_mantissa beta fexp x) - 1) *
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
apply bpow_le_F2R_m1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(bpow e <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
now rewrite <- Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
apply Rgt_not_eq, Rlt_gt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z
(0 < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
apply Rlt_trans with (2:=Hx), bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx':x <> ulp x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(bpow e <= x - ulp x)%R
(* *)
contradict Hx'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
x = ulp x
pattern x at 1; rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp x);
  Fexp := cexp beta fexp x |} = ulp x
rewrite  <- Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
F2R {| Fnum := 1; Fexp := cexp beta fexp x |} = ulp x
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
F2R {| Fnum := 1; Fexp := cexp beta fexp x |} =
bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
x <> 0%R
unfold F2R; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(1 * bpow (cexp beta fexp x))%R =
bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
x <> 0%R
now rewrite Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
x <> 0%R
apply Rgt_not_eq, Rlt_gt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Fx:F x
Hx:(bpow e < x)%R
H:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
Hm:1%Z = Ztrunc (scaled_mantissa beta fexp x)
(0 < x)%R
apply Rlt_trans with (2:=Hx), bpow_gt_0.
Qed.

(* was succ_le_bpow *)
Theorem id_p_ulp_le_bpow :
  forall x e, (0 < x)%R -> F x ->
  (x < bpow e)%R ->
  (x + ulp x <= bpow e)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (x : R) (e : Z),
(0 < x)%R ->
F x -> (x < bpow e)%R -> (x + ulp x <= bpow e)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (x : R) (e : Z),
(0 < x)%R ->
F x -> (x < bpow e)%R -> (x + ulp x <= bpow e)%R
intros x e Zx Fx Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(x + ulp x <= bpow e)%R
pattern x at 1 ; rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} + ulp x <= bpow e)%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} +
 bpow (cexp beta fexp x) <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
x <> 0%R
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(IZR
   (Fnum
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) *
 bpow
   (Fexp
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) +
 bpow (cexp beta fexp x) <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
x <> 0%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) + bpow (cexp beta fexp x) <=
 bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
x <> 0%R
pattern (bpow (cexp beta fexp x)) at 2 ; rewrite <- Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) + 1 * bpow (cexp beta fexp x) <=
 bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
x <> 0%R
rewrite <- Rmult_plus_distr_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
((IZR (Ztrunc (scaled_mantissa beta fexp x)) + 1) *
 bpow (cexp beta fexp x) <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
x <> 0%R
rewrite <- plus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x) + 1) *
 bpow (cexp beta fexp x) <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
x <> 0%R
apply F2R_p1_le_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < 
 bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
x <> 0%R
apply gt_0_F2R with beta (cexp beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < 
 bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
x <> 0%R
now rewrite <- Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
x <> 0%R
now rewrite <- Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
e:Z
Zx:(0 < x)%R
Fx:F x
Hx:(x < bpow e)%R
x <> 0%R
now apply Rgt_not_eq.
Qed.

Lemma generic_format_pred_aux1:
  forall x, (0 < x)%R -> F x ->
  x <> bpow (mag beta x - 1) ->
  F (x - ulp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x -> x <> bpow (mag beta x - 1) -> F (x - ulp x)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x -> x <> bpow (mag beta x - 1) -> F (x - ulp x)
intros x Zx Fx Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
F (x - ulp x)
destruct (mag beta x) as (ex, Ex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (Build_mag_prop beta x ex Ex - 1)
F (x - ulp x)
simpl in Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
F (x - ulp x)
specialize (Ex (Rgt_not_eq _ _ Zx)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
F (x - ulp x)
assert (Ex' : (bpow (ex - 1) < x < bpow ex)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
(bpow (ex - 1) < x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
F (x - ulp x)
rewrite Rabs_pos_eq in Ex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= x < bpow ex)%R
Hx:x <> bpow (ex - 1)
(bpow (ex - 1) < x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
F (x - ulp x)
destruct Ex as (H,H'); destruct H; split; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
H:bpow (ex - 1) = x
H':(x < bpow ex)%R
Hx:x <> bpow (ex - 1)
(bpow (ex - 1) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
F (x - ulp x)
contradict Hx; easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
F (x - ulp x)
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
F (x - ulp x)
unfold generic_format, scaled_mantissa, cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x)%R =
F2R
  {|
  Fnum := Ztrunc
            ((x - ulp x) *
             bpow (- fexp (mag beta (x - ulp x))));
  Fexp := fexp (mag beta (x - ulp x)) |}
rewrite mag_unique with beta (x - ulp x)%R ex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x)%R =
F2R
  {|
  Fnum := Ztrunc ((x - ulp x) * bpow (- fexp ex));
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
pattern x at 1 3 ; rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} - ulp x)%R =
F2R
  {|
  Fnum := Ztrunc
            ((F2R
                {|
                Fnum := Ztrunc
                          (scaled_mantissa beta fexp x);
                Fexp := cexp beta fexp x |} - ulp x) *
             bpow (- fexp ex));
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} -
 bpow (cexp beta fexp x))%R =
F2R
  {|
  Fnum := Ztrunc
            ((F2R
                {|
                Fnum := Ztrunc
                          (scaled_mantissa beta fexp x);
                Fexp := cexp beta fexp x |} -
              bpow (cexp beta fexp x)) *
             bpow (- fexp ex));
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
unfold scaled_mantissa.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (x * bpow (- cexp beta fexp x));
   Fexp := cexp beta fexp x |} -
 bpow (cexp beta fexp x))%R =
F2R
  {|
  Fnum := Ztrunc
            ((F2R
                {|
                Fnum := Ztrunc
                          (x *
                           bpow (- cexp beta fexp x));
                Fexp := cexp beta fexp x |} -
              bpow (cexp beta fexp x)) *
             bpow (- fexp ex));
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
rewrite cexp_fexp with (1 := Ex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (x * bpow (- fexp ex));
   Fexp := fexp ex |} - bpow (fexp ex))%R =
F2R
  {|
  Fnum := Ztrunc
            ((F2R
                {|
                Fnum := Ztrunc (x * bpow (- fexp ex));
                Fexp := fexp ex |} - bpow (fexp ex)) *
             bpow (- fexp ex));
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(IZR
   (Fnum
      {|
      Fnum := Ztrunc (x * bpow (- fexp ex));
      Fexp := fexp ex |}) *
 bpow
   (Fexp
      {|
      Fnum := Ztrunc (x * bpow (- fexp ex));
      Fexp := fexp ex |}) - bpow (fexp ex))%R =
(IZR
   (Fnum
      {|
      Fnum := Ztrunc
                ((IZR
                    (Fnum
                       {|
                       Fnum := Ztrunc
                                 (x * bpow (- fexp ex));
                       Fexp := fexp ex |}) *
                  bpow
                    (Fexp
                       {|
                       Fnum := Ztrunc
                                 (x * bpow (- fexp ex));
                       Fexp := fexp ex |}) -
                  bpow (fexp ex)) * bpow (- fexp ex));
      Fexp := fexp ex |}) *
 bpow
   (Fexp
      {|
      Fnum := Ztrunc
                ((IZR
                    (Fnum
                       {|
                       Fnum := Ztrunc
                                 (x * bpow (- fexp ex));
                       Fexp := fexp ex |}) *
                  bpow
                    (Fexp
                       {|
                       Fnum := Ztrunc
                                 (x * bpow (- fexp ex));
                       Fexp := fexp ex |}) -
                  bpow (fexp ex)) * bpow (- fexp ex));
      Fexp := fexp ex |}))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) -
 bpow (fexp ex))%R =
(IZR
   (Ztrunc
      ((IZR (Ztrunc (x * bpow (- fexp ex))) *
        bpow (fexp ex) - bpow (fexp ex)) *
       bpow (- fexp ex))) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
rewrite Rmult_minus_distr_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) -
 bpow (fexp ex))%R =
(IZR
   (Ztrunc
      (IZR (Ztrunc (x * bpow (- fexp ex))) *
       bpow (fexp ex) * bpow (- fexp ex) -
       bpow (fexp ex) * bpow (- fexp ex))) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
rewrite Rmult_assoc.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) -
 bpow (fexp ex))%R =
(IZR
   (Ztrunc
      (IZR (Ztrunc (x * bpow (- fexp ex))) *
       (bpow (fexp ex) * bpow (- fexp ex)) -
       bpow (fexp ex) * bpow (- fexp ex))) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
rewrite <- bpow_plus, Zplus_opp_r, Rmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) -
 bpow (fexp ex))%R =
(IZR
   (Ztrunc
      (IZR (Ztrunc (x * bpow (- fexp ex))) - bpow 0)) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
change (bpow 0) with 1%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) -
 bpow (fexp ex))%R =
(IZR
   (Ztrunc (IZR (Ztrunc (x * bpow (- fexp ex))) - 1)) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
rewrite <- minus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) -
 bpow (fexp ex))%R =
(IZR
   (Ztrunc (IZR (Ztrunc (x * bpow (- fexp ex)) - 1))) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
rewrite Ztrunc_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) -
 bpow (fexp ex))%R =
(IZR (Ztrunc (x * bpow (- fexp ex)) - 1) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
rewrite minus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) -
 bpow (fexp ex))%R =
((IZR (Ztrunc (x * bpow (- fexp ex))) - 1) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
rewrite Rmult_minus_distr_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) -
 bpow (fexp ex))%R =
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) -
 1 * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
now rewrite Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
rewrite Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) <= x - ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
apply id_m_ulp_ge_bpow; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
intro H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
H:x = bpow (cexp beta fexp x)
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
assert (ex-1 < cexp beta fexp x  < ex)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
H:x = bpow (cexp beta fexp x)
(ex - 1 < cexp beta fexp x < ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
H:x = bpow (cexp beta fexp x)
H0:(ex - 1 < cexp beta fexp x < ex)%Z
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
split ; apply (lt_bpow beta) ; rewrite <- H ; easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
H:x = bpow (cexp beta fexp x)
H0:(ex - 1 < cexp beta fexp x < ex)%Z
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
clear -H0.beta:radix
fexp:Z -> Z
x:R
ex:Z
H0:(ex - 1 < cexp beta fexp x < ex)%Z
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (ex - 1) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
apply Ex'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
apply Rle_lt_trans with (2 := proj2 Ex').beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
pattern x at 3 ; rewrite <- Rplus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(x - ulp x <= x + 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
apply Rplus_le_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(- ulp x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
rewrite <-Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(- ulp x <= - 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
apply ulp_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= x - ulp x)%R
apply Rle_0_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(ulp x <= x)%R
pattern x at 2; rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(ulp x <=
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (cexp beta fexp x) <=
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
unfold F2R; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(bpow (cexp beta fexp x) <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
pattern (bpow (cexp beta fexp x)) at 1; rewrite <- Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(1 * bpow (cexp beta fexp x) <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
apply Rmult_le_compat_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
apply IZR_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
assert (0 <  Ztrunc (scaled_mantissa beta fexp x))%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
H:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
apply gt_0_F2R with beta (cexp beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
H:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
rewrite <- Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
H:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
apply Rle_lt_trans with (2:=proj1 Ex').beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
(0 <= bpow (ex - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
H:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
H:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Hx:x <> bpow (ex - 1)
Ex':(bpow (ex - 1) < x < bpow ex)%R
x <> 0%R
now apply Rgt_not_eq.
Qed.

Lemma generic_format_pred_aux2 :
  forall x, (0 < x)%R -> F x ->
  let e := mag_val beta x (mag beta x) in
  x = bpow (e - 1) ->
  F (x - bpow (fexp (e - 1))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
let e := mag_val beta x (mag beta x) in
x = bpow (e - 1) -> F (x - bpow (fexp (e - 1)))
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
let e := mag_val beta x (mag beta x) in
x = bpow (e - 1) -> F (x - bpow (fexp (e - 1)))
intros x Zx Fx e Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
F (x - bpow (fexp (e - 1)))
pose (f:=(x - bpow (fexp (e - 1)))%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
F (x - bpow (fexp (e - 1)))
fold f.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
F f
assert (He:(fexp (e-1) <= e-1)%Z).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
(fexp (e - 1) <= e - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) <= e - 1)%Z
F f
apply generic_format_bpow_inv with beta; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
F (bpow (e - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) <= e - 1)%Z
F f
rewrite <- Hx; assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) <= e - 1)%Z
F f
case (Zle_lt_or_eq _ _ He); clear He; intros He.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
F f
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
assert (f = F2R (Float beta (Zpower beta (e-1-(fexp (e-1))) -1) (fexp (e-1))))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
F f
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
unfold f; rewrite Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 1) - bpow (fexp (e - 1)))%R =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
F f
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
unfold F2R; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 1) - bpow (fexp (e - 1)))%R =
(IZR (beta ^ (e - 1 - fexp (e - 1)) - 1) *
 bpow (fexp (e - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
F f
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
rewrite minus_IZR, IZR_Zpower.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 1) - bpow (fexp (e - 1)))%R =
((bpow (e - 1 - fexp (e - 1)) - 1) *
 bpow (fexp (e - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
(0 <= e - 1 - fexp (e - 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
F f
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
rewrite Rmult_minus_distr_r, Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 1) - bpow (fexp (e - 1)))%R =
(bpow (e - 1 - fexp (e - 1)) * bpow (fexp (e - 1)) -
 bpow (fexp (e - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
(0 <= e - 1 - fexp (e - 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
F f
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
rewrite <- bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 1) - bpow (fexp (e - 1)))%R =
(bpow (e - 1 - fexp (e - 1) + fexp (e - 1)) -
 bpow (fexp (e - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
(0 <= e - 1 - fexp (e - 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
F f
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
now replace (e - 1 - fexp (e - 1) + fexp (e - 1))%Z with (e-1)%Z by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
(0 <= e - 1 - fexp (e - 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
F f
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
F f
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
rewrite H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
F
  (F2R
     {|
     Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
     Fexp := fexp (e - 1) |})
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply generic_format_F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(beta ^ (e - 1 - fexp (e - 1)) - 1)%Z <> 0%Z ->
(cexp beta fexp
   (F2R
      {|
      Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
      Fexp := fexp (e - 1) |}) <= fexp (e - 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
intros _.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(cexp beta fexp
   (F2R
      {|
      Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
      Fexp := fexp (e - 1) |}) <= fexp (e - 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply Zeq_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
cexp beta fexp
  (F2R
     {|
     Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
     Fexp := fexp (e - 1) |}) = fexp (e - 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply cexp_fexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1 - 1) <=
 Rabs
   (F2R
      {|
      Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
      Fexp := fexp (e - 1) |}) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
rewrite <- H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1 - 1) <= Rabs f < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
unfold f; rewrite Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1 - 1) <=
 Rabs (bpow (e - 1) - bpow (fexp (e - 1))) <
 bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
rewrite Rabs_right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1 - 1) <=
 bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1 - 1) <=
 bpow (e - 1) - bpow (fexp (e - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply Rplus_le_reg_l with (bpow (fexp (e-1))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (fexp (e - 1)) + bpow (e - 1 - 1) <=
 bpow (fexp (e - 1)) +
 (bpow (e - 1) - bpow (fexp (e - 1))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (fexp (e - 1)) + bpow (e - 1 - 1) <=
 bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply Rle_trans with (bpow (e - 2) + bpow (e - 2))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (fexp (e - 1)) + bpow (e - 1 - 1) <=
 bpow (e - 2) + bpow (e - 2))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 2) + bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply Rplus_le_compat ; apply bpow_le ; omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 2) + bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply Rle_trans with (2*bpow (e - 2))%R;[right; ring|idtac].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(2 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply Rle_trans with (bpow 1*bpow (e - 2))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(2 * bpow (e - 2) <= bpow 1 * bpow (e - 2))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply Rmult_le_compat_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(0 <= bpow (e - 2))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(2 <= bpow 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(2 <= bpow 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
replace (bpow 1) with (IZR beta).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(2 <= IZR beta)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
IZR beta = bpow 1
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply IZR_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(2 <= beta)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
IZR beta = bpow 1
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply <- Zle_is_le_bool.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(2 <=? beta)%Z = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
IZR beta = bpow 1
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
now destruct beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
IZR beta = bpow 1
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
IZR beta = IZR (Z.pow_pos beta 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
unfold Zpower_pos; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
IZR beta = IZR (beta * 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
now rewrite Zmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
rewrite <- bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (1 + (e - 2)) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
replace (1+(e-2))%Z with (e-1)%Z by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
now right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
rewrite <- Rplus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1) + 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply Rplus_lt_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(- bpow (fexp (e - 1)) < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
rewrite <- Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(- bpow (fexp (e - 1)) < - 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply Ropp_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(0 < bpow (fexp (e - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply Rle_ge; apply Rle_0_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(bpow (fexp (e - 1)) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
apply bpow_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:(fexp (e - 1) < e - 1)%Z
H:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}
(fexp (e - 1) <= e - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F f
replace f with 0%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
F 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
0%R = f
apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
0%R = f
unfold f.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
0%R = (x - bpow (fexp (e - 1)))%R
rewrite Hx, He.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hx:x = bpow (e - 1)
f:=(x - bpow (fexp (e - 1)))%R:R
He:fexp (e - 1) = (e - 1)%Z
0%R = (bpow (e - 1) - bpow (e - 1))%R
ring.
Qed.

Lemma generic_format_succ_aux1 :
  forall x, (0 < x)%R -> F x ->
  F (x + ulp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 < x)%R -> F x -> F (x + ulp x)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 < x)%R -> F x -> F (x + ulp x)
intros x Zx Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
F (x + ulp x)
destruct (mag beta x) as (ex, Ex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
F (x + ulp x)
specialize (Ex (Rgt_not_eq _ _ Zx)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
F (x + ulp x)
assert (Ex' := Ex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
F (x + ulp x)
rewrite Rabs_pos_eq in Ex'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
destruct (id_p_ulp_le_bpow x ex) ; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
unfold generic_format, scaled_mantissa, cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(x + ulp x)%R =
F2R
  {|
  Fnum := Ztrunc
            ((x + ulp x) *
             bpow (- fexp (mag beta (x + ulp x))));
  Fexp := fexp (mag beta (x + ulp x)) |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite mag_unique with beta (x + ulp x)%R ex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(x + ulp x)%R =
F2R
  {|
  Fnum := Ztrunc ((x + ulp x) * bpow (- fexp ex));
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
pattern x at 1 3 ; rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} + ulp x)%R =
F2R
  {|
  Fnum := Ztrunc
            ((F2R
                {|
                Fnum := Ztrunc
                          (scaled_mantissa beta fexp x);
                Fexp := cexp beta fexp x |} + ulp x) *
             bpow (- fexp ex));
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} +
 bpow (cexp beta fexp x))%R =
F2R
  {|
  Fnum := Ztrunc
            ((F2R
                {|
                Fnum := Ztrunc
                          (scaled_mantissa beta fexp x);
                Fexp := cexp beta fexp x |} +
              bpow (cexp beta fexp x)) *
             bpow (- fexp ex));
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
unfold scaled_mantissa.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (x * bpow (- cexp beta fexp x));
   Fexp := cexp beta fexp x |} +
 bpow (cexp beta fexp x))%R =
F2R
  {|
  Fnum := Ztrunc
            ((F2R
                {|
                Fnum := Ztrunc
                          (x *
                           bpow (- cexp beta fexp x));
                Fexp := cexp beta fexp x |} +
              bpow (cexp beta fexp x)) *
             bpow (- fexp ex));
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite cexp_fexp with (1 := Ex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (x * bpow (- fexp ex));
   Fexp := fexp ex |} + bpow (fexp ex))%R =
F2R
  {|
  Fnum := Ztrunc
            ((F2R
                {|
                Fnum := Ztrunc (x * bpow (- fexp ex));
                Fexp := fexp ex |} + bpow (fexp ex)) *
             bpow (- fexp ex));
  Fexp := fexp ex |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(IZR
   (Fnum
      {|
      Fnum := Ztrunc (x * bpow (- fexp ex));
      Fexp := fexp ex |}) *
 bpow
   (Fexp
      {|
      Fnum := Ztrunc (x * bpow (- fexp ex));
      Fexp := fexp ex |}) + bpow (fexp ex))%R =
(IZR
   (Fnum
      {|
      Fnum := Ztrunc
                ((IZR
                    (Fnum
                       {|
                       Fnum := Ztrunc
                                 (x * bpow (- fexp ex));
                       Fexp := fexp ex |}) *
                  bpow
                    (Fexp
                       {|
                       Fnum := Ztrunc
                                 (x * bpow (- fexp ex));
                       Fexp := fexp ex |}) +
                  bpow (fexp ex)) * bpow (- fexp ex));
      Fexp := fexp ex |}) *
 bpow
   (Fexp
      {|
      Fnum := Ztrunc
                ((IZR
                    (Fnum
                       {|
                       Fnum := Ztrunc
                                 (x * bpow (- fexp ex));
                       Fexp := fexp ex |}) *
                  bpow
                    (Fexp
                       {|
                       Fnum := Ztrunc
                                 (x * bpow (- fexp ex));
                       Fexp := fexp ex |}) +
                  bpow (fexp ex)) * bpow (- fexp ex));
      Fexp := fexp ex |}))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) +
 bpow (fexp ex))%R =
(IZR
   (Ztrunc
      ((IZR (Ztrunc (x * bpow (- fexp ex))) *
        bpow (fexp ex) + bpow (fexp ex)) *
       bpow (- fexp ex))) * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite Rmult_plus_distr_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) +
 bpow (fexp ex))%R =
(IZR
   (Ztrunc
      (IZR (Ztrunc (x * bpow (- fexp ex))) *
       bpow (fexp ex) * bpow (- fexp ex) +
       bpow (fexp ex) * bpow (- fexp ex))) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite Rmult_assoc.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) +
 bpow (fexp ex))%R =
(IZR
   (Ztrunc
      (IZR (Ztrunc (x * bpow (- fexp ex))) *
       (bpow (fexp ex) * bpow (- fexp ex)) +
       bpow (fexp ex) * bpow (- fexp ex))) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite <- bpow_plus, Zplus_opp_r, Rmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) +
 bpow (fexp ex))%R =
(IZR
   (Ztrunc
      (IZR (Ztrunc (x * bpow (- fexp ex))) + bpow 0)) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
change (bpow 0) with 1%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) +
 bpow (fexp ex))%R =
(IZR
   (Ztrunc (IZR (Ztrunc (x * bpow (- fexp ex))) + 1)) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite <- plus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) +
 bpow (fexp ex))%R =
(IZR
   (Ztrunc (IZR (Ztrunc (x * bpow (- fexp ex)) + 1))) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite Ztrunc_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) +
 bpow (fexp ex))%R =
(IZR (Ztrunc (x * bpow (- fexp ex)) + 1) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite plus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) +
 bpow (fexp ex))%R =
((IZR (Ztrunc (x * bpow (- fexp ex))) + 1) *
 bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite Rmult_plus_distr_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) +
 bpow (fexp ex))%R =
(IZR (Ztrunc (x * bpow (- fexp ex))) * bpow (fexp ex) +
 1 * bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
now rewrite Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= x + ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(bpow (ex - 1) <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(x + ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
apply Rle_trans with (1 := proj1 Ex').beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(x <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(x + ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
pattern x at 1 ; rewrite <- Rplus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(x + 0 <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(x + ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
apply Rplus_le_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(x + ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
apply ulp_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(x + ulp x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
exact H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
apply Rplus_le_le_0_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x < bpow ex)%R
(0 <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
apply ulp_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
F (bpow ex)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
apply generic_format_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
(fexp (ex + 1) <= ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
apply valid_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
(fexp ex < ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
destruct (Zle_or_lt ex (fexp ex)) ; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
(fexp ex < ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
elim Rlt_not_le with (1 := Zx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
(x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
replace (Ztrunc (scaled_mantissa beta fexp x)) with Z0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
(F2R {| Fnum := 0; Fexp := cexp beta fexp x |} <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
0%Z = Ztrunc (scaled_mantissa beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite F2R_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
(0 <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
0%Z = Ztrunc (scaled_mantissa beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
apply Rle_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
0%Z = Ztrunc (scaled_mantissa beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
unfold scaled_mantissa.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
0%Z = Ztrunc (x * bpow (- cexp beta fexp x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite cexp_fexp with (1 := Ex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
0%Z = Ztrunc (x * bpow (- fexp ex))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
destruct (mantissa_small_pos beta fexp x ex) ; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
0%Z = Ztrunc (x * bpow (- fexp ex))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
rewrite Ztrunc_floor.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
0%Z = Zfloor (x * bpow (- fexp ex))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
(0 <= x * bpow (- fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
Zfloor (x * bpow (- fexp ex)) = 0%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
(0 <= x * bpow (- fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
apply Zfloor_imp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
(0 <= x * bpow (- fexp ex) < IZR (0 + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
(0 <= x * bpow (- fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
(0 <= x * bpow (- fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
(x * bpow (- fexp ex) < IZR (0 + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
(0 <= x * bpow (- fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
(x * bpow (- fexp ex) < IZR (0 + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
(0 <= x * bpow (- fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
exact H2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Ex':(bpow (ex - 1) <= x < bpow ex)%R
H:(x + ulp x)%R = bpow ex
H0:(ex <= fexp ex)%Z
H1:(0 < x * bpow (- fexp ex))%R
H2:(x * bpow (- fexp ex) < 1)%R
(0 <= x * bpow (- fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
now apply Rlt_le.
Qed.

Lemma generic_format_pred_pos :
  forall x, F x -> (0 < x)%R ->
  F (pred_pos x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> (0 < x)%R -> F (pred_pos x)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> (0 < x)%R -> F (pred_pos x)
intros x Fx Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(0 < x)%R
F (pred_pos x)
unfold pred_pos; case Req_bool_spec; intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(0 < x)%R
H:x = bpow (mag beta x - 1)
F (x - bpow (fexp (mag beta x - 1)))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(0 < x)%R
H:x <> bpow (mag beta x - 1)
F (x - ulp x)
now apply generic_format_pred_aux2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(0 < x)%R
H:x <> bpow (mag beta x - 1)
F (x - ulp x)
now apply generic_format_pred_aux1.
Qed.

Theorem generic_format_succ :
  forall x, F x ->
  F (succ x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> F (succ x)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> F (succ x)
intros x Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
F (succ x)
unfold succ; case Rle_bool_spec; intros Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(0 <= x)%R
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(x < 0)%R
F (- pred_pos (- x))
destruct Zx as [Zx|Zx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(0 < x)%R
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:0%R = x
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(x < 0)%R
F (- pred_pos (- x))
now apply generic_format_succ_aux1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:0%R = x
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(x < 0)%R
F (- pred_pos (- x))
rewrite <- Zx, Rplus_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:0%R = x
F (ulp 0)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(x < 0)%R
F (- pred_pos (- x))
apply generic_format_ulp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(x < 0)%R
F (- pred_pos (- x))
apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(x < 0)%R
F (pred_pos (- x))
apply generic_format_pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(x < 0)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(x < 0)%R
(0 < - x)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Zx:(x < 0)%R
(0 < - x)%R
now apply Ropp_0_gt_lt_contravar.
Qed.

Theorem generic_format_pred :
  forall x, F x ->
  F (pred x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> F (pred x)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> F (pred x)
intros x Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
F (pred x)
unfold pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
F (- succ (- x))
apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
F (succ (- x))
apply generic_format_succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
F (- x)
now apply generic_format_opp.
Qed.

Lemma pred_pos_lt_id :
  forall x, (x <> 0)%R ->
  (pred_pos x < x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, x <> 0%R -> (pred_pos x < x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, x <> 0%R -> (pred_pos x < x)%R
intros x Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
(pred_pos x < x)%R
unfold pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
((if Req_bool x (bpow (mag beta x - 1))
  then x - bpow (fexp (mag beta x - 1))
  else x - ulp x) < x)%R
case Req_bool_spec; intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x = bpow (mag beta x - 1)
(x - bpow (fexp (mag beta x - 1)) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(x - ulp x < x)%R
(* *)
rewrite <- Rplus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x = bpow (mag beta x - 1)
(x - bpow (fexp (mag beta x - 1)) < x + 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(x - ulp x < x)%R
apply Rplus_lt_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x = bpow (mag beta x - 1)
(- bpow (fexp (mag beta x - 1)) < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(x - ulp x < x)%R
rewrite <- Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x = bpow (mag beta x - 1)
(- bpow (fexp (mag beta x - 1)) < - 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(x - ulp x < x)%R
apply Ropp_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x = bpow (mag beta x - 1)
(0 < bpow (fexp (mag beta x - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(x - ulp x < x)%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(x - ulp x < x)%R
(* *)
rewrite <- Rplus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(x - ulp x < x + 0)%R
apply Rplus_lt_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(- ulp x < 0)%R
rewrite <- Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(- ulp x < - 0)%R
apply Ropp_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(0 < ulp x)%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
H:x <> bpow (mag beta x - 1)
(0 < bpow (cexp beta fexp x))%R
apply bpow_gt_0.
Qed.

Theorem succ_gt_id :
  forall x, (x <> 0)%R ->
  (x < succ x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, x <> 0%R -> (x < succ x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, x <> 0%R -> (x < succ x)%R
intros x Zx; unfold succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
(x <
 (if Rle_bool 0 x then x + ulp x else - pred_pos (- x)))%R
case Rle_bool_spec; intros Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(0 <= x)%R
(x < x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(x < 0)%R
(x < - pred_pos (- x))%R
pattern x at 1; rewrite <- (Rplus_0_r x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(0 <= x)%R
(x + 0 < x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(x < 0)%R
(x < - pred_pos (- x))%R
apply Rplus_lt_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(0 <= x)%R
(0 < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(x < 0)%R
(x < - pred_pos (- x))%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(0 <= x)%R
(0 < bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(x < 0)%R
(x < - pred_pos (- x))%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(x < 0)%R
(x < - pred_pos (- x))%R
pattern x at 1; rewrite <- (Ropp_involutive x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(x < 0)%R
(- - x < - pred_pos (- x))%R
apply Ropp_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(x < 0)%R
(pred_pos (- x) < - x)%R
apply pred_pos_lt_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
Hx:(x < 0)%R
(- x)%R <> 0%R
auto with real.
Qed.


Theorem pred_lt_id :
  forall x,  (x <> 0)%R ->
  (pred x < x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, x <> 0%R -> (pred x < x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, x <> 0%R -> (pred x < x)%R
intros x Zx; unfold pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
(- succ (- x) < x)%R
pattern x at 2; rewrite <- (Ropp_involutive x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
(- succ (- x) < - - x)%R
apply Ropp_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
(- x < succ (- x))%R
apply succ_gt_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:x <> 0%R
(- x)%R <> 0%R
auto with real.
Qed.

Theorem succ_ge_id :
  forall x, (x <= succ x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (x <= succ x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (x <= succ x)%R
intros x; case (Req_dec x 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
x = 0%R -> (x <= succ x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
x <> 0%R -> (x <= succ x)%R
intros V; rewrite V.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
V:x = 0%R
(0 <= succ 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
x <> 0%R -> (x <= succ x)%R
unfold succ; rewrite Rle_bool_true;[idtac|now right].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
V:x = 0%R
(0 <= 0 + ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
x <> 0%R -> (x <= succ x)%R
rewrite Rplus_0_l; apply ulp_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
x <> 0%R -> (x <= succ x)%R
intros; left; now apply succ_gt_id.
Qed.


Theorem pred_le_id :
  forall x, (pred x <= x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (pred x <= x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (pred x <= x)%R
intros x; unfold pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
(- succ (- x) <= x)%R
pattern x at 2; rewrite <- (Ropp_involutive x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
(- succ (- x) <= - - x)%R
apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
(- x <= succ (- x))%R
apply succ_ge_id.
Qed.


Lemma pred_pos_ge_0 :
  forall x,
  (0 < x)%R -> F x -> (0 <= pred_pos x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 < x)%R -> F x -> (0 <= pred_pos x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 < x)%R -> F x -> (0 <= pred_pos x)%R
intros x Zx Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(0 <= pred_pos x)%R
unfold pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(0 <=
 (if Req_bool x (bpow (mag beta x - 1))
  then x - bpow (fexp (mag beta x - 1))
  else x - ulp x))%R
case Req_bool_spec; intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x = bpow (mag beta x - 1)
(0 <= x - bpow (fexp (mag beta x - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(0 <= x - ulp x)%R
(* *)
apply Rle_0_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x = bpow (mag beta x - 1)
(bpow (fexp (mag beta x - 1)) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(0 <= x - ulp x)%R
rewrite H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x = bpow (mag beta x - 1)
(bpow (fexp (mag beta (bpow (mag beta x - 1)) - 1)) <=
 bpow (mag beta x - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(0 <= x - ulp x)%R
apply bpow_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x = bpow (mag beta x - 1)
(fexp (mag beta (bpow (mag beta x - 1)) - 1) <=
 mag beta x - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(0 <= x - ulp x)%R
destruct (mag beta x) as (ex,Ex) ; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
H:x = bpow (Build_mag_prop beta x ex Ex - 1)
(fexp (mag beta (bpow (ex - 1)) - 1) <= ex - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(0 <= x - ulp x)%R
rewrite mag_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
H:x = bpow (Build_mag_prop beta x ex Ex - 1)
(fexp (ex - 1 + 1 - 1) <= ex - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(0 <= x - ulp x)%R
ring_simplify (ex - 1 + 1 - 1)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
H:x = bpow (Build_mag_prop beta x ex Ex - 1)
(fexp (ex - 1) <= ex - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(0 <= x - ulp x)%R
apply generic_format_bpow_inv with beta; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
H:x = bpow (Build_mag_prop beta x ex Ex - 1)
F (bpow (ex - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(0 <= x - ulp x)%R
simpl in H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Ex:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
H:x = bpow (ex - 1)
F (bpow (ex - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(0 <= x - ulp x)%R
rewrite <- H; assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(0 <= x - ulp x)%R
apply Rle_0_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(ulp x <= x)%R
now apply ulp_le_id.
Qed.

Theorem pred_ge_0 :
  forall x,
  (0 < x)%R -> F x -> (0 <= pred x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 < x)%R -> F x -> (0 <= pred x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, (0 < x)%R -> F x -> (0 <= pred x)%R
intros x Zx Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(0 <= pred x)%R
rewrite pred_eq_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(0 <= pred_pos x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(0 <= x)%R
now apply pred_pos_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(0 <= x)%R
now left.
Qed.


Lemma pred_pos_plus_ulp_aux1 :
  forall x, (0 < x)%R -> F x ->
  x <> bpow (mag beta x - 1) ->
  ((x - ulp x) + ulp (x-ulp x) = x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
x <> bpow (mag beta x - 1) ->
(x - ulp x + ulp (x - ulp x))%R = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
x <> bpow (mag beta x - 1) ->
(x - ulp x + ulp (x - ulp x))%R = x
intros x Zx Fx Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
(x - ulp x + ulp (x - ulp x))%R = x
replace (ulp (x - ulp x)) with (ulp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
(x - ulp x + ulp x)%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
ulp x = ulp (x - ulp x)
ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
ulp x = ulp (x - ulp x)
assert (H : x <> 0%R) by now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
ulp x = ulp (x - ulp x)
assert (H' : x <> bpow (cexp beta fexp x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
x <> bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
unfold cexp ; intros M.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
M:x = bpow (fexp (mag beta x))
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
case_eq (mag beta x); intros ex Hex T.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
M:x = bpow (fexp (mag beta x))
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
T:mag beta x = Build_mag_prop beta x ex Hex
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
assert (Lex:(mag_val beta x (mag beta x) = ex)%Z).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
M:x = bpow (fexp (mag beta x))
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
T:mag beta x = Build_mag_prop beta x ex Hex
mag beta x = ex
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
M:x = bpow (fexp (mag beta x))
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
T:mag beta x = Build_mag_prop beta x ex Hex
Lex:mag beta x = ex
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
rewrite T; reflexivity.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
M:x = bpow (fexp (mag beta x))
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
T:mag beta x = Build_mag_prop beta x ex Hex
Lex:mag beta x = ex
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
rewrite Lex in *.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
T:mag beta x = Build_mag_prop beta x ex Hex
Lex:mag beta x = ex
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
clear T; simpl in *; specialize (Hex H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
rewrite Rabs_pos_eq in Hex by now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
assert (ex - 1 < fexp ex < ex)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(ex - 1 < fexp ex < ex)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
H0:(ex - 1 < fexp ex < ex)%Z
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
  split ; apply (lt_bpow beta) ; rewrite <- M by easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
H0:(ex - 1 < fexp ex < ex)%Z
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
  lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
H0:(ex - 1 < fexp ex < ex)%Z
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
  apply Hex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
M:x = bpow (fexp ex)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
H0:(ex - 1 < fexp ex < ex)%Z
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ulp x = ulp (x - ulp x)
rewrite 2!ulp_neq_0 by lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
bpow (cexp beta fexp x) =
bpow (cexp beta fexp (x - bpow (cexp beta fexp x)))
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
cexp beta fexp x =
cexp beta fexp (x - bpow (cexp beta fexp x))
unfold cexp ; apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
mag beta x = mag beta (x - bpow (fexp (mag beta x)))
case_eq (mag beta x); intros ex Hex T.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
T:mag beta x = Build_mag_prop beta x ex Hex
Build_mag_prop beta x ex Hex =
mag beta
  (x - bpow (fexp (Build_mag_prop beta x ex Hex)))
assert (Lex:(mag_val beta x (mag beta x) = ex)%Z).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
T:mag beta x = Build_mag_prop beta x ex Hex
mag beta x = ex
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
T:mag beta x = Build_mag_prop beta x ex Hex
Lex:mag beta x = ex
Build_mag_prop beta x ex Hex =
mag beta
  (x - bpow (fexp (Build_mag_prop beta x ex Hex)))
rewrite T; reflexivity.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
Hx:x <> bpow (mag beta x - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
T:mag beta x = Build_mag_prop beta x ex Hex
Lex:mag beta x = ex
Build_mag_prop beta x ex Hex =
mag beta
  (x - bpow (fexp (Build_mag_prop beta x ex Hex)))
rewrite Lex in *; simpl in *; clear T.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
ex = mag beta (x - bpow (fexp ex))
specialize (Hex H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
ex = mag beta (x - bpow (fexp ex))
apply sym_eq, mag_unique.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= Rabs (x - bpow (fexp ex)) < bpow ex)%R
rewrite Rabs_right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
rewrite Rabs_right in Hex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
2: apply Rle_ge; apply Rlt_le; easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
destruct Hex as ([H1|H1],H2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:(bpow (ex - 1) < x)%R
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:bpow (ex - 1) = x
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
apply Rle_trans with (x-ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:(bpow (ex - 1) < x)%R
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:(bpow (ex - 1) < x)%R
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(x - ulp x <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:bpow (ex - 1) = x
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
apply id_m_ulp_ge_bpow; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:(bpow (ex - 1) < x)%R
H2:(x < bpow ex)%R
Lex:mag beta x = ex
x <> ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:(bpow (ex - 1) < x)%R
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(x - ulp x <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:bpow (ex - 1) = x
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:(bpow (ex - 1) < x)%R
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(x - ulp x <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:bpow (ex - 1) = x
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:(bpow (ex - 1) < x)%R
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (cexp beta fexp x) <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:bpow (ex - 1) = x
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
right; unfold cexp; now rewrite Lex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
H1:bpow (ex - 1) = x
H2:(x < bpow ex)%R
Lex:mag beta x = ex
(bpow (ex - 1) <= x - bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
apply Rle_lt_trans with (2:=proj2 Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
rewrite <- Rplus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) <= x + 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
apply Rplus_le_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(- bpow (fexp ex) <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
rewrite <- Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(- bpow (fexp ex) <= - 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= x < bpow ex)%R
Lex:mag beta x = ex
(0 <= bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(x - bpow (fexp ex) >= 0)%R
apply Rle_ge.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(0 <= x - bpow (fexp ex))%R
apply Rle_0_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(bpow (fexp ex) <= x)%R
rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(bpow (fexp ex) <=
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
unfold F2R, cexp; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(bpow (fexp ex) <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (fexp (mag beta x)))%R
rewrite Lex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(bpow (fexp ex) <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (fexp ex))%R
pattern (bpow (fexp ex)) at 1; rewrite <- Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(1 * bpow (fexp ex) <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (fexp ex))%R
apply Rmult_le_compat_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(0 <= bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R
apply IZR_le, (Zlt_le_succ 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
apply gt_0_F2R with beta (cexp beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
ex:Z
Hx:x <> bpow (ex - 1)
H:x <> 0%R
H':x <> bpow (cexp beta fexp x)
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
Lex:mag beta x = ex
(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
now rewrite <- Fx.
Qed.

Lemma pred_pos_plus_ulp_aux2 :
  forall x, (0 < x)%R -> F x ->
  let e := mag_val beta x (mag beta x) in
  x =  bpow (e - 1) ->
  (x - bpow (fexp (e-1)) <> 0)%R ->
  ((x - bpow (fexp (e-1))) + ulp (x - bpow (fexp (e-1))) = x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
let e := mag_val beta x (mag beta x) in
x = bpow (e - 1) ->
(x - bpow (fexp (e - 1)))%R <> 0%R ->
(x - bpow (fexp (e - 1)) +
 ulp (x - bpow (fexp (e - 1))))%R = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
let e := mag_val beta x (mag beta x) in
x = bpow (e - 1) ->
(x - bpow (fexp (e - 1)))%R <> 0%R ->
(x - bpow (fexp (e - 1)) +
 ulp (x - bpow (fexp (e - 1))))%R = x
intros x Zx Fx e Hxe Zp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
(x - bpow (fexp (e - 1)) +
 ulp (x - bpow (fexp (e - 1))))%R = x
replace (ulp (x - bpow (fexp (e - 1)))) with (bpow (fexp (e - 1))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
(x - bpow (fexp (e - 1)) + bpow (fexp (e - 1)))%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
assert (He:(fexp (e-1) <= e-1)%Z).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
(fexp (e - 1) <= e - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) <= e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply generic_format_bpow_inv with beta; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
F (bpow (e - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) <= e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
rewrite <- Hxe; assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) <= e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
case (Zle_lt_or_eq _ _ He); clear He; intros He.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
(* *)
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
bpow (fexp (e - 1)) =
bpow (cexp beta fexp (x - bpow (fexp (e - 1))))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
fexp (e - 1) =
cexp beta fexp (x - bpow (fexp (e - 1)))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
unfold cexp ; apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(e - 1)%Z = mag beta (x - bpow (fexp (e - 1)))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
mag beta (x - bpow (fexp (e - 1))) = (e - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply mag_unique.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 1 - 1) <= Rabs (x - bpow (fexp (e - 1))) <
 bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
rewrite Rabs_right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 1 - 1) <= x - bpow (fexp (e - 1)) <
 bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 1 - 1) <= x - bpow (fexp (e - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply Rplus_le_reg_l with (bpow (fexp (e-1))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (fexp (e - 1)) + bpow (e - 1 - 1) <=
 bpow (fexp (e - 1)) + (x - bpow (fexp (e - 1))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (fexp (e - 1)) + bpow (e - 1 - 1) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply Rle_trans with (bpow (e - 2) + bpow (e - 2))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (fexp (e - 1)) + bpow (e - 1 - 1) <=
 bpow (e - 2) + bpow (e - 2))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 2) + bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply Rplus_le_compat; apply bpow_le; omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 2) + bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply Rle_trans with (2*bpow (e - 2))%R;[right; ring|idtac].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(2 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply Rle_trans with (bpow 1*bpow (e - 2))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(2 * bpow (e - 2) <= bpow 1 * bpow (e - 2))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow 1 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply Rmult_le_compat_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(0 <= bpow (e - 2))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(2 <= bpow 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow 1 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(2 <= bpow 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow 1 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
replace (bpow 1) with (IZR beta).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(2 <= IZR beta)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
IZR beta = bpow 1
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow 1 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply IZR_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(2 <= beta)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
IZR beta = bpow 1
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow 1 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply <- Zle_is_le_bool.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(2 <=? beta)%Z = true
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
IZR beta = bpow 1
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow 1 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
now destruct beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
IZR beta = bpow 1
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow 1 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
IZR beta = IZR (Z.pow_pos beta 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow 1 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
unfold Zpower_pos; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
IZR beta = IZR (beta * 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow 1 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
now rewrite Zmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow 1 * bpow (e - 2) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
rewrite <- bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (1 + (e - 2)) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
replace (1+(e-2))%Z with (e-1)%Z by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 1) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
now right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) < bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
rewrite <- Rplus_0_r, Hxe.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1) + 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply Rplus_lt_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(- bpow (fexp (e - 1)) < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
rewrite <- Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(- bpow (fexp (e - 1)) < - 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply Ropp_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(0 < bpow (fexp (e - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(x - bpow (fexp (e - 1)) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply Rle_ge; apply Rle_0_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (fexp (e - 1)) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
rewrite Hxe.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(bpow (fexp (e - 1)) <= bpow (e - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
apply bpow_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:(fexp (e - 1) < e - 1)%Z
(fexp (e - 1) <= e - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
Zp:(x - bpow (fexp (e - 1)))%R <> 0%R
He:fexp (e - 1) = (e - 1)%Z
bpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))
(* *)
contradict Zp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
Hxe:x = bpow (e - 1)
He:fexp (e - 1) = (e - 1)%Z
(x - bpow (fexp (e - 1)))%R = 0%R
rewrite Hxe, He; ring.
Qed.

Lemma pred_pos_plus_ulp_aux3 :
  forall x, (0 < x)%R -> F x ->
  let e := mag_val beta x (mag beta x) in
  x =  bpow (e - 1) ->
  (x - bpow (fexp (e-1)) = 0)%R ->
  (ulp 0 = x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
let e := mag_val beta x (mag beta x) in
x = bpow (e - 1) ->
(x - bpow (fexp (e - 1)))%R = 0%R -> ulp 0 = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
let e := mag_val beta x (mag beta x) in
x = bpow (e - 1) ->
(x - bpow (fexp (e - 1)))%R = 0%R -> ulp 0 = x
intros x Hx Fx e H1 H2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
ulp 0 = x
assert (H3:(x = bpow (fexp (e - 1)))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
x = bpow (fexp (e - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
ulp 0 = x
now apply Rminus_diag_uniq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
ulp 0 = x
assert (H4: (fexp (e-1) = e-1)%Z).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
fexp (e - 1) = (e - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
ulp 0 = x
apply bpow_inj with beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
bpow (fexp (e - 1)) = bpow (e - 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
ulp 0 = x
now rewrite <- H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
ulp 0 = x
unfold ulp; rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
match negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end = x
case negligible_exp_spec.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
(forall n : Z, (fexp n < n)%Z) -> 0%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
forall n : Z, (n <= fexp n)%Z -> bpow (fexp n) = x
intros K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
K:forall n : Z, (fexp n < n)%Z
0%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
forall n : Z, (n <= fexp n)%Z -> bpow (fexp n) = x
specialize (K (e-1)%Z).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
K:(fexp (e - 1) < e - 1)%Z
0%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
forall n : Z, (n <= fexp n)%Z -> bpow (fexp n) = x
contradict K; omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
forall n : Z, (n <= fexp n)%Z -> bpow (fexp n) = x
intros n Hn.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
n:Z
Hn:(n <= fexp n)%Z
bpow (fexp n) = x
rewrite H3; apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
n:Z
Hn:(n <= fexp n)%Z
fexp n = fexp (e - 1)
case (Zle_or_lt n (e-1)); intros H6.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
n:Z
Hn:(n <= fexp n)%Z
H6:(n <= e - 1)%Z
fexp n = fexp (e - 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
n:Z
Hn:(n <= fexp n)%Z
H6:(e - 1 < n)%Z
fexp n = fexp (e - 1)
apply valid_exp; omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
e:=mag_val beta x (mag beta x):Z
H1:x = bpow (e - 1)
H2:(x - bpow (fexp (e - 1)))%R = 0%R
H3:x = bpow (fexp (e - 1))
H4:fexp (e - 1) = (e - 1)%Z
n:Z
Hn:(n <= fexp n)%Z
H6:(e - 1 < n)%Z
fexp n = fexp (e - 1)
apply sym_eq, valid_exp; omega.
Qed.

The following one is false for x = 0 in FTZ

Lemma pred_pos_plus_ulp :
  forall x, (0 < x)%R -> F x ->
  (pred_pos x + ulp (pred_pos x) = x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x -> (pred_pos x + ulp (pred_pos x))%R = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x -> (pred_pos x + ulp (pred_pos x))%R = x
intros x Zx Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
(pred_pos x + ulp (pred_pos x))%R = x
unfold pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
((if Req_bool x (bpow (mag beta x - 1))
  then x - bpow (fexp (mag beta x - 1))
  else x - ulp x) +
 ulp
   (if Req_bool x (bpow (mag beta x - 1))
    then x - bpow (fexp (mag beta x - 1))
    else x - ulp x))%R = x
case Req_bool_spec; intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x = bpow (mag beta x - 1)
(x - bpow (fexp (mag beta x - 1)) +
 ulp (x - bpow (fexp (mag beta x - 1))))%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(x - ulp x + ulp (x - ulp x))%R = x
case (Req_EM_T (x - bpow (fexp (mag_val beta x (mag beta x) -1))) 0); intros H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x = bpow (mag beta x - 1)
H1:(x - bpow (fexp (mag beta x - 1)))%R = 0%R
(x - bpow (fexp (mag beta x - 1)) +
 ulp (x - bpow (fexp (mag beta x - 1))))%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x = bpow (mag beta x - 1)
H1:(x - bpow (fexp (mag beta x - 1)))%R <> 0%R
(x - bpow (fexp (mag beta x - 1)) +
 ulp (x - bpow (fexp (mag beta x - 1))))%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(x - ulp x + ulp (x - ulp x))%R = x
rewrite H1, Rplus_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x = bpow (mag beta x - 1)
H1:(x - bpow (fexp (mag beta x - 1)))%R = 0%R
ulp 0 = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x = bpow (mag beta x - 1)
H1:(x - bpow (fexp (mag beta x - 1)))%R <> 0%R
(x - bpow (fexp (mag beta x - 1)) +
 ulp (x - bpow (fexp (mag beta x - 1))))%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(x - ulp x + ulp (x - ulp x))%R = x
now apply pred_pos_plus_ulp_aux3.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x = bpow (mag beta x - 1)
H1:(x - bpow (fexp (mag beta x - 1)))%R <> 0%R
(x - bpow (fexp (mag beta x - 1)) +
 ulp (x - bpow (fexp (mag beta x - 1))))%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(x - ulp x + ulp (x - ulp x))%R = x
now apply pred_pos_plus_ulp_aux2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
H:x <> bpow (mag beta x - 1)
(x - ulp x + ulp (x - ulp x))%R = x
now apply pred_pos_plus_ulp_aux1.
Qed.

Theorem pred_plus_ulp :
  forall x, (0 < x)%R -> F x ->
  (pred x + ulp (pred x))%R = x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R -> F x -> (pred x + ulp (pred x))%R = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R -> F x -> (pred x + ulp (pred x))%R = x
intros x Hx Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
(pred x + ulp (pred x))%R = x
rewrite pred_eq_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
(pred_pos x + ulp (pred_pos x))%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
(0 <= x)%R
now apply pred_pos_plus_ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
(0 <= x)%R
now apply Rlt_le.
Qed.

Rounding x + small epsilon

Theorem mag_plus_eps :
  forall x, (0 < x)%R -> F x ->
  forall eps, (0 <= eps < ulp x)%R ->
  mag beta (x + eps) = mag beta x :> Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
forall eps : R,
(0 <= eps < ulp x)%R ->
mag beta (x + eps) = mag beta x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
forall eps : R,
(0 <= eps < ulp x)%R ->
mag beta (x + eps) = mag beta x
intros x Zx Fx eps Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
mag beta (x + eps) = mag beta x
destruct (mag beta x) as (ex, He).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
mag beta (x + eps) = Build_mag_prop beta x ex He
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%R
mag beta (x + eps) = ex
specialize (He (Rgt_not_eq _ _ Zx)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
mag beta (x + eps) = ex
apply mag_unique.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= Rabs (x + eps) < bpow ex)%R
rewrite Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) <= x + eps < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
rewrite Rabs_pos_eq in He.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(bpow (ex - 1) <= x + eps < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(bpow (ex - 1) <= x + eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(x + eps < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
apply Rle_trans with (1 := proj1 He).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(x <= x + eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(x + eps < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
pattern x at 1 ; rewrite <- Rplus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(x + 0 <= x + eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(x + eps < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
now apply Rplus_le_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(x + eps < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
apply Rlt_le_trans with (x + ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(x + eps < x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(x + ulp x <= bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
now apply Rplus_lt_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(x + ulp x <= bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
pattern x at 1 ; rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} + ulp x <= bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} +
 bpow (cexp beta fexp x) <= bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(IZR
   (Fnum
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) *
 bpow
   (Fexp
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) +
 bpow (cexp beta fexp x) <= bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) + bpow (cexp beta fexp x) <=
 bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
pattern (bpow (cexp beta fexp x)) at 2 ; rewrite <- Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) + 1 * bpow (cexp beta fexp x) <=
 bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
rewrite <- Rmult_plus_distr_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
((IZR (Ztrunc (scaled_mantissa beta fexp x)) + 1) *
 bpow (cexp beta fexp x) <= bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
rewrite <- plus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x) + 1) *
 bpow (cexp beta fexp x) <= bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
apply F2R_p1_le_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < 
 bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
apply gt_0_F2R with beta (cexp beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < 
 bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
now rewrite <- Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
now rewrite <- Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= x < bpow ex)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x + eps)%R
apply Rplus_le_le_0_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= eps)%R
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
ex:Z
He:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 <= eps)%R
apply Heps.
Qed.

Theorem round_DN_plus_eps_pos :
  forall x, (0 <= x)%R -> F x ->
  forall eps, (0 <= eps < ulp x)%R ->
  round beta fexp Zfloor (x + eps) = x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 <= x)%R ->
F x ->
forall eps : R,
(0 <= eps < ulp x)%R ->
round beta fexp Zfloor (x + eps) = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 <= x)%R ->
F x ->
forall eps : R,
(0 <= eps < ulp x)%R ->
round beta fexp Zfloor (x + eps) = x
intros x Zx Fx eps Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
destruct Zx as [Zx|Zx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
(* . 0 < x *)
pattern x at 2 ; rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) =
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp x);
  Fexp := cexp beta fexp x |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
unfold round.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
F2R
  {|
  Fnum := Zfloor (scaled_mantissa beta fexp (x + eps));
  Fexp := cexp beta fexp (x + eps) |} =
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp x);
  Fexp := cexp beta fexp x |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
unfold scaled_mantissa.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
F2R
  {|
  Fnum := Zfloor
            ((x + eps) *
             bpow (- cexp beta fexp (x + eps)));
  Fexp := cexp beta fexp (x + eps) |} =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- cexp beta fexp x));
  Fexp := cexp beta fexp x |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
F2R
  {|
  Fnum := Zfloor
            ((x + eps) *
             bpow (- cexp beta fexp (x + eps)));
  Fexp := cexp beta fexp (x + eps) |} =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- cexp beta fexp x));
  Fexp := cexp beta fexp x |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
unfold cexp at 1 2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
F2R
  {|
  Fnum := Zfloor
            ((x + eps) *
             bpow (- fexp (mag beta (x + eps))));
  Fexp := fexp (mag beta (x + eps)) |} =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- cexp beta fexp x));
  Fexp := cexp beta fexp x |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite mag_plus_eps ; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
F2R
  {|
  Fnum := Zfloor
            ((x + eps) * bpow (- fexp (mag beta x)));
  Fexp := fexp (mag beta x) |} =
F2R
  {|
  Fnum := Ztrunc (x * bpow (- cexp beta fexp x));
  Fexp := cexp beta fexp x |}
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply (f_equal (fun m => F2R (Float beta m _))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
Zfloor ((x + eps) * bpow (- fexp (mag beta x))) =
Ztrunc (x * bpow (- cexp beta fexp x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite Ztrunc_floor.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
Zfloor ((x + eps) * bpow (- fexp (mag beta x))) =
Zfloor (x * bpow (- cexp beta fexp x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply Zfloor_imp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(IZR (Zfloor (x * bpow (- cexp beta fexp x))) <=
 (x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(IZR (Zfloor (x * bpow (- cexp beta fexp x))) <=
 (x + eps) * bpow (- fexp (mag beta x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply (Rle_trans _ _ _ (Zfloor_lb _)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(x * bpow (- cexp beta fexp x) <=
 (x + eps) * bpow (- fexp (mag beta x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply Rmult_le_compat_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= bpow (- fexp (mag beta x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(x <= x + eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(x <= x + eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
pattern x at 1 ; rewrite <- Rplus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(x + 0 <= x + eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
now apply Rplus_le_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply Rlt_le_trans with ((x + ulp x) * bpow (- cexp beta fexp x))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + eps) * bpow (- fexp (mag beta x)) <
 (x + ulp x) * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + ulp x) * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply Rmult_lt_compat_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 < bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(x + eps < x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + ulp x) * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(x + eps < x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + ulp x) * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
now apply Rplus_lt_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
((x + ulp x) * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite Rmult_plus_distr_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(x * bpow (- cexp beta fexp x) +
 ulp x * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite plus_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(x * bpow (- cexp beta fexp x) +
 ulp x * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x))) + 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply Rplus_le_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(x * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(ulp x * bpow (- cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
pattern x at 1 3 ; rewrite Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} *
 bpow (- cexp beta fexp x) <=
 IZR
   (Zfloor
      (F2R
         {|
         Fnum := Ztrunc (scaled_mantissa beta fexp x);
         Fexp := cexp beta fexp x |} *
       bpow (- cexp beta fexp x))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(ulp x * bpow (- cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
unfold F2R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(IZR
   (Fnum
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) *
 bpow
   (Fexp
      {|
      Fnum := Ztrunc (scaled_mantissa beta fexp x);
      Fexp := cexp beta fexp x |}) *
 bpow (- cexp beta fexp x) <=
 IZR
   (Zfloor
      (IZR
         (Fnum
            {|
            Fnum := Ztrunc
                      (scaled_mantissa beta fexp x);
            Fexp := cexp beta fexp x |}) *
       bpow
         (Fexp
            {|
            Fnum := Ztrunc
                      (scaled_mantissa beta fexp x);
            Fexp := cexp beta fexp x |}) *
       bpow (- cexp beta fexp x))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(ulp x * bpow (- cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) * bpow (- cexp beta fexp x) <=
 IZR
   (Zfloor
      (IZR (Ztrunc (scaled_mantissa beta fexp x)) *
       bpow (cexp beta fexp x) *
       bpow (- cexp beta fexp x))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(ulp x * bpow (- cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite Rmult_assoc.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 (bpow (cexp beta fexp x) * bpow (- cexp beta fexp x)) <=
 IZR
   (Zfloor
      (IZR (Ztrunc (scaled_mantissa beta fexp x)) *
       (bpow (cexp beta fexp x) *
        bpow (- cexp beta fexp x)))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(ulp x * bpow (- cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite <- bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x + - cexp beta fexp x) <=
 IZR
   (Zfloor
      (IZR (Ztrunc (scaled_mantissa beta fexp x)) *
       bpow (cexp beta fexp x + - cexp beta fexp x))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(ulp x * bpow (- cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite Zplus_opp_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x)) * bpow 0 <=
 IZR
   (Zfloor
      (IZR (Ztrunc (scaled_mantissa beta fexp x)) *
       bpow 0)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(ulp x * bpow (- cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite Rmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x)) <=
 IZR
   (Zfloor
      (IZR (Ztrunc (scaled_mantissa beta fexp x)))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(ulp x * bpow (- cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite Zfloor_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(IZR (Ztrunc (scaled_mantissa beta fexp x)) <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(ulp x * bpow (- cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply Rle_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(ulp x * bpow (- cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(bpow (cexp beta fexp x) * bpow (- cexp beta fexp x) <=
 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
2: now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(bpow (cexp beta fexp x) * bpow (- cexp beta fexp x) <=
 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite <- bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(bpow (cexp beta fexp x + - cexp beta fexp x) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
rewrite Zplus_opp_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(bpow 0 <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply Rle_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x * bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply Rmult_le_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
(0 <= bpow (- cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zfloor (x + eps) = x
(* . x=0 *)
rewrite <- Zx, Rplus_0_l; rewrite <- Zx in Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
round beta fexp Zfloor eps = 0%R
case (proj1 Heps); intros P.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:(0 < eps)%R
round beta fexp Zfloor eps = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
unfold round, scaled_mantissa, cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:(0 < eps)%R
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
revert Heps; unfold ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
(0 <= eps <
 (if Req_bool 0 0
  then
   match negligible_exp with
   | Some n => bpow (fexp n)
   | None => 0
   end
  else bpow (cexp beta fexp 0)))%R ->
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
(0 <= eps <
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R ->
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
case negligible_exp_spec.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
(forall n : Z, (fexp n < n)%Z) ->
(0 <= eps < 0)%R ->
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
forall n : Z,
(n <= fexp n)%Z ->
(0 <= eps < bpow (fexp n))%R ->
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
intros _ (H1,H2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
H1:(0 <= eps)%R
H2:(eps < 0)%R
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
forall n : Z,
(n <= fexp n)%Z ->
(0 <= eps < bpow (fexp n))%R ->
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
exfalso ; lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
forall n : Z,
(n <= fexp n)%Z ->
(0 <= eps < bpow (fexp n))%R ->
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
intros n Hn H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
assert (fexp (mag beta eps) = fexp n).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
fexp (mag beta eps) = fexp n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
apply valid_exp; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
(mag beta eps <= fexp n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
assert(mag beta eps-1 < fexp n)%Z;[idtac|omega].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
(mag beta eps - 1 < fexp n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
apply lt_bpow with beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
(bpow (mag beta eps - 1) < bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
apply Rle_lt_trans with (2:=proj2 H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
(bpow (mag beta eps - 1) <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
destruct (mag beta eps) as (e,He).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
e:Z
He:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R
(bpow (Build_mag_prop beta eps e He - 1) <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
simpl; rewrite Rabs_pos_eq in He.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
e:Z
He:eps <> 0%R -> (bpow (e - 1) <= eps < bpow e)%R
(bpow (e - 1) <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
e:Z
He:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R
(0 <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
now apply He, Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
e:Z
He:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R
(0 <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
now left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
replace (Zfloor (eps * bpow (- fexp (mag beta eps)))) with 0%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R {| Fnum := 0; Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
0%Z = Zfloor (eps * bpow (- fexp (mag beta eps)))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
unfold F2R; simpl; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
0%Z = Zfloor (eps * bpow (- fexp (mag beta eps)))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
apply sym_eq, Zfloor_imp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(0 <= eps * bpow (- fexp (mag beta eps)) < IZR (0 + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(0 <= eps * bpow (- fexp (mag beta eps)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) < IZR (0 + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
apply Rmult_le_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(0 <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(0 <= bpow (- fexp (mag beta eps)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) < IZR (0 + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
now left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(0 <= bpow (- fexp (mag beta eps)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) < IZR (0 + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) < IZR (0 + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
apply Rmult_lt_reg_r with (bpow (fexp n)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(0 < bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) * bpow (fexp n) <
 IZR (0 + 1) * bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) * bpow (fexp n) <
 IZR (0 + 1) * bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
rewrite Rmult_assoc, <- bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps) + fexp n) <
 IZR (0 + 1) * bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
rewrite H0; ring_simplify (-fexp n + fexp n)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow 0 < IZR (0 + 1) * bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
simpl; rewrite Rmult_1_l, Rmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
P:(0 < eps)%R
n:Z
Hn:(n <= fexp n)%Z
H:(0 <= eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps < bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
apply H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
round beta fexp Zfloor eps = 0%R
rewrite <- P, round_0; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:0%R = x
Fx:F x
eps:R
Heps:(0 <= eps < ulp 0)%R
P:0%R = eps
Valid_rnd Zfloor
apply valid_rnd_DN.
Qed.

Theorem round_UP_plus_eps_pos :
  forall x, (0 <= x)%R -> F x ->
  forall eps, (0 < eps <= ulp x)%R ->
  round beta fexp Zceil (x + eps) = (x + ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 <= x)%R ->
F x ->
forall eps : R,
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 <= x)%R ->
F x ->
forall eps : R,
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
intros x Zx Fx eps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
case Zx; intros Zx1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
(* . 0 < x *)
intros (Heps1,[Heps2|Heps2]).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
assert (Heps: (0 <= eps < ulp x)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
(0 <= eps < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
(0 <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
(eps < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
(eps < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
exact Heps2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
assert (Hd := round_DN_plus_eps_pos x Zx Fx eps Heps).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
rewrite round_UP_DN_ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
(round beta fexp Zfloor (x + eps) + ulp (x + eps))%R =
(x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
~ F (x + eps)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
rewrite Hd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
(x + ulp (x + eps))%R = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
~ F (x + eps)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
rewrite 2!ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
(x + bpow (cexp beta fexp (x + eps)))%R =
(x + bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
(x + eps)%R <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
~ F (x + eps)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
(x + bpow (fexp (mag beta (x + eps))))%R =
(x + bpow (fexp (mag beta x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
(x + eps)%R <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
~ F (x + eps)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
now rewrite mag_plus_eps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
(x + eps)%R <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
~ F (x + eps)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
(x + eps)%R <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
~ F (x + eps)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
now apply Rgt_not_eq, Rplus_lt_0_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
~ F (x + eps)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
intros Fs.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:round beta fexp Zfloor (x + eps) = x
Fs:F (x + eps)
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
rewrite round_generic in Hd...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:(x + eps)%R = x
Fs:F (x + eps)
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
apply Rgt_not_eq with (2 := Hd).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:(x + eps)%R = x
Fs:F (x + eps)
(x + eps > x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
pattern x at 2 ; rewrite <- Rplus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:(eps < ulp x)%R
Heps:(0 <= eps < ulp x)%R
Hd:(x + eps)%R = x
Fs:F (x + eps)
(x + eps > x + 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
now apply Rplus_lt_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
rewrite Heps2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
round beta fexp Zceil (x + ulp x) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
apply round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:(0 < x)%R
Heps1:(0 < eps)%R
Heps2:eps = ulp x
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
now apply generic_format_succ_aux1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R
(* . x=0 *)
rewrite <- Zx1, 2!Rplus_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
(0 < eps <= ulp 0)%R ->
round beta fexp Zceil eps = ulp 0
intros Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
round beta fexp Zceil eps = ulp 0
case (proj2 Heps).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
(eps < ulp 0)%R -> round beta fexp Zceil eps = ulp 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
unfold round, scaled_mantissa, cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
(eps < ulp 0)%R ->
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = ulp 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
unfold ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
(eps <
 (if Req_bool 0 0
  then
   match negligible_exp with
   | Some n => bpow (fexp n)
   | None => 0
   end
  else bpow (cexp beta fexp 0)))%R ->
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} =
(if Req_bool 0 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp 0))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
(eps <
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R ->
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} =
match negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
case negligible_exp_spec.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
(forall n : Z, (fexp n < n)%Z) ->
(eps < 0)%R ->
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
forall n : Z,
(n <= fexp n)%Z ->
(eps < bpow (fexp n))%R ->
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
forall n : Z,
(n <= fexp n)%Z ->
(eps < bpow (fexp n))%R ->
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
intros n Hn H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
assert (fexp (mag beta eps) = fexp n).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
fexp (mag beta eps) = fexp n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
apply valid_exp; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
(mag beta eps <= fexp n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
assert(mag beta eps-1 < fexp n)%Z;[idtac|omega].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
(mag beta eps - 1 < fexp n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
apply lt_bpow with beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
(bpow (mag beta eps - 1) < bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
apply Rle_lt_trans with (2:=H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
(bpow (mag beta eps - 1) <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
destruct (mag beta eps) as (e,He).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
e:Z
He:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R
(bpow (Build_mag_prop beta eps e He - 1) <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
simpl; rewrite Rabs_pos_eq in He.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
e:Z
He:eps <> 0%R -> (bpow (e - 1) <= eps < bpow e)%R
(bpow (e - 1) <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
e:Z
He:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R
(0 <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
now apply He, Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
e:Z
He:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R
(0 <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
now left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
replace (Zceil (eps * bpow (- fexp (mag beta eps)))) with 1%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
F2R {| Fnum := 1; Fexp := fexp (mag beta eps) |} =
bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
1%Z = Zceil (eps * bpow (- fexp (mag beta eps)))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
unfold F2R; simpl; rewrite H0; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
1%Z = Zceil (eps * bpow (- fexp (mag beta eps)))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
apply sym_eq, Zceil_imp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(IZR (1 - 1) < eps * bpow (- fexp (mag beta eps)) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(IZR (1 - 1) < eps * bpow (- fexp (mag beta eps)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
simpl; apply Rmult_lt_0_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(0 < eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(0 < bpow (- fexp (mag beta eps)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
apply Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(0 < bpow (- fexp (mag beta eps)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) <= 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
apply Rmult_le_reg_r with (bpow (fexp n)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(0 < bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) * bpow (fexp n) <=
 1 * bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps)) * bpow (fexp n) <=
 1 * bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
rewrite Rmult_assoc, <- bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow (- fexp (mag beta eps) + fexp n) <=
 1 * bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
rewrite H0; ring_simplify (-fexp n + fexp n)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps * bpow 0 <= 1 * bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
simpl; rewrite Rmult_1_l, Rmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
n:Z
Hn:(n <= fexp n)%Z
H:(eps < bpow (fexp n))%R
H0:fexp (mag beta eps) = fexp n
(eps <= bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
now left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
eps = ulp 0 -> round beta fexp Zceil eps = ulp 0
intros P; rewrite P.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
P:eps = ulp 0
round beta fexp Zceil (ulp 0) = ulp 0
apply round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Zx:(0 <= x)%R
Fx:F x
eps:R
Zx1:0%R = x
Heps:(0 < eps <= ulp 0)%R
P:eps = ulp 0
F (ulp 0)
apply generic_format_ulp_0.
Qed.

Theorem round_UP_pred_plus_eps_pos :
  forall x, (0 < x)%R -> F x ->
  forall eps, (0 < eps <= ulp (pred x) )%R ->
  round beta fexp Zceil (pred x + eps) = x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
forall eps : R,
(0 < eps <= ulp (pred x))%R ->
round beta fexp Zceil (pred x + eps) = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
forall eps : R,
(0 < eps <= ulp (pred x))%R ->
round beta fexp Zceil (pred x + eps) = x
intros x Hx Fx eps Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
round beta fexp Zceil (pred x + eps) = x
rewrite round_UP_plus_eps_pos; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
(pred x + ulp (pred x))%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
(0 <= pred x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
F (pred x)
rewrite pred_eq_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
(pred_pos x + ulp (pred_pos x))%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
(0 <= pred x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
F (pred x)
apply pred_pos_plus_ulp; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
(0 <= pred x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
F (pred x)
now left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
(0 <= pred x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
F (pred x)
now apply pred_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred x))%R
F (pred x)
apply generic_format_pred; trivial.
Qed.

Theorem round_DN_minus_eps_pos :
  forall x,  (0 < x)%R -> F x ->
  forall eps, (0 < eps <= ulp (pred x))%R ->
  round beta fexp Zfloor (x - eps) = pred x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
forall eps : R,
(0 < eps <= ulp (pred x))%R ->
round beta fexp Zfloor (x - eps) = pred x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 < x)%R ->
F x ->
forall eps : R,
(0 < eps <= ulp (pred x))%R ->
round beta fexp Zfloor (x - eps) = pred x
intros x Hpx Fx eps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
(0 < eps <= ulp (pred x))%R ->
round beta fexp Zfloor (x - eps) = pred x
rewrite pred_eq_pos;[intros Heps|now left].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
round beta fexp Zfloor (x - eps) = pred_pos x
replace (x-eps)%R with (pred_pos x + (ulp (pred_pos x)-eps))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
round beta fexp Zfloor
  (pred_pos x + (ulp (pred_pos x) - eps)) = pred_pos x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(pred_pos x + (ulp (pred_pos x) - eps))%R =
(x - eps)%R
2: pattern x at 3; rewrite <- (pred_pos_plus_ulp x); trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
round beta fexp Zfloor
  (pred_pos x + (ulp (pred_pos x) - eps)) = pred_pos x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(pred_pos x + (ulp (pred_pos x) - eps))%R =
(pred_pos x + ulp (pred_pos x) - eps)%R
2: ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
round beta fexp Zfloor
  (pred_pos x + (ulp (pred_pos x) - eps)) = pred_pos x
rewrite round_DN_plus_eps_pos; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(0 <= pred_pos x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
F (pred_pos x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(0 <= ulp (pred_pos x) - eps < ulp (pred_pos x))%R
now apply pred_pos_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
F (pred_pos x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(0 <= ulp (pred_pos x) - eps < ulp (pred_pos x))%R
now apply generic_format_pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(0 <= ulp (pred_pos x) - eps < ulp (pred_pos x))%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(0 <= ulp (pred_pos x) - eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(ulp (pred_pos x) - eps < ulp (pred_pos x))%R
apply Rle_0_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(eps <= ulp (pred_pos x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(ulp (pred_pos x) - eps < ulp (pred_pos x))%R
now apply Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(ulp (pred_pos x) - eps < ulp (pred_pos x))%R
rewrite <- Rplus_0_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(ulp (pred_pos x) - eps < ulp (pred_pos x) + 0)%R
apply Rplus_lt_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(- eps < 0)%R
rewrite <- Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(- eps < - 0)%R
apply Ropp_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hpx:(0 < x)%R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred_pos x))%R
(0 < eps)%R
now apply Heps.
Qed.

Theorem round_DN_plus_eps:
  forall x, F x ->
  forall eps, (0 <= eps < if (Rle_bool 0 x) then (ulp x)
                                     else (ulp (pred (-x))))%R ->
  round beta fexp Zfloor (x + eps) = x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
forall eps : R,
(0 <= eps <
 (if Rle_bool 0 x then ulp x else ulp (pred (- x))))%R ->
round beta fexp Zfloor (x + eps) = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
forall eps : R,
(0 <= eps <
 (if Rle_bool 0 x then ulp x else ulp (pred (- x))))%R ->
round beta fexp Zfloor (x + eps) = x
intros x Fx eps Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
round beta fexp Zfloor (x + eps) = x
case (Rle_or_lt 0 x); intros Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(0 <= x)%R
round beta fexp Zfloor (x + eps) = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(x < 0)%R
round beta fexp Zfloor (x + eps) = x
apply round_DN_plus_eps_pos; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(0 <= x)%R
(0 <= eps < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(x < 0)%R
round beta fexp Zfloor (x + eps) = x
split; try apply Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(0 <= x)%R
(eps < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(x < 0)%R
round beta fexp Zfloor (x + eps) = x
rewrite Rle_bool_true in Heps; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp x)%R
Zx:(0 <= x)%R
(eps < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(x < 0)%R
round beta fexp Zfloor (x + eps) = x
now apply Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(x < 0)%R
round beta fexp Zfloor (x + eps) = x
(* *)
rewrite Rle_bool_false in Heps; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
round beta fexp Zfloor (x + eps) = x
rewrite <- (Ropp_involutive (x+eps)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
round beta fexp Zfloor (- - (x + eps)) = x
pattern x at 2; rewrite <- (Ropp_involutive x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
round beta fexp Zfloor (- - (x + eps)) = (- - x)%R
rewrite round_DN_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(- round beta fexp Zceil (- (x + eps)))%R = (- - x)%R
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
round beta fexp Zceil (- (x + eps)) = (- x)%R
replace (-(x+eps))%R with (pred (-x) + (ulp (pred (-x)) - eps))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
round beta fexp Zceil
  (pred (- x) + (ulp (pred (- x)) - eps)) = (- x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R
rewrite round_UP_pred_plus_eps_pos; try reflexivity.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < ulp (pred (- x)) - eps <= ulp (pred (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R
now apply Ropp_0_gt_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < ulp (pred (- x)) - eps <= ulp (pred (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < ulp (pred (- x)) - eps <= ulp (pred (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < ulp (pred (- x)) - eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(ulp (pred (- x)) - eps <= ulp (pred (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R
apply Rplus_lt_reg_l with eps; ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(eps < ulp (pred (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(ulp (pred (- x)) - eps <= ulp (pred (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R
apply Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(ulp (pred (- x)) - eps <= ulp (pred (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R
apply Rplus_le_reg_l with (eps-ulp (pred (- x)))%R; ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R
apply Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R
unfold pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ (- - x) + (ulp (- succ (- - x)) - eps))%R =
(- (x + eps))%R
rewrite Ropp_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (ulp (- succ x) - eps))%R =
(- (x + eps))%R
unfold succ; rewrite Rle_bool_false; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(- - pred_pos (- x) + (ulp (- - pred_pos (- x)) - eps))%R =
(- (x + eps))%R
rewrite Ropp_involutive; unfold Rminus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(pred_pos (- x) + (ulp (pred_pos (- x)) + - eps))%R =
(- (x + eps))%R
rewrite <- Rplus_assoc, pred_pos_plus_ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(- x + - eps)%R = (- (x + eps))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
now apply Ropp_0_gt_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 <= eps < ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
now apply generic_format_opp.
Qed.

Theorem round_UP_plus_eps :
  forall x, F x ->
  forall eps, (0 < eps <= if (Rle_bool 0 x) then (ulp x)
                                     else (ulp (pred (-x))))%R ->
  round beta fexp Zceil (x + eps) = (succ x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
forall eps : R,
(0 < eps <=
 (if Rle_bool 0 x then ulp x else ulp (pred (- x))))%R ->
round beta fexp Zceil (x + eps) = succ x
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
forall eps : R,
(0 < eps <=
 (if Rle_bool 0 x then ulp x else ulp (pred (- x))))%R ->
round beta fexp Zceil (x + eps) = succ x
intros x Fx eps Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
round beta fexp Zceil (x + eps) = succ x
case (Rle_or_lt 0 x); intros Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(0 <= x)%R
round beta fexp Zceil (x + eps) = succ x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(x < 0)%R
round beta fexp Zceil (x + eps) = succ x
rewrite succ_eq_pos; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(0 <= x)%R
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(x < 0)%R
round beta fexp Zceil (x + eps) = succ x
rewrite Rle_bool_true in Heps; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp x)%R
Zx:(0 <= x)%R
round beta fexp Zceil (x + eps) = (x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(x < 0)%R
round beta fexp Zceil (x + eps) = succ x
apply round_UP_plus_eps_pos; assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%R
Zx:(x < 0)%R
round beta fexp Zceil (x + eps) = succ x
(* *)
rewrite Rle_bool_false in Heps; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
round beta fexp Zceil (x + eps) = succ x
rewrite <- (Ropp_involutive (x+eps)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
round beta fexp Zceil (- - (x + eps)) = succ x
rewrite <- (Ropp_involutive (succ x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
round beta fexp Zceil (- - (x + eps)) = (- - succ x)%R
rewrite round_UP_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- round beta fexp Zfloor (- (x + eps)))%R =
(- - succ x)%R
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
round beta fexp Zfloor (- (x + eps)) = (- succ x)%R
replace (-(x+eps))%R with (-succ x + (-eps + ulp (pred (-x))))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
round beta fexp Zfloor
  (- succ x + (- eps + ulp (pred (- x)))) =
(- succ x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
apply round_DN_plus_eps_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= - succ x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- succ x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
rewrite <- pred_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= pred (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- succ x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
apply pred_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- succ x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
now apply Ropp_0_gt_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- succ x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- succ x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
now apply generic_format_opp, generic_format_succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= - eps + ulp (pred (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- eps + ulp (pred (- x)) < ulp (- succ x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
apply Rplus_le_reg_l with eps; ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(eps <= ulp (pred (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- eps + ulp (pred (- x)) < ulp (- succ x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
apply Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- eps + ulp (pred (- x)) < ulp (- succ x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
unfold pred; rewrite Ropp_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- eps + ulp (- succ x) < ulp (- succ x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
apply Rplus_lt_reg_l with (eps-ulp (- succ x))%R; ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
apply Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
unfold succ; rewrite Rle_bool_false; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- - pred_pos (- x) + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R
apply trans_eq with (-x +-eps)%R;[idtac|ring].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- - pred_pos (- x) + (- eps + ulp (pred (- x))))%R =
(- x + - eps)%R
pattern (-x)%R at 3; rewrite <- (pred_pos_plus_ulp (-x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- - pred_pos (- x) + (- eps + ulp (pred (- x))))%R =
(pred_pos (- x) + ulp (pred_pos (- x)) + - eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
rewrite pred_eq_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(- - pred_pos (- x) + (- eps + ulp (pred_pos (- x))))%R =
(pred_pos (- x) + ulp (pred_pos (- x)) + - eps)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 <= - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
left; now apply Ropp_0_gt_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
now apply Ropp_0_gt_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <= ulp (pred (- x)))%R
Zx:(x < 0)%R
F (- x)
now apply generic_format_opp.
Qed.


Lemma le_pred_pos_lt :
  forall x y,
  F x -> F y ->
  (0 <= x < y)%R ->
  (x <= pred_pos y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (0 <= x < y)%R -> (x <= pred_pos y)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (0 <= x < y)%R -> (x <= pred_pos y)%R
intros x y Fx Fy H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
(x <= pred_pos y)%R
case (proj1 H); intros V.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
assert (Zy:(0 < y)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
(0 < y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Rle_lt_trans with (1:=proj1 H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
(x < y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
(* *)
assert (Zp: (0 < pred y)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
(0 < pred y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
assert (Zp:(0 <= pred y)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
(0 <= pred y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 <= pred y)%R
(0 < pred y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply pred_ge_0 ; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 <= pred y)%R
(0 < pred y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
destruct Zp; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
(0 < pred y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
generalize H0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
0%R = pred y -> (0 < pred y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite pred_eq_pos;[idtac|now left].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
0%R = pred_pos y -> (0 < pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
unfold pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
0%R =
(if Req_bool y (bpow (mag beta y - 1))
 then (y - bpow (fexp (mag beta y - 1)))%R
 else (y - ulp y)%R) ->
(0 <
 (if Req_bool y (bpow (mag beta y - 1))
  then y - bpow (fexp (mag beta y - 1))
  else y - ulp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
destruct (mag beta y) as (ey,Hey); simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
0%R =
(if Req_bool y (bpow (ey - 1))
 then (y - bpow (fexp (ey - 1)))%R
 else (y - ulp y)%R) ->
(0 <
 (if Req_bool y (bpow (ey - 1))
  then y - bpow (fexp (ey - 1))
  else y - ulp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
case Req_bool_spec; intros Hy2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
0%R = (y - bpow (fexp (ey - 1)))%R ->
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
(* . *)
intros Hy3.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
assert (ey-1 = fexp (ey -1))%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
(ey - 1)%Z = fexp (ey - 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply bpow_inj with beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
bpow (ey - 1) = bpow (fexp (ey - 1))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite <- Hy2, <- Rplus_0_l, Hy3.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
y =
(y - bpow (fexp (ey - 1)) + bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
assert (Zx: (x <> 0)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
destruct (mag beta x) as (ex,Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
specialize (Hex Zx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
assert (ex <= ey)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(ex <= ey)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply bpow_lt_bpow with beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(bpow (ex - 1) < bpow ey)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Rle_lt_trans with (1:=proj1 Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs x < bpow ey)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Rlt_trans with (Rabs y).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs x < Rabs y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs y < bpow ey)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite 2!Rabs_right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(x < y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(y >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(x >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs y < bpow ey)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(y >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(x >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs y < bpow ey)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rgt_ge.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(x >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs y < bpow ey)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rgt_ge.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
(Rabs y < bpow ey)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Hey.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
case (Zle_lt_or_eq _ _ H2); intros Hexy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
assert (fexp ex = fexp (ey-1))%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
fexp ex = fexp (ey - 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply valid_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
(ey - 1 <= fexp (ey - 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
(ex <= fexp (ey - 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
(ex <= fexp (ey - 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite <- H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
(ex <= ey - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
absurd (0 < Ztrunc (scaled_mantissa beta fexp x) < 1)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
~ (0 < Ztrunc (scaled_mantissa beta fexp x) < 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(0 < Ztrunc (scaled_mantissa beta fexp x) < 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(0 < Ztrunc (scaled_mantissa beta fexp x) < 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(0 < Ztrunc (scaled_mantissa beta fexp x))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(Ztrunc (scaled_mantissa beta fexp x) < 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply gt_0_F2R with beta (cexp beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(Ztrunc (scaled_mantissa beta fexp x) < 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now rewrite <- Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(Ztrunc (scaled_mantissa beta fexp x) < 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply lt_IZR.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(IZR (Ztrunc (scaled_mantissa beta fexp x)) < 1)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Rmult_lt_reg_r with (bpow (cexp beta fexp x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(0 < bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) < 
 1 * bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) < 1 * bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
replace (IZR (Ztrunc (scaled_mantissa beta fexp x)) *
  bpow (cexp beta fexp x))%R with x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(x < 1 * bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(x < bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(x < bpow (fexp (mag beta x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite mag_unique with beta x ex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(x < bpow (fexp ex))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(bpow (ex - 1) <= Rabs x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite H3,<-H1, <- Hy2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(x < y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(bpow (ex - 1) <= Rabs x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:(ex < ey)%Z
H3:fexp ex = fexp (ey - 1)
(bpow (ex - 1) <= Rabs x < bpow ex)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
exact Hex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(0 < y - bpow (fexp (ey - 1)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
absurd (y <= x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
~ (y <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(y <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rlt_not_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(y <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite Rabs_right in Hex.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(y <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(x >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Rle_trans with (2:=proj1 Hex).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(y <= bpow (ex - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(x >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite Hexy, Hy2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(bpow (ey - 1) <= bpow (ey - 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(x >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rle_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y = bpow (ey - 1)
Hy3:0%R = (y - bpow (fexp (ey - 1)))%R
H1:(ey - 1)%Z = fexp (ey - 1)
Zx:x <> 0%R
ex:Z
Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%R
H2:(ex <= ey)%Z
Hexy:ex = ey
(x >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rgt_ge.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
0%R = (y - ulp y)%R -> (0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
(* . *)
intros Hy3.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
(0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
assert (y = bpow (fexp ey))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
y = bpow (fexp ey)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Rminus_diag_uniq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
(y - bpow (fexp ey))%R = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite Hy3.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
(y - bpow (fexp ey))%R = (y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite ulp_neq_0;[idtac|now apply Rgt_not_eq].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
(y - bpow (fexp ey))%R =
(y - bpow (cexp beta fexp y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
(y - bpow (fexp ey))%R =
(y - bpow (fexp (mag beta y)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite (mag_unique beta y ey); trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
(bpow (ey - 1) <= Rabs y < bpow ey)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Hey.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy2:y <> bpow (ey - 1)
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(0 < y - ulp y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
contradict Hy2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
y = bpow (ey - 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
bpow (fexp ey) = bpow (ey - 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
fexp ey = (ey - 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Zplus_reg_l with 1%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(1 + fexp ey)%Z = (1 + (ey - 1))%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(fexp ey + 1)%Z = ey
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply trans_eq with (mag beta y).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(fexp ey + 1)%Z = mag beta y
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
mag beta y = ey
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply sym_eq; apply mag_unique.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey + 1 - 1) <= Rabs y < bpow (fexp ey + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
mag beta y = ey
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite H1, Rabs_right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey + 1 - 1) <= bpow (fexp ey) <
 bpow (fexp ey + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
mag beta y = ey
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey + 1 - 1) <= bpow (fexp ey))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey) < bpow (fexp ey + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
mag beta y = ey
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply bpow_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(fexp ey + 1 - 1 <= fexp ey)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey) < bpow (fexp ey + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
mag beta y = ey
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey) < bpow (fexp ey + 1))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
mag beta y = ey
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply bpow_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(fexp ey < fexp ey + 1)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
mag beta y = ey
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
omega.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (fexp ey) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
mag beta y = ey
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Rle_ge; apply bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
mag beta y = ey
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply mag_unique.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
(bpow (ey - 1) <= Rabs y < bpow ey)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Hey.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
H0:0%R = pred y
ey:Z
Hey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R
Hy3:0%R = (y - ulp y)%R
H1:y = bpow (fexp ey)
y <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
(* *)
case (Rle_or_lt (ulp (pred_pos y)) (y-x)); intros H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(ulp (pred_pos y) <= y - x)%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
(* . *)
apply Rplus_le_reg_r with (-x + ulp (pred_pos y))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(ulp (pred_pos y) <= y - x)%R
(x + (- x + ulp (pred_pos y)) <=
 pred_pos y + (- x + ulp (pred_pos y)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
ring_simplify (x+(-x+ulp (pred_pos y)))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(ulp (pred_pos y) <= y - x)%R
(ulp (pred_pos y) <=
 pred_pos y + (- x + ulp (pred_pos y)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Rle_trans with (1:=H1).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(ulp (pred_pos y) <= y - x)%R
(y - x <= pred_pos y + (- x + ulp (pred_pos y)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite <- (pred_pos_plus_ulp y) at 1; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(ulp (pred_pos y) <= y - x)%R
(pred_pos y + ulp (pred_pos y) - x <=
 pred_pos y + (- x + ulp (pred_pos y)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Req_le; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(x <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
(* . *)
replace x with (y-(y-x))%R by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(y - (y - x) <= pred_pos y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite <- pred_eq_pos;[idtac|now left].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(y - (y - x) <= pred y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite <- round_DN_minus_eps_pos with (eps:=(y-x)%R); try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(y - (y - x) <= round beta fexp Zfloor (y - (y - x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(0 < y - x <= ulp (pred y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
ring_simplify (y-(y-x))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(x <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(0 < y - x <= ulp (pred y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply Req_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
x = round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(0 < y - x <= ulp (pred y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
round beta fexp Zfloor x = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(0 < y - x <= ulp (pred y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
apply round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(0 < y - x <= ulp (pred y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
split; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(0 < y - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(y - x <= ulp (pred y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rlt_Rminus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(y - x <= ulp (pred y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite pred_eq_pos;[idtac|now left].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:(0 < x)%R
Zy:(0 < y)%R
Zp:(0 < pred y)%R
H1:(y - x < ulp (pred_pos y))%R
(y - x <= ulp (pred_pos y))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(x <= pred_pos y)%R
rewrite <- V; apply pred_pos_ge_0; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(0 <= x < y)%R
V:0%R = x
(0 < y)%R
apply Rle_lt_trans with (1:=proj1 H); apply H.
Qed.

Lemma succ_le_lt_aux:
  forall x y,
  F x -> F y ->
  (0 <= x)%R -> (x < y)%R ->
  (succ x <= y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x ->
F y -> (0 <= x)%R -> (x < y)%R -> (succ x <= y)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x ->
F y -> (0 <= x)%R -> (x < y)%R -> (succ x <= y)%R
intros x y Hx Hy Zx H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
(succ x <= y)%R
rewrite succ_eq_pos; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
(x + ulp x <= y)%R
case (Rle_or_lt (ulp x) (y-x)); intros H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(ulp x <= y - x)%R
(x + ulp x <= y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
(x + ulp x <= y)%R
apply Rplus_le_reg_r with (-x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(ulp x <= y - x)%R
(x + ulp x + - x <= y + - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
(x + ulp x <= y)%R
now ring_simplify (x+ulp x + -x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
(x + ulp x <= y)%R
replace y with (x+(y-x))%R by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
(x + ulp x <= x + (y - x))%R
absurd (x < y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
~ (x < y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
(x < y)%R
2: apply H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
~ (x < y)%R
apply Rle_not_lt; apply Req_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
y = x
rewrite <- round_DN_plus_eps_pos with (eps:=(y-x)%R); try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
y = round beta fexp Zfloor (x + (y - x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
(0 <= y - x < ulp x)%R
ring_simplify (x+(y-x))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
y = round beta fexp Zfloor y
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
(0 <= y - x < ulp x)%R
apply sym_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
round beta fexp Zfloor y = y
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
(0 <= y - x < ulp x)%R
apply round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
(0 <= y - x < ulp x)%R
split; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Hx:F x
Hy:F y
Zx:(0 <= x)%R
H:(x < y)%R
H1:(y - x < ulp x)%R
(0 <= y - x)%R
apply Rlt_le; now apply Rlt_Rminus.
Qed.

Theorem succ_le_lt:
  forall x y,
  F x -> F y ->
  (x < y)%R ->
  (succ x <= y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x < y)%R -> (succ x <= y)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x < y)%R -> (succ x <= y)%R
intros x y Fx Fy H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
(succ x <= y)%R
destruct (Rle_or_lt 0 x) as [Hx|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(0 <= x)%R
(succ x <= y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
(succ x <= y)%R
now apply succ_le_lt_aux.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
(succ x <= y)%R
unfold succ; rewrite Rle_bool_false; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
(- pred_pos (- x) <= y)%R
case (Rle_or_lt y 0); intros Hy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
(- pred_pos (- x) <= y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- pred_pos (- x) <= y)%R
rewrite <- (Ropp_involutive y).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
(- pred_pos (- x) <= - - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- pred_pos (- x) <= y)%R
apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
(- y <= pred_pos (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- pred_pos (- x) <= y)%R
apply le_pred_pos_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
F (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
(0 <= - y < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- pred_pos (- x) <= y)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
(0 <= - y < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- pred_pos (- x) <= y)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
(0 <= - y < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- pred_pos (- x) <= y)%R
split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
(0 <= - y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
(- y < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- pred_pos (- x) <= y)%R
rewrite <- Ropp_0; now apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(y <= 0)%R
(- y < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- pred_pos (- x) <= y)%R
now apply Ropp_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- pred_pos (- x) <= y)%R
apply Rle_trans with (-0)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- pred_pos (- x) <= - 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- 0 <= y)%R
apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(0 <= pred_pos (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- 0 <= y)%R
apply pred_pos_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- 0 <= y)%R
rewrite <- Ropp_0; now apply Ropp_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- 0 <= y)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
H:(x < y)%R
Hx:(x < 0)%R
Hy:(0 < y)%R
(- 0 <= y)%R
rewrite Ropp_0; now left.
Qed.

Theorem pred_ge_gt :
  forall x y,
  F x -> F y ->
  (x < y)%R ->
  (x <= pred y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x < y)%R -> (x <= pred y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x < y)%R -> (x <= pred y)%R
intros x y Fx Fy Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(x <= pred y)%R
rewrite <- (Ropp_involutive x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(- - x <= pred y)%R
unfold pred; apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(succ (- y) <= - x)%R
apply succ_le_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
F (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(- y < - x)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(- y < - x)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(- y < - x)%R
now apply Ropp_lt_contravar.
Qed.

Theorem succ_gt_ge :
  forall x y,
  (y <> 0)%R ->
  (x <= y)%R ->
  (x < succ y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
y <> 0%R -> (x <= y)%R -> (x < succ y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
y <> 0%R -> (x <= y)%R -> (x < succ y)%R
intros x y Zy Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Zy:y <> 0%R
Hxy:(x <= y)%R
(x < succ y)%R
apply Rle_lt_trans with (1 := Hxy).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Zy:y <> 0%R
Hxy:(x <= y)%R
(y < succ y)%R
now apply succ_gt_id.
Qed.

Theorem pred_lt_le :
  forall x y,
  (x <> 0)%R ->
  (x <= y)%R ->
  (pred x < y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
x <> 0%R -> (x <= y)%R -> (pred x < y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
x <> 0%R -> (x <= y)%R -> (pred x < y)%R
intros x y Zy Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Zy:x <> 0%R
Hxy:(x <= y)%R
(pred x < y)%R
apply Rlt_le_trans with (2 := Hxy).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Zy:x <> 0%R
Hxy:(x <= y)%R
(pred x < x)%R
now apply pred_lt_id.
Qed.

Lemma succ_pred_pos :
  forall x, F x -> (0 < x)%R -> succ (pred x) = x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> (0 < x)%R -> succ (pred x) = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> (0 < x)%R -> succ (pred x) = x
intros x Fx Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
succ (pred x) = x
rewrite pred_eq_pos by now left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
succ (pred_pos x) = x
rewrite succ_eq_pos by now apply pred_pos_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(pred_pos x + ulp (pred_pos x))%R = x
now apply pred_pos_plus_ulp.
Qed.

Theorem pred_ulp_0 :
  pred (ulp 0) = 0%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
pred (ulp 0) = 0%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
pred (ulp 0) = 0%R
rewrite pred_eq_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
pred_pos (ulp 0) = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(0 <= ulp 0)%R
2: apply ulp_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
pred_pos (ulp 0) = 0%R
unfold ulp; rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
pred_pos
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end = 0%R
case negligible_exp_spec'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
negligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) ->
pred_pos
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
pred_pos
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end = 0%R
(* *)
intros [H1 _]; rewrite H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H1:negligible_exp = None
pred_pos 0 = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
pred_pos
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end = 0%R
unfold pred_pos; rewrite Req_bool_false.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H1:negligible_exp = None
(0 - ulp 0)%R = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H1:negligible_exp = None
0%R <> bpow (mag beta 0 - 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
pred_pos
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end = 0%R
2: apply Rlt_not_eq, bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H1:negligible_exp = None
(0 - ulp 0)%R = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
pred_pos
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end = 0%R
unfold ulp; rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H1:negligible_exp = None
(0 -
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
pred_pos
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end = 0%R
rewrite H1; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
pred_pos
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end = 0%R
(* *)
intros (n,(H1,H2)); rewrite H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
pred_pos (bpow (fexp n)) = 0%R
unfold pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
(if
  Req_bool (bpow (fexp n))
    (bpow (mag beta (bpow (fexp n)) - 1))
 then
  (bpow (fexp n) -
   bpow (fexp (mag beta (bpow (fexp n)) - 1)))%R
 else (bpow (fexp n) - ulp (bpow (fexp n)))%R) = 0%R
rewrite mag_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
(if Req_bool (bpow (fexp n)) (bpow (fexp n + 1 - 1))
 then (bpow (fexp n) - bpow (fexp (fexp n + 1 - 1)))%R
 else (bpow (fexp n) - ulp (bpow (fexp n)))%R) = 0%R
replace (fexp n + 1 - 1)%Z with (fexp n) by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
(if Req_bool (bpow (fexp n)) (bpow (fexp n))
 then (bpow (fexp n) - bpow (fexp (fexp n)))%R
 else (bpow (fexp n) - ulp (bpow (fexp n)))%R) = 0%R
rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
(bpow (fexp n) - bpow (fexp (fexp n)))%R = 0%R
apply Rminus_diag_eq, f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
n:Z
H1:negligible_exp = Some n
H2:(n <= fexp n)%Z
fexp n = fexp (fexp n)
apply sym_eq, valid_exp; omega.
Qed.

Theorem succ_0 :
  succ 0 = ulp 0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
succ 0 = ulp 0
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
succ 0 = ulp 0
unfold succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(if Rle_bool 0 0
 then (0 + ulp 0)%R
 else (- pred_pos (- 0))%R) = ulp 0
rewrite Rle_bool_true.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(0 + ulp 0)%R = ulp 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(0 <= 0)%R
apply Rplus_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
(0 <= 0)%R
apply Rle_refl.
Qed.

Theorem pred_0 :
  pred 0 = Ropp (ulp 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
pred 0 = (- ulp 0)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
pred 0 = (- ulp 0)%R
rewrite <- succ_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
pred 0 = (- succ 0)%R
rewrite <- Ropp_0 at 1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
pred (- 0) = (- succ 0)%R
apply pred_opp.
Qed.

Lemma pred_succ_pos :
  forall x, F x -> (0 < x)%R ->
  pred (succ x) = x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> (0 < x)%R -> pred (succ x) = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> (0 < x)%R -> pred (succ x) = x
intros x Fx Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
pred (succ x) = x
apply Rle_antisym.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(pred (succ x) <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(x <= pred (succ x))%R
-beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(pred (succ x) <= x)%R apply Rnot_lt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
~ (x < pred (succ x))%R
  intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
H:(x < pred (succ x))%R
False
  apply succ_le_lt with (1 := Fx) in H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
H:(succ x <= pred (succ x))%R
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
H:(x < pred (succ x))%R
F (pred (succ x))
  revert H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(succ x <= pred (succ x))%R -> False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
H:(x < pred (succ x))%R
F (pred (succ x))
  apply Rlt_not_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(pred (succ x) < succ x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
H:(x < pred (succ x))%R
F (pred (succ x))
  apply pred_lt_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
succ x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
H:(x < pred (succ x))%R
F (pred (succ x))
  apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(succ x > 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
H:(x < pred (succ x))%R
F (pred (succ x))
  apply Rlt_le_trans with (1 := Hx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(x <= succ x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
H:(x < pred (succ x))%R
F (pred (succ x))
  apply succ_ge_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
H:(x < pred (succ x))%R
F (pred (succ x))
  now apply generic_format_pred, generic_format_succ.
-beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(x <= pred (succ x))%R apply pred_ge_gt with (1 := Fx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
F (succ x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(x < succ x)%R
  now apply generic_format_succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
(x < succ x)%R
  apply succ_gt_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
x <> 0%R
  now apply Rgt_not_eq.
Qed.

Theorem succ_pred :
  forall x, F x ->
  succ (pred x) = x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> succ (pred x) = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> succ (pred x) = x
intros x Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
succ (pred x) = x
destruct (Rle_or_lt 0 x) as [[Hx|Hx]|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
succ (pred x) = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:0%R = x
succ (pred x) = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
succ (pred x) = x
now apply succ_pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:0%R = x
succ (pred x) = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
succ (pred x) = x
rewrite <- Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:0%R = x
succ (pred 0) = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
succ (pred x) = x
rewrite pred_0, succ_opp, pred_ulp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:0%R = x
(- 0)%R = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
succ (pred x) = x
apply Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
succ (pred x) = x
unfold pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
succ (- succ (- x)) = x
rewrite succ_opp, pred_succ_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
(- - x)%R = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
(0 < - x)%R
apply Ropp_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
(0 < - x)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(x < 0)%R
(0 < - x)%R
now apply Ropp_0_gt_lt_contravar.
Qed.

Theorem pred_succ :
  forall x, F x ->
  pred (succ x) = x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> pred (succ x) = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R, F x -> pred (succ x) = x
intros x Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
pred (succ x) = x
rewrite <- (Ropp_involutive x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
pred (succ (- - x)) = (- - x)%R
rewrite succ_opp, pred_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
(- succ (pred (- x)))%R = (- - x)%R
apply f_equal, succ_pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
F (- x)
now apply generic_format_opp.
Qed.

Theorem round_UP_pred_plus_eps :
  forall x, F x ->
  forall eps, (0 < eps <= if Rle_bool x 0 then ulp x
                          else ulp (pred x))%R ->
  round beta fexp Zceil (pred x + eps) = x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
forall eps : R,
(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R ->
round beta fexp Zceil (pred x + eps) = x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
forall eps : R,
(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R ->
round beta fexp Zceil (pred x + eps) = x
intros x Fx eps Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
round beta fexp Zceil (pred x + eps) = x
rewrite round_UP_plus_eps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
succ (pred x) = x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (pred x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(0 < eps <=
 (if Rle_bool 0 (pred x)
  then ulp (pred x)
  else ulp (pred (- pred x))))%R
now apply succ_pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (pred x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(0 < eps <=
 (if Rle_bool 0 (pred x)
  then ulp (pred x)
  else ulp (pred (- pred x))))%R
now apply generic_format_pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(0 < eps <=
 (if Rle_bool 0 (pred x)
  then ulp (pred x)
  else ulp (pred (- pred x))))%R
unfold pred at 4.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(0 < eps <=
 (if Rle_bool 0 (pred x)
  then ulp (pred x)
  else ulp (pred (- - succ (- x)))))%R
rewrite Ropp_involutive, pred_succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(0 < eps <=
 (if Rle_bool 0 (pred x)
  then ulp (pred x)
  else ulp (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite ulp_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
generalize Heps; case (Rle_bool_spec x 0); intros H1 H2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite Rle_bool_false; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
(pred x < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
case H1; intros H1'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':(x < 0)%R
(pred x < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(pred x < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
apply Rlt_le_trans with (2:=H1).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':(x < 0)%R
(pred x < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(pred x < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
apply pred_lt_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':(x < 0)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(pred x < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
now apply Rlt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(pred x < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite H1'; unfold pred, succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(-
 (if Rle_bool 0 (- 0)
  then - 0 + ulp (- 0)
  else - pred_pos (- - 0)) < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite Ropp_0; rewrite Rle_bool_true;[idtac|now right].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(- (0 + ulp 0) < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite Rplus_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(- ulp 0 < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite <- Ropp_0; apply Ropp_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(0 < ulp (- 0))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
apply Rlt_le_trans with (1:=proj1 H2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(eps <= ulp (- 0))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
apply Rle_trans with (1:=proj2 H2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(ulp x <= ulp (- 0))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite Ropp_0, H1'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(ulp 0 <= ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
now right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite Rle_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 <= pred x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
now apply pred_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
now apply generic_format_opp.
Qed.

Theorem round_DN_minus_eps:
  forall x,  F x ->
  forall eps, (0 < eps <= if (Rle_bool x 0) then (ulp x)
                                     else (ulp (pred x)))%R ->
  round beta fexp Zfloor (x - eps) = pred x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
forall eps : R,
(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R ->
round beta fexp Zfloor (x - eps) = pred x
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
forall eps : R,
(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R ->
round beta fexp Zfloor (x - eps) = pred x
intros x Fx eps Heps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
round beta fexp Zfloor (x - eps) = pred x
replace (x-eps)%R with (-(-x+eps))%R by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
round beta fexp Zfloor (- (- x + eps)) = pred x
rewrite round_DN_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(- round beta fexp Zceil (- x + eps))%R = pred x
unfold pred; apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
round beta fexp Zceil (- x + eps) = succ (- x)
pattern (-x)%R at 1; rewrite <- (pred_succ (-x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
round beta fexp Zceil (pred (succ (- x)) + eps) =
succ (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
apply round_UP_pred_plus_eps.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (succ (- x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp (pred (succ (- x)))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
now apply generic_format_succ, generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp (pred (succ (- x)))))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite pred_succ.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite ulp_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
generalize Heps; case (Rle_bool_spec x 0); intros H1 H2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite Rle_bool_false; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
(0 < succ (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
case H1; intros H1'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':(x < 0)%R
(0 < succ (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(0 < succ (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
apply Rlt_le_trans with (-x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':(x < 0)%R
(- x <= succ (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(0 < succ (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
now apply Ropp_0_gt_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':(x < 0)%R
(- x <= succ (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(0 < succ (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
apply succ_ge_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(0 < succ (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite H1', Ropp_0, succ_eq_pos;[idtac|now right].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(0 < 0 + ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite Rplus_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(0 < ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
apply Rlt_le_trans with (1:=proj1 H2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(x <= 0)%R
H2:(0 < eps <= ulp x)%R
H1':x = 0%R
(eps <= ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite H1' in H2; apply H2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite Rle_bool_true.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 < eps <= ulp (succ (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(succ (- x) <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
now rewrite succ_opp, ulp_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(succ (- x) <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite succ_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(- pred x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
rewrite <- Ropp_0; apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
H1:(0 < x)%R
H2:(0 < eps <= ulp (pred x))%R
(0 <= pred x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
now apply pred_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
eps:R
Heps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%R
F (- x)
now apply generic_format_opp.
Qed.

Error of a rounding, expressed in number of ulps

false for x=0 in the FLX format

(* was ulp_error *)
Theorem error_lt_ulp :
  forall rnd { Zrnd : Valid_rnd rnd } x,
  (x <> 0)%R ->
  (Rabs (round beta fexp rnd x - x) < ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < ulp x)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) < ulp x)%R
intros rnd Zrnd x Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
(Rabs (round beta fexp rnd x - x) < ulp x)%R
destruct (generic_format_EM beta fexp x) as [Hx|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:F x
(Rabs (round beta fexp rnd x - x) < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp rnd x - x) < ulp x)%R
(* x = rnd x *)
rewrite round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:F x
(Rabs (x - x) < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp rnd x - x) < ulp x)%R
unfold Rminus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:F x
(Rabs (x + - x) < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp rnd x - x) < ulp x)%R
rewrite Rplus_opp_r, Rabs_R0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:F x
(0 < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp rnd x - x) < ulp x)%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:F x
(0 < bpow (cexp beta fexp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp rnd x - x) < ulp x)%R
apply bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp rnd x - x) < ulp x)%R
(* x <> rnd x *)
destruct (round_DN_or_UP beta fexp rnd x) as [H|H] ; rewrite H ; clear H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zfloor x - x) < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
(* . *)
rewrite Rabs_left1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(- (round beta fexp Zfloor x - x) < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
rewrite Ropp_minus_distr.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(x - round beta fexp Zfloor x < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
apply Rplus_lt_reg_l with (round beta fexp Zfloor x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x +
 (x - round beta fexp Zfloor x) <
 round beta fexp Zfloor x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
rewrite <- round_UP_DN_ulp with (1 := Hx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x +
 (x - round beta fexp Zfloor x) <
 round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(x < round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
assert (Hu: (x <= round beta fexp Zceil x)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(x <= round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hu:(x <= round beta fexp Zceil x)%R
(x < round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
apply round_UP_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hu:(x <= round beta fexp Zceil x)%R
(x < round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
destruct Hu as [Hu|Hu].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hu:(x < round beta fexp Zceil x)%R
(x < round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hu:x = round beta fexp Zceil x
(x < round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
exact Hu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hu:x = round beta fexp Zceil x
(x < round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
elim Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hu:x = round beta fexp Zceil x
F x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
rewrite Hu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hu:x = round beta fexp Zceil x
F (round beta fexp Zceil x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
apply generic_format_round...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
apply Rle_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
apply round_DN_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(Rabs (round beta fexp Zceil x - x) < ulp x)%R
(* . *)
rewrite Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zceil x - x < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
rewrite round_UP_DN_ulp with (1 := Hx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x + ulp x - x < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
apply Rplus_lt_reg_r with (x - ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x + ulp x - x + (x - ulp x) <
 ulp x + (x - ulp x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
assert (Hd: (round beta fexp Zfloor x <= x)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(round beta fexp Zfloor x <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hd:(round beta fexp Zfloor x <= x)%R
(round beta fexp Zfloor x < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
apply round_DN_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hd:(round beta fexp Zfloor x <= x)%R
(round beta fexp Zfloor x < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
destruct Hd as [Hd|Hd].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hd:(round beta fexp Zfloor x < x)%R
(round beta fexp Zfloor x < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hd:round beta fexp Zfloor x = x
(round beta fexp Zfloor x < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
exact Hd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hd:round beta fexp Zfloor x = x
(round beta fexp Zfloor x < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
elim Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hd:round beta fexp Zfloor x = x
F x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
rewrite <- Hd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
Hd:round beta fexp Zfloor x = x
F (round beta fexp Zfloor x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
apply generic_format_round...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(0 <= round beta fexp Zceil x - x)%R
apply Rle_0_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
Hx:~ F x
(x <= round beta fexp Zceil x)%R
apply round_UP_pt...
Qed.

(* was ulp_error_le *)
Theorem error_le_ulp :
  forall rnd { Zrnd : Valid_rnd rnd } x,
  (Rabs (round beta fexp rnd x - x) <= ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(Rabs (round beta fexp rnd x - x) <= ulp x)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(Rabs (round beta fexp rnd x - x) <= ulp x)%R
intros  rnd Zrnd x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
(Rabs (round beta fexp rnd x - x) <= ulp x)%R
case (Req_dec x 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
x = 0%R ->
(Rabs (round beta fexp rnd x - x) <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <= ulp x)%R
intros Zx; rewrite Zx, round_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R
(Rabs (0 - 0) <= ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <= ulp x)%R
unfold Rminus; rewrite Rplus_0_l, Ropp_0, Rabs_R0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R
(0 <= ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <= ulp x)%R
apply ulp_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <= ulp x)%R
intros Zx; left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
(Rabs (round beta fexp rnd x - x) < ulp x)%R
now apply error_lt_ulp.
Qed.

Theorem error_le_half_ulp :
  forall choice x,
  (Rabs (round beta fexp (Znearest choice) x - x) <= /2 * ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (x : R),
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (x : R),
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
intros choice x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
destruct (generic_format_EM beta fexp x) as [Hx|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
(* x = rnd x *)
rewrite round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(Rabs (x - x) <= / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
unfold Rminus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(Rabs (x + - x) <= / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
rewrite Rplus_opp_r, Rabs_R0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(0 <= / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply Rmult_le_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(0 <= / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(0 <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(0 < / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(0 <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply Rinv_0_lt_compat.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(0 < 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(0 <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
now apply IZR_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:F x
(0 <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply ulp_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
(* x <> rnd x *)
set (d := round beta fexp Zfloor x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
destruct (round_N_pt beta fexp choice x) as (Hr1, Hr2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
destruct (Rle_or_lt (x - d) (d + ulp x - x)) as [H|H].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
(* . rnd(x) = rndd(x) *)
apply Rle_trans with (Rabs (d - x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (d - x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(Rabs (d - x) <= / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply Hr2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
F d
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(Rabs (d - x) <= / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply (round_DN_pt beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(Rabs (d - x) <= / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
rewrite Rabs_left1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(- (d - x) <= / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
rewrite Ropp_minus_distr.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(x - d <= / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply Rmult_le_reg_r with 2%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(0 < 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
((x - d) * 2 <= / 2 * ulp x * 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
now apply IZR_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
((x - d) * 2 <= / 2 * ulp x * 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply Rplus_le_reg_r with (d - x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
((x - d) * 2 + (d - x) <= / 2 * ulp x * 2 + (d - x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(x - d <= - x + d + 2 * / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply Rle_trans with (1 := H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d + ulp x - x <= - x + d + 2 * / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d + ulp x - x)%R = (- x + d + 2 * / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R field.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d - x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply Rle_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(x - d <= d + ulp x - x)%R
(d <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply (round_DN_pt beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
(* . rnd(x) = rndu(x) *)
assert (Hu: (d + ulp x)%R = round beta fexp Zceil x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(d + ulp x)%R = round beta fexp Zceil x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
unfold d.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
(round beta fexp Zfloor x + ulp x)%R =
round beta fexp Zceil x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
now rewrite <- round_UP_DN_ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
apply Rle_trans with (Rabs (d + ulp x - x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (d + ulp x - x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(Rabs (d + ulp x - x) <= / 2 * ulp x)%R
apply Hr2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
F (d + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(Rabs (d + ulp x - x) <= / 2 * ulp x)%R
rewrite Hu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
F (round beta fexp Zceil x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(Rabs (d + ulp x - x) <= / 2 * ulp x)%R
apply (round_UP_pt beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(Rabs (d + ulp x - x) <= / 2 * ulp x)%R
rewrite Rabs_pos_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(d + ulp x - x <= / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(0 <= d + ulp x - x)%R
apply Rmult_le_reg_r with 2%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(0 < 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
((d + ulp x - x) * 2 <= / 2 * ulp x * 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(0 <= d + ulp x - x)%R
now apply IZR_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
((d + ulp x - x) * 2 <= / 2 * ulp x * 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(0 <= d + ulp x - x)%R
apply Rplus_le_reg_r with (- (d + ulp x - x))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
((d + ulp x - x) * 2 + - (d + ulp x - x) <=
 / 2 * ulp x * 2 + - (d + ulp x - x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(0 <= d + ulp x - x)%R
ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(d + ulp x - x <= - d + 2 * ulp x * / 2 - ulp x + x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(0 <= d + ulp x - x)%R
apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(d + ulp x - x < - d + 2 * ulp x * / 2 - ulp x + x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(0 <= d + ulp x - x)%R
apply Rlt_le_trans with (1 := H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(x - d <= - d + 2 * ulp x * / 2 - ulp x + x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(0 <= d + ulp x - x)%R
right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(x - d)%R = (- d + 2 * ulp x * / 2 - ulp x + x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(0 <= d + ulp x - x)%R field.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(0 <= d + ulp x - x)%R
apply Rle_0_minus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(x <= d + ulp x)%R
rewrite Hu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
Hx:~ F x
d:=round beta fexp Zfloor x:R
Hr1:F (round beta fexp (Znearest choice) x)
Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%R
H:(d + ulp x - x < x - d)%R
Hu:(d + ulp x)%R = round beta fexp Zceil x
(x <= round beta fexp Zceil x)%R
apply (round_UP_pt beta fexp x).
Qed.

Theorem ulp_DN :
  forall x, (0 <= x)%R ->
  ulp (round beta fexp Zfloor x) = ulp x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 <= x)%R -> ulp (round beta fexp Zfloor x) = ulp x
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(0 <= x)%R -> ulp (round beta fexp Zfloor x) = ulp x
intros x [Hx|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
ulp (round beta fexp Zfloor x) = ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:0%R = x
ulp (round beta fexp Zfloor x) = ulp x
-beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
ulp (round beta fexp Zfloor x) = ulp x rewrite (ulp_neq_0 x) by now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
ulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)
  destruct (round_ge_generic beta fexp Zfloor 0 x) as [Hd|Hd].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
F 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:(0 < round beta fexp Zfloor x)%R
ulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)
    apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:(0 < round beta fexp Zfloor x)%R
ulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)
    now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:(0 < round beta fexp Zfloor x)%R
ulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)
  +beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:(0 < round beta fexp Zfloor x)%R
ulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x) rewrite ulp_neq_0 by now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:(0 < round beta fexp Zfloor x)%R
bpow (cexp beta fexp (round beta fexp Zfloor x)) =
bpow (cexp beta fexp x)
    now rewrite cexp_DN with (2 := Hd).
  +beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x) rewrite <- Hd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:0%R = round beta fexp Zfloor x
ulp 0 = bpow (cexp beta fexp x)
    unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:0%R = round beta fexp Zfloor x
ulp 0 = bpow (fexp (mag beta x))
    destruct (mag beta x) as [e He].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:0%R = round beta fexp Zfloor x
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
ulp 0 = bpow (fexp (Build_mag_prop beta x e He))
    simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:0%R = round beta fexp Zfloor x
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
ulp 0 = bpow (fexp e)
    specialize (He (Rgt_not_eq _ _ Hx)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:0%R = round beta fexp Zfloor x
e:Z
He:(bpow (e - 1) <= Rabs x < bpow e)%R
ulp 0 = bpow (fexp e)
    apply sym_eq in Hd.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:round beta fexp Zfloor x = 0%R
e:Z
He:(bpow (e - 1) <= Rabs x < bpow e)%R
ulp 0 = bpow (fexp e)
    assert (H := exp_small_round_0 _ _ _ _ _ He Hd).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:round beta fexp Zfloor x = 0%R
e:Z
He:(bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
ulp 0 = bpow (fexp e)
    unfold ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:round beta fexp Zfloor x = 0%R
e:Z
He:(bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
(if Req_bool 0 0
 then
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0%R
  end
 else bpow (cexp beta fexp 0)) = bpow (fexp e)
    rewrite Req_bool_true by easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:round beta fexp Zfloor x = 0%R
e:Z
He:(bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
match negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end = bpow (fexp e)
    destruct negligible_exp_spec as [H0|k Hk].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:round beta fexp Zfloor x = 0%R
e:Z
He:(bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
H0:forall n : Z, (fexp n < n)%Z
0%R = bpow (fexp e)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:round beta fexp Zfloor x = 0%R
e:Z
He:(bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
k:Z
Hk:(k <= fexp k)%Z
bpow (fexp k) = bpow (fexp e)
    now elim Zlt_not_le with (1 := H0 e).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:(0 < x)%R
Hd:round beta fexp Zfloor x = 0%R
e:Z
He:(bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
k:Z
Hk:(k <= fexp k)%Z
bpow (fexp k) = bpow (fexp e)
    now apply f_equal, fexp_negligible_exp_eq.
-beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Hx:0%R = x
ulp (round beta fexp Zfloor x) = ulp x rewrite <- Hx, round_0...
Qed.

Theorem round_neq_0_negligible_exp :
  negligible_exp = None -> forall rnd { Zrnd : Valid_rnd rnd } x,
  (x <> 0)%R -> (round beta fexp rnd x <> 0)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
negligible_exp = None ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, x <> 0%R -> round beta fexp rnd x <> 0%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
negligible_exp = None ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, x <> 0%R -> round beta fexp rnd x <> 0%R
intros H rndn Hrnd x Hx K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
False
case negligible_exp_spec'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
negligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) -> False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False
intros (_,Hn).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
Hn:forall n : Z, (fexp n < n)%Z
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False
destruct (mag beta x) as (e,He).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
Hn:forall n : Z, (fexp n < n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
False
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False
absurd (fexp e < e)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
Hn:forall n : Z, (fexp n < n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
~ (fexp e < e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
Hn:forall n : Z, (fexp n < n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(fexp e < e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False
apply Zle_not_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
Hn:forall n : Z, (fexp n < n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(e <= fexp e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
Hn:forall n : Z, (fexp n < n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(fexp e < e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False
apply exp_small_round_0 with beta rndn x...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
Hn:forall n : Z, (fexp n < n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(fexp e < e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False
apply (Hn e).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False
intros (n,(H1,_)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:negligible_exp = None
rndn:R -> Z
Hrnd:Valid_rnd rndn
x:R
Hx:x <> 0%R
K:round beta fexp rndn x = 0%R
n:Z
H1:negligible_exp = Some n
False
rewrite H in H1; discriminate.
Qed.

allows rnd x to be 0

Theorem error_lt_ulp_round :
  forall { Hm : Monotone_exp fexp } rnd { Zrnd : Valid_rnd rnd } x,
  (x <> 0)%R ->
  (Rabs (round beta fexp rnd x - x) < ulp (round beta fexp rnd x))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
intros Hm.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
(* wlog *)
cut (forall rnd : R -> Z, Valid_rnd rnd -> forall x : R, (0 < x)%R  ->
    (Rabs (round beta fexp rnd x - x) < ulp (round beta fexp rnd x))%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
(forall rnd : R -> Z,
 Valid_rnd rnd ->
 forall x : R,
 (0 < x)%R ->
 (Rabs (round beta fexp rnd x - x) <
  ulp (round beta fexp rnd x))%R) ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
x <> 0%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
intros M rnd Hrnd x Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
case (Rle_or_lt 0 x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
(0 <= x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
(x < 0)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
intros H; destruct H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:(0 < x)%R
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:0%R = x
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
(x < 0)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
now apply M.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:0%R = x
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
(x < 0)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
contradict H; now apply sym_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
(x < 0)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:(x < 0)%R
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
rewrite <- (Ropp_involutive x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:(x < 0)%R
(Rabs (round beta fexp rnd (- - x) - - - x) <
 ulp (round beta fexp rnd (- - x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
rewrite round_opp, ulp_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:(x < 0)%R
(Rabs (- round beta fexp (Zrnd_opp rnd) (- x) - - - x) <
 ulp (round beta fexp (Zrnd_opp rnd) (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
replace (- round beta fexp (Zrnd_opp rnd) (- x) - - - x)%R with
    (-(round beta fexp (Zrnd_opp rnd) (- x) - (-x)))%R by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:(x < 0)%R
(Rabs (- (round beta fexp (Zrnd_opp rnd) (- x) - - x)) <
 ulp (round beta fexp (Zrnd_opp rnd) (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
rewrite Rabs_Ropp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:(x < 0)%R
(Rabs (round beta fexp (Zrnd_opp rnd) (- x) - - x) <
 ulp (round beta fexp (Zrnd_opp rnd) (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
apply M.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:(x < 0)%R
Valid_rnd (Zrnd_opp rnd)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
now apply valid_rnd_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
M:forall rnd0 : R -> Z,
Valid_rnd rnd0 ->
forall x0 : R,
(0 < x0)%R ->
(Rabs (round beta fexp rnd0 x0 - x0) <
 ulp (round beta fexp rnd0 x0))%R
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Zx:x <> 0%R
H:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
now apply Ropp_0_gt_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
(* 0 < x *)
intros rnd Hrnd x Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R
apply Rlt_le_trans with (ulp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(Rabs (round beta fexp rnd x - x) < ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(ulp x <= ulp (round beta fexp rnd x))%R
apply error_lt_ulp...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(ulp x <= ulp (round beta fexp rnd x))%R
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(ulp x <= ulp (round beta fexp rnd x))%R
rewrite <- ulp_DN; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(ulp (round beta fexp Zfloor x) <=
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
apply ulp_le_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(round beta fexp Zfloor x <= round beta fexp rnd x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
apply round_ge_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
F 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(round beta fexp Zfloor x <= round beta fexp rnd x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(round beta fexp Zfloor x <= round beta fexp rnd x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(round beta fexp Zfloor x <= round beta fexp rnd x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
case (round_DN_or_UP beta fexp rnd x); intros V; rewrite V.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
V:round beta fexp rnd x = round beta fexp Zfloor x
(round beta fexp Zfloor x <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
V:round beta fexp rnd x = round beta fexp Zceil x
(round beta fexp Zfloor x <= round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
apply Rle_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
V:round beta fexp rnd x = round beta fexp Zceil x
(round beta fexp Zfloor x <= round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
apply Rle_trans with x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
V:round beta fexp rnd x = round beta fexp Zceil x
(round beta fexp Zfloor x <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
V:round beta fexp rnd x = round beta fexp Zceil x
(x <= round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
apply round_DN_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
V:round beta fexp rnd x = round beta fexp Zceil x
(x <= round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
apply round_UP_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
rnd:R -> Z
Hrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
(0 <= x)%R
now apply Rlt_le.
Qed.

Lemma error_le_ulp_round :
  forall { Hm : Monotone_exp fexp } rnd { Zrnd : Valid_rnd rnd } x,
  (Rabs (round beta fexp rnd x - x) <= ulp (round beta fexp rnd x))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R
intros Mexp rnd Vrnd x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R
destruct (Req_dec x 0) as [Zx|Nzx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Zx:x = 0%R
(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Nzx:x <> 0%R
(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R
{beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Zx:x = 0%R
(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R rewrite Zx, round_0; [|exact Vrnd].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Zx:x = 0%R
(Rabs (0 - 0) <= ulp 0)%R
  unfold Rminus; rewrite Ropp_0, Rplus_0_l, Rabs_R0; apply ulp_ge_0. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Nzx:x <> 0%R
(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R
now apply Rlt_le, error_lt_ulp_round.
Qed.

allows both x and rnd x to be 0

Theorem error_le_half_ulp_round :
  forall { Hm : Monotone_exp fexp },
  forall choice x,
  (Rabs (round beta fexp (Znearest choice) x - x) <= /2 * ulp (round beta fexp (Znearest choice) x))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall (choice : Z -> bool) (x : R),
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall (choice : Z -> bool) (x : R),
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
intros Hm choice x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
case (Req_dec (round beta fexp (Znearest choice) x) 0); intros Hfx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
(* *)
case (Req_dec x 0); intros Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x = 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rle_trans with (1:=error_le_half_ulp _ _).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x = 0%R
(/ 2 * ulp x <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite Hx, round_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x = 0%R
(/ 2 * ulp 0 <= / 2 * ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
right; ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
generalize (error_le_half_ulp choice x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite Hfx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
(Rabs (0 - x) <= / 2 * ulp x)%R ->
(Rabs (0 - x) <= / 2 * ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
unfold Rminus; rewrite Rplus_0_l, Rabs_Ropp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
(Rabs x <= / 2 * ulp x)%R -> (Rabs x <= / 2 * ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
intros N.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
(Rabs x <= / 2 * ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
unfold ulp; rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
(Rabs x <=
 / 2 *
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
case negligible_exp_spec'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
negligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) ->
(Rabs x <=
 / 2 *
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(Rabs x <=
 / 2 *
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
intros (H1,H2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
H1:negligible_exp = None
H2:forall n : Z, (fexp n < n)%Z
(Rabs x <=
 / 2 *
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(Rabs x <=
 / 2 *
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
contradict Hfx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
H1:negligible_exp = None
H2:forall n : Z, (fexp n < n)%Z
round beta fexp (Znearest choice) x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(Rabs x <=
 / 2 *
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply round_neq_0_negligible_exp...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(Rabs x <=
 / 2 *
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
intros (n,(H1,Hn)); rewrite H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
(Rabs x <= / 2 * bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rle_trans with (1:=N).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
(/ 2 * ulp x <= / 2 * bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
right; apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
ulp x = bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite ulp_neq_0; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
bpow (cexp beta fexp x) = bpow (fexp n)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
cexp beta fexp x = fexp n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
fexp (mag beta x) = fexp n
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply valid_exp; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
(mag beta x <= fexp n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
assert (mag beta x -1 < fexp n)%Z;[idtac|omega].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
(mag beta x - 1 < fexp n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply lt_bpow with beta.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
(bpow (mag beta x - 1) < bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
destruct (mag beta x) as (e,He).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (Build_mag_prop beta x e He - 1) < bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (e - 1) < bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rle_lt_trans with (Rabs x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (e - 1) <= Rabs x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(Rabs x < bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
now apply He.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(Rabs x < bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rle_lt_trans with (Rabs (round beta fexp (Znearest choice) x - x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(Rabs x <=
 Rabs (round beta fexp (Znearest choice) x - x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(Rabs (round beta fexp (Znearest choice) x - x) <
 bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
right; rewrite Hfx; unfold Rminus; rewrite Rplus_0_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
Rabs x = Rabs (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(Rabs (round beta fexp (Znearest choice) x - x) <
 bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply sym_eq, Rabs_Ropp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(Rabs (round beta fexp (Znearest choice) x - x) <
 bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rlt_le_trans with (ulp 0).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(Rabs (round beta fexp (Znearest choice) x - x) <
 ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(ulp 0 <= bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite <- Hfx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(Rabs (round beta fexp (Znearest choice) x - x) <
 ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(ulp 0 <= bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply error_lt_ulp_round...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(ulp 0 <= bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
unfold ulp; rewrite Req_bool_true, H1; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x = 0%R
Hx:x <> 0%R
N:(Rabs x <= / 2 * ulp x)%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (fexp n) <= bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
now right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
(* *)
case (round_DN_or_UP beta fexp (Znearest choice) x); intros Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
(* . *)
destruct (Rle_or_lt 0 x) as [H|H].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(0 <= x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite Hx at 2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(0 <= x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp Zfloor x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite ulp_DN by easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(0 <= x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply error_le_half_ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rle_trans with (1:=error_le_half_ulp _ _).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(/ 2 * ulp x <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rmult_le_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(0 <= / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(ulp x <= ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rlt_le, pos_half_prf.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(ulp x <= ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply ulp_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(Rabs x <= Rabs (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite Rabs_left1 by now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(- x <= Rabs (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(- x <= Rabs (round beta fexp Zfloor x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite Rabs_left1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(- x <= - round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(round beta fexp Zfloor x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(round beta fexp Zfloor x <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(round beta fexp Zfloor x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply round_DN_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(round beta fexp Zfloor x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply round_le_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
F 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
H:(x < 0)%R
(x <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
(* . *)
destruct (Rle_or_lt 0 x) as [H|H].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(0 <= x)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rle_trans with (1:=error_le_half_ulp _ _).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(0 <= x)%R
(/ 2 * ulp x <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rmult_le_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(0 <= x)%R
(0 <= / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(0 <= x)%R
(ulp x <= ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply Rlt_le, pos_half_prf.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(0 <= x)%R
(ulp x <= ulp (round beta fexp (Znearest choice) x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
apply ulp_le_pos; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(0 <= x)%R
(x <= round beta fexp (Znearest choice) x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite Hx; apply (round_UP_pt beta fexp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R
rewrite Hx at 2; rewrite <- (ulp_opp (round beta fexp Zceil x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (- round beta fexp Zceil x))%R
rewrite <- round_DN_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp Zfloor (- x)))%R
rewrite ulp_DN; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(0 <= - x)%R
pattern x at 1 2; rewrite <- Ropp_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs
   (round beta fexp (Znearest choice) (- - x) - - - x) <=
 / 2 * ulp (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(0 <= - x)%R
rewrite round_N_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs
   (-
    round beta fexp
      (Znearest
         (fun t : Z => negb (choice (- (t + 1))%Z)))
      (- x) - - - x) <= / 2 * ulp (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(0 <= - x)%R
unfold Rminus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs
   (-
    round beta fexp
      (Znearest
         (fun t : Z => negb (choice (- (t + 1))%Z)))
      (- x) + - - - x) <= / 2 * ulp (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(0 <= - x)%R
rewrite <- Ropp_plus_distr, Rabs_Ropp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(Rabs
   (round beta fexp
      (Znearest
         (fun t : Z => negb (choice (- (t + 1))%Z)))
      (- x) + - - x) <= / 2 * ulp (- x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(0 <= - x)%R
apply error_le_half_ulp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(0 <= - x)%R
rewrite <- Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(- 0 <= - x)%R
apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice:Z -> bool
x:R
Hfx:round beta fexp (Znearest choice) x <> 0%R
Hx:round beta fexp (Znearest choice) x =
round beta fexp Zceil x
H:(x < 0)%R
(x <= 0)%R
now apply Rlt_le.
Qed.

Theorem pred_le :
  forall x y, F x -> F y -> (x <= y)%R ->
  (pred x <= pred y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x <= y)%R -> (pred x <= pred y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x <= y)%R -> (pred x <= pred y)%R
intros x y Fx Fy [Hxy| ->].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(pred x <= pred y)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
y:R
Fx, Fy:F y
(pred y <= pred y)%R
2: apply Rle_refl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(pred x <= pred y)%R
apply pred_ge_gt with (2 := Fy).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
F (pred x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(pred x < y)%R
now apply generic_format_pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(pred x < y)%R
apply Rle_lt_trans with (2 := Hxy).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(pred x <= x)%R
apply pred_le_id.
Qed.

Theorem succ_le :
  forall x y, F x -> F y -> (x <= y)%R ->
  (succ x <= succ y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x <= y)%R -> (succ x <= succ y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x <= y)%R -> (succ x <= succ y)%R
intros x y Fx Fy Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x <= y)%R
(succ x <= succ y)%R
apply Ropp_le_cancel.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x <= y)%R
(- succ y <= - succ x)%R
rewrite <- 2!pred_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x <= y)%R
(pred (- y) <= pred (- x))%R
apply pred_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x <= y)%R
F (- y)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x <= y)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x <= y)%R
(- y <= - x)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x <= y)%R
F (- x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x <= y)%R
(- y <= - x)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x <= y)%R
(- y <= - x)%R
now apply Ropp_le_contravar.
Qed.

Theorem pred_le_inv: forall x y, F x -> F y
   -> (pred x <= pred y)%R -> (x <= y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (pred x <= pred y)%R -> (x <= y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (pred x <= pred y)%R -> (x <= y)%R
intros x y Fx Fy Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(pred x <= pred y)%R
(x <= y)%R
rewrite <- (succ_pred x), <- (succ_pred y); try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(pred x <= pred y)%R
(succ (pred x) <= succ (pred y))%R
apply succ_le; trivial; now apply generic_format_pred.
Qed.

Theorem succ_le_inv: forall x y, F x -> F y
   -> (succ x <= succ y)%R -> (x <= y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (succ x <= succ y)%R -> (x <= y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (succ x <= succ y)%R -> (x <= y)%R
intros x y Fx Fy Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(succ x <= succ y)%R
(x <= y)%R
rewrite <- (pred_succ x), <- (pred_succ y); try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(succ x <= succ y)%R
(pred (succ x) <= pred (succ y))%R
apply pred_le; trivial; now apply generic_format_succ.
Qed.

Theorem pred_lt :
  forall x y, F x -> F y -> (x < y)%R ->
  (pred x < pred y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x < y)%R -> (pred x < pred y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x < y)%R -> (pred x < pred y)%R
intros x y Fx Fy Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(pred x < pred y)%R
apply Rnot_le_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
~ (pred y <= pred x)%R
intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
H:(pred y <= pred x)%R
False
apply Rgt_not_le with (1 := Hxy).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
H:(pred y <= pred x)%R
(y <= x)%R
now apply pred_le_inv.
Qed.

Theorem succ_lt :
  forall x y, F x -> F y -> (x < y)%R ->
  (succ x < succ y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x < y)%R -> (succ x < succ y)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F x -> F y -> (x < y)%R -> (succ x < succ y)%R
intros x y Fx Fy Hxy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
(succ x < succ y)%R
apply Rnot_le_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
~ (succ y <= succ x)%R
intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
H:(succ y <= succ x)%R
False
apply Rgt_not_le with (1 := Hxy).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fx:F x
Fy:F y
Hxy:(x < y)%R
H:(succ y <= succ x)%R
(y <= x)%R
now apply succ_le_inv.
Qed.

Adding ulp is a, somewhat reasonable, overapproximation of succ.

Lemma succ_le_plus_ulp :
  forall { Hm : Monotone_exp fexp } x,
  (succ x <= x + ulp x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall x : R, (succ x <= x + ulp x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall x : R, (succ x <= x + ulp x)%R
intros Mexp x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
(succ x <= x + ulp x)%R
destruct (Rle_or_lt 0 x) as [Px|Nx]; [now right; apply succ_eq_pos|].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
(succ x <= x + ulp x)%R
replace (_ + _)%R with (- (-x - ulp x))%R by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
(succ x <= - (- x - ulp x))%R
unfold succ; rewrite (Rle_bool_false _ _ Nx), <-ulp_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
(- pred_pos (- x) <= - (- x - ulp (- x)))%R
apply Ropp_le_contravar; unfold pred_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
(- x - ulp (- x) <=
 (if Req_bool (- x) (bpow (mag beta (- x) - 1))
  then - x - bpow (fexp (mag beta (- x) - 1))
  else - x - ulp (- x)))%R
destruct (Req_dec (-x) (bpow (mag beta (-x) - 1))) as [Hx|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
(- x - ulp (- x) <=
 (if Req_bool (- x) (bpow (mag beta (- x) - 1))
  then - x - bpow (fexp (mag beta (- x) - 1))
  else - x - ulp (- x)))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
Hx:(- x)%R <> bpow (mag beta (- x) - 1)
(- x - ulp (- x) <=
 (if Req_bool (- x) (bpow (mag beta (- x) - 1))
  then - x - bpow (fexp (mag beta (- x) - 1))
  else - x - ulp (- x)))%R
{beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
(- x - ulp (- x) <=
 (if Req_bool (- x) (bpow (mag beta (- x) - 1))
  then - x - bpow (fexp (mag beta (- x) - 1))
  else - x - ulp (- x)))%R rewrite (Req_bool_true _ _ Hx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
(- x - ulp (- x) <=
 - x - bpow (fexp (mag beta (- x) - 1)))%R
   apply (Rplus_le_reg_r x); ring_simplify; apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
(bpow (fexp (mag beta (- x) - 1)) <= ulp (- x))%R
   unfold ulp; rewrite Req_bool_false; [|lra].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
(bpow (fexp (mag beta (- x) - 1)) <=
 bpow (cexp beta fexp (- x)))%R
   apply bpow_le, Mexp; lia. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Nx:(x < 0)%R
Hx:(- x)%R <> bpow (mag beta (- x) - 1)
(- x - ulp (- x) <=
 (if Req_bool (- x) (bpow (mag beta (- x) - 1))
  then - x - bpow (fexp (mag beta (- x) - 1))
  else - x - ulp (- x)))%R
 now rewrite (Req_bool_false _ _ Hx); right.
Qed.

And it also lies in the format.

Lemma generic_format_plus_ulp :
  forall { Hm : Monotone_exp fexp } x,
  generic_format beta fexp x ->
  generic_format beta fexp (x + ulp x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall x : R, F x -> F (x + ulp x)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall x : R, F x -> F (x + ulp x)
intros Mexp x Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F x
F (x + ulp x)
destruct (Rle_or_lt 0 x) as [Px|Nx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F x
Px:(0 <= x)%R
F (x + ulp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F x
Nx:(x < 0)%R
F (x + ulp x)
{beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F x
Px:(0 <= x)%R
F (x + ulp x) now rewrite <-(succ_eq_pos _ Px); apply generic_format_succ. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F x
Nx:(x < 0)%R
F (x + ulp x)
apply generic_format_opp in Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
F (x + ulp x)
replace (_ + _)%R with (- (-x - ulp x))%R by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
F (- (- x - ulp x))
apply generic_format_opp; rewrite <-ulp_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
F (- x - ulp (- x))
destruct (Req_dec (-x) (bpow (mag beta (-x) - 1))) as [Hx|Hx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
F (- x - ulp (- x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R <> bpow (mag beta (- x) - 1)
F (- x - ulp (- x))
{beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
F (- x - ulp (- x)) unfold ulp; rewrite Req_bool_false; [|lra].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
F (- x - bpow (cexp beta fexp (- x)))
  rewrite Hx at 1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
F
  (bpow (mag beta (- x) - 1) -
   bpow (cexp beta fexp (- x)))
  unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
F
  (bpow (mag beta (- x) - 1) -
   bpow (fexp (mag beta (- x))))
  set (e := mag _ _).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
F (bpow (e - 1) - bpow (fexp e))
  assert (Hfe : (fexp e < e)%Z).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
(fexp e < e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
F (bpow (e - 1) - bpow (fexp e))
  {beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
(fexp e < e)%Z now apply mag_generic_gt; [|lra|]. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
F (bpow (e - 1) - bpow (fexp e))
  replace (e - 1)%Z with (e - 1 - fexp e + fexp e)%Z by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
F (bpow (e - 1 - fexp e + fexp e) - bpow (fexp e))
  rewrite bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
F
  (bpow (e - 1 - fexp e) * bpow (fexp e) -
   bpow (fexp e))
  set (m := bpow (_ - _)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
F (m * bpow (fexp e) - bpow (fexp e))
  replace (_ - _)%R with ((m - 1) * bpow (fexp e))%R; [|unfold m; ring].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
F ((m - 1) * bpow (fexp e))
  case_eq (e - 1 - fexp e)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
(e - 1 - fexp e)%Z = 0%Z ->
F ((m - 1) * bpow (fexp e))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
forall p : positive,
(e - 1 - fexp e)%Z = Z.pos p ->
F ((m - 1) * bpow (fexp e))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
forall p : positive,
(e - 1 - fexp e)%Z = Z.neg p ->
F ((m - 1) * bpow (fexp e))
  {beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
(e - 1 - fexp e)%Z = 0%Z ->
F ((m - 1) * bpow (fexp e)) intro He; unfold m; rewrite He; simpl; ring_simplify (1 - 1)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
He:(e - 1 - fexp e)%Z = 0%Z
F (0 * bpow (fexp e))
    rewrite Rmult_0_l; apply generic_format_0. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
forall p : positive,
(e - 1 - fexp e)%Z = Z.pos p ->
F ((m - 1) * bpow (fexp e))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
forall p : positive,
(e - 1 - fexp e)%Z = Z.neg p ->
F ((m - 1) * bpow (fexp e))
  {beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
forall p : positive,
(e - 1 - fexp e)%Z = Z.pos p ->
F ((m - 1) * bpow (fexp e)) intros p Hp; unfold m; rewrite Hp; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
F ((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))
    pose (f := {| Defs.Fnum := (Z.pow_pos beta p - 1)%Z;
                  Defs.Fexp := fexp e |} : Defs.float beta).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
F ((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))
    apply (generic_format_F2R' _ _ _ f); [|intro Hm'; unfold f; simpl].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
F2R f =
((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
(cexp beta fexp
   ((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e)) <=
 fexp e)%Z
    {beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
F2R f =
((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R now unfold Defs.F2R; simpl; rewrite minus_IZR. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
(cexp beta fexp
   ((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e)) <=
 fexp e)%Z
    unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
(fexp
   (mag beta
      ((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))) <=
 fexp e)%Z
    replace (IZR _) with (bpow (Z.pos p)); [|now simpl].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
(fexp
   (mag beta ((bpow (Z.pos p) - 1) * bpow (fexp e))) <=
 fexp e)%Z
    rewrite <-Hp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
(fexp
   (mag beta
      ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))) <=
 fexp e)%Z
    assert (He : (1 <= e - 1 - fexp e)%Z); [lia|].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
(fexp
   (mag beta
      ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))) <=
 fexp e)%Z
    set (e' := mag _ (_ * _)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
(fexp e' <= fexp e)%Z
    assert (H : (e' = e - 1 :> Z)%Z); [|rewrite H; apply Mexp; lia].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
e' = (e - 1)%Z
    unfold e'; apply mag_unique.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
(bpow (e - 1 - 1) <=
 Rabs ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)) <
 bpow (e - 1))%R
    rewrite Rabs_mult, (Rabs_pos_eq (bpow _)); [|apply bpow_ge_0].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
(bpow (e - 1 - 1) <=
 Rabs (bpow (e - 1 - fexp e) - 1) * bpow (fexp e) <
 bpow (e - 1))%R
    rewrite Rabs_pos_eq;
      [|apply (Rplus_le_reg_r 1); ring_simplify;
        change 1%R with (bpow 0); apply bpow_le; lia].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
(bpow (e - 1 - 1) <=
 (bpow (e - 1 - fexp e) - 1) * bpow (fexp e) <
 bpow (e - 1))%R
    assert (beta_pos : (0 < IZR beta)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
(0 < IZR beta)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - 1) <=
 (bpow (e - 1 - fexp e) - 1) * bpow (fexp e) <
 bpow (e - 1))%R
    {beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
(0 < IZR beta)%R apply (Rlt_le_trans _ 2); [lra|].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
(2 <= IZR beta)%R
      apply IZR_le, Z.leb_le, radix_prop. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - 1) <=
 (bpow (e - 1 - fexp e) - 1) * bpow (fexp e) <
 bpow (e - 1))%R
    split.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - 1) <=
 (bpow (e - 1 - fexp e) - 1) * bpow (fexp e))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
((bpow (e - 1 - fexp e) - 1) * bpow (fexp e) <
 bpow (e - 1))%R
    {beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - 1) <=
 (bpow (e - 1 - fexp e) - 1) * bpow (fexp e))%R replace (e - 1 - 1)%Z with (e - 1 - fexp e + -1  + fexp e)%Z by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e + -1 + fexp e) <=
 (bpow (e - 1 - fexp e) - 1) * bpow (fexp e))%R
      rewrite bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e + -1) * bpow (fexp e) <=
 (bpow (e - 1 - fexp e) - 1) * bpow (fexp e))%R
      apply Rmult_le_compat_r; [apply bpow_ge_0|].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e + -1) <=
 bpow (e - 1 - fexp e) - 1)%R
      rewrite bpow_plus; simpl; unfold Z.pow_pos; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e) * / IZR (beta * 1) <=
 bpow (e - 1 - fexp e) - 1)%R
      rewrite Zmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e) * / IZR beta <=
 bpow (e - 1 - fexp e) - 1)%R
      apply (Rmult_le_reg_r _ _ _ beta_pos).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e) * / IZR beta * IZR beta <=
 (bpow (e - 1 - fexp e) - 1) * IZR beta)%R
      rewrite Rmult_assoc, Rinv_l; [|lra]; rewrite Rmult_1_r.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e) <=
 (bpow (e - 1 - fexp e) - 1) * IZR beta)%R
      apply (Rplus_le_reg_r (IZR beta)); ring_simplify.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e) + IZR beta <=
 bpow (e - 1 - fexp e) * IZR beta)%R
      apply (Rle_trans _ (2 * bpow (e - 1 - fexp e))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e) + IZR beta <=
 2 * bpow (e - 1 - fexp e))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(2 * bpow (e - 1 - fexp e) <=
 bpow (e - 1 - fexp e) * IZR beta)%R
      {beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e) + IZR beta <=
 2 * bpow (e - 1 - fexp e))%R change 2%R with (1 + 1)%R; rewrite Rmult_plus_distr_r, Rmult_1_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(bpow (e - 1 - fexp e) + IZR beta <=
 bpow (e - 1 - fexp e) + bpow (e - 1 - fexp e))%R
        apply Rplus_le_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(IZR beta <= bpow (e - 1 - fexp e))%R
        rewrite <-bpow_1; apply bpow_le; lia. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(2 * bpow (e - 1 - fexp e) <=
 bpow (e - 1 - fexp e) * IZR beta)%R
      rewrite Rmult_comm; apply Rmult_le_compat_l; [apply bpow_ge_0|].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
(2 <= IZR beta)%R
      apply IZR_le, Z.leb_le, radix_prop. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
((bpow (e - 1 - fexp e) - 1) * bpow (fexp e) <
 bpow (e - 1))%R
    apply (Rmult_lt_reg_r (bpow (- fexp e))); [apply bpow_gt_0|].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
((bpow (e - 1 - fexp e) - 1) * bpow (fexp e) *
 bpow (- fexp e) < bpow (e - 1) * bpow (- fexp e))%R
    rewrite Rmult_assoc, <-!bpow_plus.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
((bpow (e - 1 - fexp e) - 1) *
 bpow (fexp e + - fexp e) < bpow (e - 1 + - fexp e))%R
    replace (fexp e + - fexp e)%Z with 0%Z by ring; simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
p:positive
Hp:(e - 1 - fexp e)%Z = Z.pos p
f:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float beta
Hm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%R
He:(1 <= e - 1 - fexp e)%Z
e':=mag beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e)):mag_prop beta
  ((bpow (e - 1 - fexp e) - 1) * bpow (fexp e))
beta_pos:(0 < IZR beta)%R
((bpow (e - 1 - fexp e) - 1) * 1 <
 bpow (e - 1 + - fexp e))%R
    rewrite Rmult_1_r; unfold Zminus; lra. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R = bpow (mag beta (- x) - 1)
e:=mag beta (- x):mag_prop beta (- x)
Hfe:(fexp e < e)%Z
m:=bpow (e - 1 - fexp e):R
forall p : positive,
(e - 1 - fexp e)%Z = Z.neg p ->
F ((m - 1) * bpow (fexp e))
  intros p Hp; exfalso; lia. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R <> bpow (mag beta (- x) - 1)
F (- x - ulp (- x))
replace (_ - _)%R with (pred_pos (-x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R <> bpow (mag beta (- x) - 1)
F (pred_pos (- x))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R <> bpow (mag beta (- x) - 1)
pred_pos (- x) = (- x - ulp (- x))%R
{beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R <> bpow (mag beta (- x) - 1)
F (pred_pos (- x)) now apply generic_format_pred_pos; [|lra]. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Mexp:Monotone_exp fexp
x:R
Fx:F (- x)
Nx:(x < 0)%R
Hx:(- x)%R <> bpow (mag beta (- x) - 1)
pred_pos (- x) = (- x - ulp (- x))%R
now unfold pred_pos; rewrite Req_bool_false.
Qed.

Theorem round_DN_ge_UP_gt :
  forall x y, F y ->
  (y < round beta fexp Zceil x -> y <= round beta fexp Zfloor x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F y ->
(y < round beta fexp Zceil x)%R ->
(y <= round beta fexp Zfloor x)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F y ->
(y < round beta fexp Zceil x)%R ->
(y <= round beta fexp Zfloor x)%R
intros x y Fy Hlt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(y < round beta fexp Zceil x)%R
(y <= round beta fexp Zfloor x)%R
apply round_DN_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(y < round beta fexp Zceil x)%R
(y <= x)%R
apply Rnot_lt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(y < round beta fexp Zceil x)%R
~ (x < y)%R
contradict Hlt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(x < y)%R
~ (y < round beta fexp Zceil x)%R
apply RIneq.Rle_not_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(x < y)%R
(round beta fexp Zceil x <= y)%R
apply round_UP_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(x < y)%R
(x <= y)%R
now apply Rlt_le.
Qed.

Theorem round_UP_le_DN_lt :
  forall x y, F y ->
  (round beta fexp Zfloor x < y -> round beta fexp Zceil x <= y)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F y ->
(round beta fexp Zfloor x < y)%R ->
(round beta fexp Zceil x <= y)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x y : R,
F y ->
(round beta fexp Zfloor x < y)%R ->
(round beta fexp Zceil x <= y)%R
intros x y Fy Hlt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(round beta fexp Zfloor x < y)%R
(round beta fexp Zceil x <= y)%R
apply round_UP_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(round beta fexp Zfloor x < y)%R
(x <= y)%R
apply Rnot_lt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(round beta fexp Zfloor x < y)%R
~ (y < x)%R
contradict Hlt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(y < x)%R
~ (round beta fexp Zfloor x < y)%R
apply RIneq.Rle_not_lt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(y < x)%R
(y <= round beta fexp Zfloor x)%R
apply round_DN_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, y:R
Fy:F y
Hlt:(y < x)%R
(y <= x)%R
now apply Rlt_le.
Qed.

Theorem pred_UP_le_DN :
  forall x, (pred (round beta fexp Zceil x) <= round beta fexp Zfloor x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
intros x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
destruct (generic_format_EM beta fexp x) as [Fx|Fx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
rewrite !round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
(pred x <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
apply pred_le_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
case (Req_dec (round beta fexp Zceil x) 0); intros Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
rewrite Zx; unfold pred; rewrite Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
(- succ 0 <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
unfold succ; rewrite Rle_bool_true;[idtac|now right].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
(- (0 + ulp 0) <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
rewrite Rplus_0_l; unfold ulp; rewrite Req_bool_true; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
(-
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
case negligible_exp_spec'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
negligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) ->
(-
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(-
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
intros (H1,H2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
H1:negligible_exp = None
H2:forall n : Z, (fexp n < n)%Z
(-
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(-
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
contradict Zx; apply round_neq_0_negligible_exp...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
H1:negligible_exp = None
H2:forall n : Z, (fexp n < n)%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(-
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
intros L; apply Fx; rewrite L; apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(-
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
intros (n,(H1,Hn)); rewrite H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
(- bpow (fexp n) <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
case (Rle_or_lt (- bpow (fexp n)) (round beta fexp Zfloor x)); trivial; intros K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
K:(round beta fexp Zfloor x < - bpow (fexp n))%R
(- bpow (fexp n) <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
absurd (round beta fexp Zceil x <= - bpow (fexp n))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
K:(round beta fexp Zfloor x < - bpow (fexp n))%R
~ (round beta fexp Zceil x <= - bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
K:(round beta fexp Zfloor x < - bpow (fexp n))%R
(round beta fexp Zceil x <= - bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
apply Rlt_not_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
K:(round beta fexp Zfloor x < - bpow (fexp n))%R
(- bpow (fexp n) < round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
K:(round beta fexp Zfloor x < - bpow (fexp n))%R
(round beta fexp Zceil x <= - bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
rewrite Zx, <- Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
K:(round beta fexp Zfloor x < - bpow (fexp n))%R
(- bpow (fexp n) < - 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
K:(round beta fexp Zfloor x < - bpow (fexp n))%R
(round beta fexp Zceil x <= - bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
apply Ropp_lt_contravar, bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
K:(round beta fexp Zfloor x < - bpow (fexp n))%R
(round beta fexp Zceil x <= - bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
apply round_UP_le_DN_lt; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
K:(round beta fexp Zfloor x < - bpow (fexp n))%R
F (- bpow (fexp n))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
apply generic_format_opp, generic_format_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x = 0%R
n:Z
H1:negligible_exp = Some n
Hn:(n <= fexp n)%Z
K:(round beta fexp Zfloor x < - bpow (fexp n))%R
(fexp (fexp n + 1) <= fexp n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
now apply valid_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
assert (let u := round beta fexp Zceil x in pred u < u)%R as Hup.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
let u := round beta fexp Zceil x in (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
Hup:let u := round beta fexp Zceil x in
(pred u < u)%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
now apply pred_lt_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
Hup:let u := round beta fexp Zceil x in
(pred u < u)%R
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
apply round_DN_ge_UP_gt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
Hup:let u := round beta fexp Zceil x in
(pred u < u)%R
F (pred (round beta fexp Zceil x))
apply generic_format_pred...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
Zx:round beta fexp Zceil x <> 0%R
Hup:let u := round beta fexp Zceil x in
(pred u < u)%R
F (round beta fexp Zceil x)
now apply round_UP_pt.
Qed.

Theorem UP_le_succ_DN :
  forall x, (round beta fexp Zceil x <= succ (round beta fexp Zfloor x))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(round beta fexp Zceil x <=
 succ (round beta fexp Zfloor x))%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
(round beta fexp Zceil x <=
 succ (round beta fexp Zfloor x))%R
intros x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
(round beta fexp Zceil x <=
 succ (round beta fexp Zfloor x))%R
rewrite <- (Ropp_involutive x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
(round beta fexp Zceil (- - x) <=
 succ (round beta fexp Zfloor (- - x)))%R
rewrite round_DN_opp, round_UP_opp, succ_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
(- round beta fexp Zfloor (- x) <=
 - pred (round beta fexp Zceil (- x)))%R
apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
(pred (round beta fexp Zceil (- x)) <=
 round beta fexp Zfloor (- x))%R
apply pred_UP_le_DN.
Qed.

Theorem pred_UP_eq_DN :
  forall x,  ~ F x ->
  (pred (round beta fexp Zceil x) = round beta fexp Zfloor x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
~ F x ->
pred (round beta fexp Zceil x) =
round beta fexp Zfloor x
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
~ F x ->
pred (round beta fexp Zceil x) =
round beta fexp Zfloor x
intros x Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
pred (round beta fexp Zceil x) =
round beta fexp Zfloor x
apply Rle_antisym.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
(round beta fexp Zfloor x <=
 pred (round beta fexp Zceil x))%R
now apply pred_UP_le_DN.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
(round beta fexp Zfloor x <=
 pred (round beta fexp Zceil x))%R
apply pred_ge_gt; try apply generic_format_round...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
(round beta fexp Zfloor x < round beta fexp Zceil x)%R
pose proof round_DN_UP_lt _ _ _ Fx as HE.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
HE:(round beta fexp Zfloor x < x <
 round beta fexp Zceil x)%R
(round beta fexp Zfloor x < round beta fexp Zceil x)%R
now apply Rlt_trans with (1 := proj1 HE) (2 := proj2 HE).
Qed.

Theorem succ_DN_eq_UP :
  forall x,  ~ F x ->
  (succ (round beta fexp Zfloor x) = round beta fexp Zceil x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
~ F x ->
succ (round beta fexp Zfloor x) =
round beta fexp Zceil x
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
~ F x ->
succ (round beta fexp Zfloor x) =
round beta fexp Zceil x
intros x Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
succ (round beta fexp Zfloor x) =
round beta fexp Zceil x
rewrite <- pred_UP_eq_DN; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
succ (pred (round beta fexp Zceil x)) =
round beta fexp Zceil x
rewrite succ_pred; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:~ F x
F (round beta fexp Zceil x)
apply generic_format_round...
Qed.

Theorem round_DN_eq :
  forall x d, F d -> (d <= x < succ d)%R ->
  round beta fexp Zfloor x = d.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x d : R,
F d ->
(d <= x < succ d)%R -> round beta fexp Zfloor x = d
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x d : R,
F d ->
(d <= x < succ d)%R -> round beta fexp Zfloor x = d
intros x d Fd (Hxd1,Hxd2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
round beta fexp Zfloor x = d
generalize (round_DN_pt beta fexp x); intros (T1,(T2,T3)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
round beta fexp Zfloor x = d
apply sym_eq, Rle_antisym.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
(d <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
(round beta fexp Zfloor x <= d)%R
now apply T3.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
(round beta fexp Zfloor x <= d)%R
destruct (generic_format_EM beta fexp x) as [Fx|NFx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
Fx:F x
(round beta fexp Zfloor x <= d)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
NFx:~ F x
(round beta fexp Zfloor x <= d)%R
rewrite round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
Fx:F x
(x <= d)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
NFx:~ F x
(round beta fexp Zfloor x <= d)%R
apply succ_le_inv; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
Fx:F x
(succ x <= succ d)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
NFx:~ F x
(round beta fexp Zfloor x <= d)%R
apply succ_le_lt; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
Fx:F x
F (succ d)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
NFx:~ F x
(round beta fexp Zfloor x <= d)%R
apply generic_format_succ...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
NFx:~ F x
(round beta fexp Zfloor x <= d)%R
apply succ_le_inv; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
NFx:~ F x
(succ (round beta fexp Zfloor x) <= succ d)%R
rewrite succ_DN_eq_UP; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
NFx:~ F x
(round beta fexp Zceil x <= succ d)%R
apply round_UP_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
NFx:~ F x
F (succ d)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
NFx:~ F x
(x <= succ d)%R
apply generic_format_succ...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, d:R
Fd:F d
Hxd1:(d <= x)%R
Hxd2:(x < succ d)%R
T1:F (round beta fexp Zfloor x)
T2:(round beta fexp Zfloor x <= x)%R
T3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R
NFx:~ F x
(x <= succ d)%R
now left.
Qed.

Theorem round_UP_eq :
  forall x u, F u -> (pred u < x <= u)%R ->
  round beta fexp Zceil x = u.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x u : R,
F u ->
(pred u < x <= u)%R -> round beta fexp Zceil x = u
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x u : R,
F u ->
(pred u < x <= u)%R -> round beta fexp Zceil x = u
intros x u Fu Hux.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, u:R
Fu:F u
Hux:(pred u < x <= u)%R
round beta fexp Zceil x = u
rewrite <- (Ropp_involutive (round beta fexp Zceil x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, u:R
Fu:F u
Hux:(pred u < x <= u)%R
(- - round beta fexp Zceil x)%R = u
rewrite <- round_DN_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, u:R
Fu:F u
Hux:(pred u < x <= u)%R
(- round beta fexp Zfloor (- x))%R = u
rewrite <- (Ropp_involutive u).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, u:R
Fu:F u
Hux:(pred u < x <= u)%R
(- round beta fexp Zfloor (- x))%R = (- - u)%R
apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, u:R
Fu:F u
Hux:(pred u < x <= u)%R
round beta fexp Zfloor (- x) = (- u)%R
apply round_DN_eq; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, u:R
Fu:F u
Hux:(pred u < x <= u)%R
F (- u)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, u:R
Fu:F u
Hux:(pred u < x <= u)%R
(- u <= - x < succ (- u))%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, u:R
Fu:F u
Hux:(pred u < x <= u)%R
(- u <= - x < succ (- u))%R
split;[now apply Ropp_le_contravar|idtac].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, u:R
Fu:F u
Hux:(pred u < x <= u)%R
(- x < succ (- u))%R
rewrite succ_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x, u:R
Fu:F u
Hux:(pred u < x <= u)%R
(- x < - pred u)%R
now apply Ropp_lt_contravar.
Qed.

Lemma ulp_ulp_0 : forall {H : Exp_not_FTZ fexp},
  ulp (ulp 0) = ulp 0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Exp_not_FTZ fexp -> ulp (ulp 0) = ulp 0
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Exp_not_FTZ fexp -> ulp (ulp 0) = ulp 0
intros H; case (negligible_exp_spec').beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
negligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) -> ulp (ulp 0) = ulp 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp (ulp 0) = ulp 0
intros (K1,K2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
K1:negligible_exp = None
K2:forall n : Z, (fexp n < n)%Z
ulp (ulp 0) = ulp 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp (ulp 0) = ulp 0
replace (ulp 0) with 0%R at 1; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
K1:negligible_exp = None
K2:forall n : Z, (fexp n < n)%Z
0%R = ulp 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp (ulp 0) = ulp 0
apply sym_eq; unfold ulp; rewrite Req_bool_true; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
K1:negligible_exp = None
K2:forall n : Z, (fexp n < n)%Z
match negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp (ulp 0) = ulp 0
now rewrite K1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp (ulp 0) = ulp 0
intros (n,(Hn1,Hn2)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
ulp (ulp 0) = ulp 0
apply Rle_antisym.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(ulp (ulp 0) <= ulp 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(ulp 0 <= ulp (ulp 0))%R
replace (ulp 0) with (bpow (fexp n)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(ulp (bpow (fexp n)) <= bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) = ulp 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(ulp 0 <= ulp (ulp 0))%R
rewrite ulp_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(bpow (fexp (fexp n + 1)) <= bpow (fexp n))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) = ulp 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(ulp 0 <= ulp (ulp 0))%R
apply bpow_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(fexp (fexp n + 1) <= fexp n)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) = ulp 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(ulp 0 <= ulp (ulp 0))%R
now apply valid_exp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) = ulp 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(ulp 0 <= ulp (ulp 0))%R
unfold ulp; rewrite Req_bool_true; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) =
match negligible_exp with
| Some n0 => bpow (fexp n0)
| None => 0%R
end
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(ulp 0 <= ulp (ulp 0))%R
rewrite Hn1; easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
H:Exp_not_FTZ fexp
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
(ulp 0 <= ulp (ulp 0))%R
now apply ulp_ge_ulp_0.
Qed.

Lemma ulp_succ_pos :
  forall x, F x -> (0 < x)%R ->
  ulp (succ x) = ulp x \/ succ x = bpow (mag beta x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
(0 < x)%R ->
ulp (succ x) = ulp x \/ succ x = bpow (mag beta x)
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
(0 < x)%R ->
ulp (succ x) = ulp x \/ succ x = bpow (mag beta x)
intros x Fx Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
ulp (succ x) = ulp x \/ succ x = bpow (mag beta x)
generalize (Rlt_le _ _ Hx); intros Hx'.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
ulp (succ x) = ulp x \/ succ x = bpow (mag beta x)
rewrite succ_eq_pos;[idtac|now left].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
ulp (x + ulp x) = ulp x \/
(x + ulp x)%R = bpow (mag beta x)
destruct (mag beta x) as (e,He); simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
ulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e
rewrite Rabs_pos_eq in He; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= x < bpow e)%R
ulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e
specialize (He (Rgt_not_eq _ _ Hx)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
ulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e
assert (H:(x+ulp x <= bpow e)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
(x + ulp x <= bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x <= bpow e)%R
ulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e
apply id_p_ulp_le_bpow; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
(x < bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x <= bpow e)%R
ulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e
apply He.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x <= bpow e)%R
ulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e
destruct H;[left|now right].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
ulp (x + ulp x) = ulp x
rewrite ulp_neq_0 at 1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
bpow (cexp beta fexp (x + ulp x)) = ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
(x + ulp x)%R <> 0%R
2: apply Rgt_not_eq, Rgt_lt, Rlt_le_trans with x...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
bpow (cexp beta fexp (x + ulp x)) = ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
(x <= x + ulp x)%R
2: rewrite <- (Rplus_0_r x) at 1; apply Rplus_le_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
bpow (cexp beta fexp (x + ulp x)) = ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
(0 <= ulp x)%R
2: apply ulp_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
bpow (cexp beta fexp (x + ulp x)) = ulp x
rewrite ulp_neq_0 at 2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
bpow (cexp beta fexp (x + ulp x)) =
bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
x <> 0%R
2: now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
bpow (cexp beta fexp (x + ulp x)) =
bpow (cexp beta fexp x)
f_equal; unfold cexp; f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
mag beta (x + ulp x) = mag beta x
apply trans_eq with e.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
mag beta (x + ulp x) = e
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
e = mag beta x
apply mag_unique_pos; split; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
(bpow (e - 1) <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
e = mag beta x
apply Rle_trans with (1:=proj1 He).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
(x <= x + ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
e = mag beta x
rewrite <- (Rplus_0_r x) at 1; apply Rplus_le_compat_l.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
(0 <= ulp x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
e = mag beta x
apply ulp_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < x)%R
Hx':(0 <= x)%R
e:Z
He:(bpow (e - 1) <= x < bpow e)%R
H:(x + ulp x < bpow e)%R
e = mag beta x
now apply sym_eq, mag_unique_pos.
Qed.

Theorem ulp_pred_pos :
  forall x, F x -> (0 < pred x)%R ->
  ulp (pred x) = ulp x \/ x = bpow (mag beta x - 1).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
(0 < pred x)%R ->
ulp (pred x) = ulp x \/ x = bpow (mag beta x - 1)
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall x : R,
F x ->
(0 < pred x)%R ->
ulp (pred x) = ulp x \/ x = bpow (mag beta x - 1)
intros x Fx Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
ulp (pred x) = ulp x \/ x = bpow (mag beta x - 1)
assert (Hx': (0 < x)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
(0 < x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
ulp (pred x) = ulp x \/ x = bpow (mag beta x - 1)
  apply Rlt_le_trans with (1 := Hx).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
(pred x <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
ulp (pred x) = ulp x \/ x = bpow (mag beta x - 1)
  apply pred_le_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
ulp (pred x) = ulp x \/ x = bpow (mag beta x - 1)
assert (Zx : x <> 0%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
ulp (pred x) = ulp x \/ x = bpow (mag beta x - 1)
  now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
ulp (pred x) = ulp x \/ x = bpow (mag beta x - 1)
rewrite (ulp_neq_0 x) by easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
ulp (pred x) = bpow (cexp beta fexp x) \/
x = bpow (mag beta x - 1)
unfold cexp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
ulp (pred x) = bpow (fexp (mag beta x)) \/
x = bpow (mag beta x - 1)
destruct (mag beta x) as [e He].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
ulp (pred x) =
bpow (fexp (Build_mag_prop beta x e He)) \/
x = bpow (Build_mag_prop beta x e He - 1)
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
ulp (pred x) = bpow (fexp e) \/ x = bpow (e - 1)
assert (bpow (e - 1) <= x < bpow e)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (e - 1) <= x < bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
ulp (pred x) = bpow (fexp e) \/ x = bpow (e - 1)
  rewrite <- (Rabs_pos_eq x) by now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (e - 1) <= Rabs x < bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
ulp (pred x) = bpow (fexp e) \/ x = bpow (e - 1)
  now apply He.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
ulp (pred x) = bpow (fexp e) \/ x = bpow (e - 1)
destruct (proj1 H) as [H1|H1].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
ulp (pred x) = bpow (fexp e) \/ x = bpow (e - 1)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:bpow (e - 1) = x
ulp (pred x) = bpow (fexp e) \/ x = bpow (e - 1)
2: now right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
ulp (pred x) = bpow (fexp e) \/ x = bpow (e - 1)
left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
ulp (pred x) = bpow (fexp e)
apply pred_ge_gt with (2 := Fx) in H1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) <= pred x)%R
ulp (pred x) = bpow (fexp e)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
F (bpow (e - 1))
rewrite ulp_neq_0 by now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) <= pred x)%R
bpow (cexp beta fexp (pred x)) = bpow (fexp e)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
F (bpow (e - 1))
apply (f_equal (fun e => bpow (fexp e))).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) <= pred x)%R
mag beta (pred x) = e
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
F (bpow (e - 1))
apply mag_unique_pos.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) <= pred x)%R
(bpow (e - 1) <= pred x < bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
F (bpow (e - 1))
apply (conj H1).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) <= pred x)%R
(pred x < bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
F (bpow (e - 1))
apply Rle_lt_trans with (2 := proj2 H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) <= pred x)%R
(pred x <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
F (bpow (e - 1))
apply pred_le_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
F (bpow (e - 1))
apply generic_format_bpow.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
(fexp (e - 1 + 1) <= e - 1)%Z
apply Z.lt_le_pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
(fexp (e - 1 + 1) < e)%Z
replace (_ + 1)%Z with e by ring.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
(fexp e < e)%Z
rewrite <- (mag_unique_pos _ _ _ H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
x:R
Fx:F x
Hx:(0 < pred x)%R
Hx':(0 < x)%R
Zx:x <> 0%R
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(bpow (e - 1) <= x < bpow e)%R
H1:(bpow (e - 1) < x)%R
(fexp (mag beta x) < mag beta x)%Z
now apply mag_generic_gt.
Qed.

Lemma ulp_round_pos :
  forall { Not_FTZ_ : Exp_not_FTZ fexp},
   forall rnd { Zrnd : Valid_rnd rnd } x,
  (0 < x)%R -> ulp (round beta fexp rnd x) = ulp x
     \/ round beta fexp rnd x = bpow (mag beta x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Exp_not_FTZ fexp ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
ulp (round beta fexp rnd x) = ulp x \/
round beta fexp rnd x = bpow (mag beta x)
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Exp_not_FTZ fexp ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(0 < x)%R ->
ulp (round beta fexp rnd x) = ulp x \/
round beta fexp rnd x = bpow (mag beta x)
intros Not_FTZ_ rnd Zrnd x Hx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
ulp (round beta fexp rnd x) = ulp x \/
round beta fexp rnd x = bpow (mag beta x)
case (generic_format_EM beta fexp x); intros Fx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:F x
ulp (round beta fexp rnd x) = ulp x \/
round beta fexp rnd x = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
ulp (round beta fexp rnd x) = ulp x \/
round beta fexp rnd x = bpow (mag beta x)
rewrite round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
ulp (round beta fexp rnd x) = ulp x \/
round beta fexp rnd x = bpow (mag beta x)
case (round_DN_or_UP beta fexp rnd x); intros Hr; rewrite Hr.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zfloor x
ulp (round beta fexp Zfloor x) = ulp x \/
round beta fexp Zfloor x = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zfloor x
ulp (round beta fexp Zfloor x) = ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
apply ulp_DN; now left...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
assert (M:(0 <= round beta fexp Zfloor x)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
(0 <= round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 <= round beta fexp Zfloor x)%R
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
apply round_ge_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
F 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 <= round beta fexp Zfloor x)%R
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
apply generic_format_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 <= round beta fexp Zfloor x)%R
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
apply Rlt_le...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 <= round beta fexp Zfloor x)%R
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
destruct M as [M|M].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
rewrite <- (succ_DN_eq_UP x)...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
ulp (succ (round beta fexp Zfloor x)) = ulp x \/
succ (round beta fexp Zfloor x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
case (ulp_succ_pos (round beta fexp Zfloor x)); try intros Y.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
F (round beta fexp Zfloor x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
(0 < round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
Y:ulp (succ (round beta fexp Zfloor x)) =
ulp (round beta fexp Zfloor x)
ulp (succ (round beta fexp Zfloor x)) = ulp x \/
succ (round beta fexp Zfloor x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
Y:succ (round beta fexp Zfloor x) =
bpow (mag beta (round beta fexp Zfloor x))
ulp (succ (round beta fexp Zfloor x)) = ulp x \/
succ (round beta fexp Zfloor x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
apply generic_format_round...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
(0 < round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
Y:ulp (succ (round beta fexp Zfloor x)) =
ulp (round beta fexp Zfloor x)
ulp (succ (round beta fexp Zfloor x)) = ulp x \/
succ (round beta fexp Zfloor x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
Y:succ (round beta fexp Zfloor x) =
bpow (mag beta (round beta fexp Zfloor x))
ulp (succ (round beta fexp Zfloor x)) = ulp x \/
succ (round beta fexp Zfloor x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
Y:ulp (succ (round beta fexp Zfloor x)) =
ulp (round beta fexp Zfloor x)
ulp (succ (round beta fexp Zfloor x)) = ulp x \/
succ (round beta fexp Zfloor x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
Y:succ (round beta fexp Zfloor x) =
bpow (mag beta (round beta fexp Zfloor x))
ulp (succ (round beta fexp Zfloor x)) = ulp x \/
succ (round beta fexp Zfloor x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
rewrite ulp_DN in Y...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
Y:ulp (succ (round beta fexp Zfloor x)) =
ulp (round beta fexp Zfloor x)
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
Y:succ (round beta fexp Zfloor x) =
bpow (mag beta (round beta fexp Zfloor x))
ulp (succ (round beta fexp Zfloor x)) = ulp x \/
succ (round beta fexp Zfloor x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
now apply Rlt_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
Y:succ (round beta fexp Zfloor x) =
bpow (mag beta (round beta fexp Zfloor x))
ulp (succ (round beta fexp Zfloor x)) = ulp x \/
succ (round beta fexp Zfloor x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
right; rewrite Y.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:(0 < round beta fexp Zfloor x)%R
Y:succ (round beta fexp Zfloor x) =
bpow (mag beta (round beta fexp Zfloor x))
bpow (mag beta (round beta fexp Zfloor x)) =
bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
apply f_equal, mag_DN...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)
left; rewrite <- (succ_DN_eq_UP x)...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (succ (round beta fexp Zfloor x)) = ulp x
rewrite <- M, succ_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp (ulp 0) = ulp x
rewrite ulp_ulp_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
ulp 0 = ulp x
case (negligible_exp_spec').beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
negligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) -> ulp 0 = ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp 0 = ulp x
intros (K1,K2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
K1:negligible_exp = None
K2:forall n : Z, (fexp n < n)%Z
ulp 0 = ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp 0 = ulp x
absurd (x = 0)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
K1:negligible_exp = None
K2:forall n : Z, (fexp n < n)%Z
x <> 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
K1:negligible_exp = None
K2:forall n : Z, (fexp n < n)%Z
x = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp 0 = ulp x
now apply Rgt_not_eq.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
K1:negligible_exp = None
K2:forall n : Z, (fexp n < n)%Z
x = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp 0 = ulp x
apply eq_0_round_0_negligible_exp with Zfloor...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp 0 = ulp x
intros (n,(Hn1,Hn2)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
ulp 0 = ulp x
replace (ulp 0) with (bpow (fexp n)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) = ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) = ulp 0
2: unfold ulp; rewrite Req_bool_true; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) = ulp x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) =
match negligible_exp with
| Some n0 => bpow (fexp n0)
| None => 0%R
end
2: now rewrite Hn1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) = ulp x
rewrite ulp_neq_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) = bpow (cexp beta fexp x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
x <> 0%R
2: apply Rgt_not_eq...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
bpow (fexp n) = bpow (cexp beta fexp x)
unfold cexp; f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
fexp n = fexp (mag beta x)
destruct (mag beta x) as (e,He); simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
fexp n = fexp e
apply sym_eq, valid_exp...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(e <= fexp n)%Z
assert (e <= fexp e)%Z.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(e <= fexp e)%Z
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
(e <= fexp n)%Z
apply exp_small_round_0_pos with beta Zfloor x...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (e - 1) <= x < bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
(e <= fexp n)%Z
rewrite <- (Rabs_pos_eq x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(bpow (e - 1) <= Rabs x < bpow e)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
(e <= fexp n)%Z
apply He, Rgt_not_eq...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
(0 <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
(e <= fexp n)%Z
apply Rlt_le...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
(e <= fexp n)%Z
replace (fexp n) with (fexp e); try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Hx:(0 < x)%R
Fx:~ F x
Hr:round beta fexp rnd x = round beta fexp Zceil x
M:0%R = round beta fexp Zfloor x
n:Z
Hn1:negligible_exp = Some n
Hn2:(n <= fexp n)%Z
e:Z
He:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R
H:(e <= fexp e)%Z
fexp e = fexp n
now apply fexp_negligible_exp_eq.
Qed.

Theorem ulp_round : forall { Not_FTZ_ : Exp_not_FTZ fexp},
   forall rnd { Zrnd : Valid_rnd rnd } x,
     ulp (round beta fexp rnd x) = ulp x
         \/ Rabs (round beta fexp rnd x) = bpow (mag beta x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Exp_not_FTZ fexp ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Exp_not_FTZ fexp ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
intros Not_FTZ_ rnd Zrnd x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
case (Rtotal_order x 0); intros Zx.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
case (ulp_round_pos (Zrnd_opp rnd) (-x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
(0 < - x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
ulp (round beta fexp (Zrnd_opp rnd) (- x)) = ulp (- x) ->
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta (- x)) ->
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
now apply Ropp_0_gt_lt_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
ulp (round beta fexp (Zrnd_opp rnd) (- x)) = ulp (- x) ->
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta (- x)) ->
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
rewrite ulp_opp, <- ulp_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
ulp (- round beta fexp (Zrnd_opp rnd) (- x)) = ulp x ->
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta (- x)) ->
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
rewrite <- round_opp, Ropp_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
ulp (round beta fexp rnd x) = ulp x ->
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta (- x)) ->
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
intros Y;now left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta (- x)) ->
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
rewrite mag_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta x) ->
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
intros Y; right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
Y:round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta x)
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
rewrite <- (Ropp_involutive x) at 1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
Y:round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta x)
Rabs (round beta fexp rnd (- - x)) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
rewrite round_opp, Y.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
Y:round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta x)
Rabs (- bpow (mag beta x)) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
rewrite Rabs_Ropp, Rabs_right...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x < 0)%R
Y:round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta x)
(bpow (mag beta x) >= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
apply Rle_ge, bpow_ge_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R \/ (x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
destruct Zx as [Zx|Zx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:x = 0%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
left; rewrite Zx; rewrite round_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)
rewrite Rabs_right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x > 0)%R
ulp (round beta fexp rnd x) = ulp x \/
round beta fexp rnd x = bpow (mag beta x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x > 0)%R
(round beta fexp rnd x >= 0)%R
apply ulp_round_pos...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x > 0)%R
(round beta fexp rnd x >= 0)%R
apply Rle_ge, round_ge_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x > 0)%R
F 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x > 0)%R
(0 <= x)%R
apply generic_format_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Not_FTZ_:Exp_not_FTZ fexp
rnd:R -> Z
Zrnd:Valid_rnd rnd
x:R
Zx:(x > 0)%R
(0 <= x)%R
now apply Rlt_le.
Qed.

Lemma succ_round_ge_id :
  forall rnd { Zrnd : Valid_rnd rnd } x,
  (x <= succ (round beta fexp rnd x))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, (x <= succ (round beta fexp rnd x))%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, (x <= succ (round beta fexp rnd x))%R
intros rnd Vrnd x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
(x <= succ (round beta fexp rnd x))%R
apply (Rle_trans _ (round beta fexp Raux.Zceil x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
(x <= round beta fexp Zceil x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
(round beta fexp Zceil x <=
 succ (round beta fexp rnd x))%R
{beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
(x <= round beta fexp Zceil x)%R now apply round_UP_pt. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
(round beta fexp Zceil x <=
 succ (round beta fexp rnd x))%R
destruct (round_DN_or_UP beta fexp rnd x) as [Hr|Hr]; rewrite Hr.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Hr:round beta fexp rnd x = round beta fexp Zfloor x
(round beta fexp Zceil x <=
 succ (round beta fexp Zfloor x))%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Hr:round beta fexp rnd x = round beta fexp Zceil x
(round beta fexp Zceil x <=
 succ (round beta fexp Zceil x))%R
{beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Hr:round beta fexp rnd x = round beta fexp Zfloor x
(round beta fexp Zceil x <=
 succ (round beta fexp Zfloor x))%R now apply UP_le_succ_DN. }beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Hr:round beta fexp rnd x = round beta fexp Zceil x
(round beta fexp Zceil x <=
 succ (round beta fexp Zceil x))%R
apply succ_ge_id.
Qed.

Lemma pred_round_le_id :
  forall rnd { Zrnd : Valid_rnd rnd } x,
  (pred (round beta fexp rnd x) <= x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, (pred (round beta fexp rnd x) <= x)%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, (pred (round beta fexp rnd x) <= x)%R
intros rnd Vrnd x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
(pred (round beta fexp rnd x) <= x)%R
apply (Rle_trans _ (round beta fexp Raux.Zfloor x)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
(pred (round beta fexp rnd x) <=
 round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
(round beta fexp Zfloor x <= x)%R
2: now apply round_DN_pt.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
(pred (round beta fexp rnd x) <=
 round beta fexp Zfloor x)%R
destruct (round_DN_or_UP beta fexp rnd x) as [Hr|Hr]; rewrite Hr.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Hr:round beta fexp rnd x = round beta fexp Zfloor x
(pred (round beta fexp Zfloor x) <=
 round beta fexp Zfloor x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Hr:round beta fexp rnd x = round beta fexp Zceil x
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R
2: now apply pred_UP_le_DN.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
rnd:R -> Z
Vrnd:Valid_rnd rnd
x:R
Hr:round beta fexp rnd x = round beta fexp Zfloor x
(pred (round beta fexp Zfloor x) <=
 round beta fexp Zfloor x)%R
apply pred_le_id.
Qed.

Properties of rounding to nearest and ulp

Theorem round_N_le_midp: forall choice u v,
  F u -> (v < (u + succ u)/2)%R
      -> (round beta fexp (Znearest choice)  v <= u)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (u v : R),
F u ->
(v < (u + succ u) / 2)%R ->
(round beta fexp (Znearest choice) v <= u)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (u v : R),
F u ->
(v < (u + succ u) / 2)%R ->
(round beta fexp (Znearest choice) v <= u)%R
intros choice u v Fu H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
(round beta fexp (Znearest choice) v <= u)%R
(* . *)
assert (V: ((succ u = 0 /\ u = 0) \/ u < succ u)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
specialize (succ_ge_id u); intros P; destruct P as [P|P].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
P:(u < succ u)%R
succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
P:u = succ u
succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
now right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
P:u = succ u
succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
case (Req_dec u 0); intros Zu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
P:u = succ u
Zu:u = 0%R
succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
P:u = succ u
Zu:u <> 0%R
succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
left; split; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
P:u = succ u
Zu:u = 0%R
succ u = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
P:u = succ u
Zu:u <> 0%R
succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
now rewrite <- P.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
P:u = succ u
Zu:u <> 0%R
succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
right; now apply succ_gt_id.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:succ u = 0%R /\ u = 0%R \/ (u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
(* *)
destruct V as [(V1,V2)|V].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V1:succ u = 0%R
V2:u = 0%R
(round beta fexp (Znearest choice) v <= u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
rewrite V2; apply round_le_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V1:succ u = 0%R
V2:u = 0%R
F 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V1:succ u = 0%R
V2:u = 0%R
(v <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
apply generic_format_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V1:succ u = 0%R
V2:u = 0%R
(v <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
left; apply Rlt_le_trans with (1:=H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V1:succ u = 0%R
V2:u = 0%R
((u + succ u) / 2 <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
rewrite V1,V2; right; field.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
(* *)
assert (T: (u < (u + succ u) / 2 < succ u)%R) by lra.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T:(u < (u + succ u) / 2 < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
destruct T as (T1,T2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
(round beta fexp (Znearest choice) v <= u)%R
apply Rnd_N_pt_monotone with F v ((u + succ u) / 2)%R...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_N_pt F v (round beta fexp (Znearest choice) v)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_N_pt F ((u + succ u) / 2) u
apply round_N_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_N_pt F ((u + succ u) / 2) u
apply Rnd_N_pt_DN with (succ u)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_DN_pt F ((u + succ u) / 2) u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_UP_pt F ((u + succ u) / 2) (succ u)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
pattern u at 3; replace u with (round beta fexp Zfloor ((u + succ u) / 2)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_DN_pt F ((u + succ u) / 2)
  (round beta fexp Zfloor ((u + succ u) / 2))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
round beta fexp Zfloor ((u + succ u) / 2) = u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_UP_pt F ((u + succ u) / 2) (succ u)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
apply round_DN_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
round beta fexp Zfloor ((u + succ u) / 2) = u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_UP_pt F ((u + succ u) / 2) (succ u)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
apply round_DN_eq; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
(u <= (u + succ u) / 2 < succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_UP_pt F ((u + succ u) / 2) (succ u)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
split; try left; assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_UP_pt F ((u + succ u) / 2) (succ u)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
pattern (succ u) at 2; replace (succ u) with (round beta fexp Zceil ((u + succ u) / 2)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
Rnd_UP_pt F ((u + succ u) / 2)
  (round beta fexp Zceil ((u + succ u) / 2))
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
round beta fexp Zceil ((u + succ u) / 2) = succ u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
apply round_UP_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
round beta fexp Zceil ((u + succ u) / 2) = succ u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
apply round_UP_eq; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
F (succ u)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
(pred (succ u) < (u + succ u) / 2 <= succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
apply generic_format_succ...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
(pred (succ u) < (u + succ u) / 2 <= succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
rewrite pred_succ; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
(u < (u + succ u) / 2 <= succ u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
split; try left; assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:(v < (u + succ u) / 2)%R
V:(u < succ u)%R
T1:(u < (u + succ u) / 2)%R
T2:((u + succ u) / 2 < succ u)%R
((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R
right; field.
Qed.


Theorem round_N_ge_midp: forall choice u v,
 F u ->  ((u + pred u)/2 < v)%R
      -> (u <= round beta fexp (Znearest choice)  v)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (u v : R),
F u ->
((u + pred u) / 2 < v)%R ->
(u <= round beta fexp (Znearest choice) v)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (u v : R),
F u ->
((u + pred u) / 2 < v)%R ->
(u <= round beta fexp (Znearest choice) v)%R
intros choice u v Fu H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(u <= round beta fexp (Znearest choice) v)%R
rewrite <- (Ropp_involutive v).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(u <= round beta fexp (Znearest choice) (- - v))%R
rewrite round_N_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(u <=
 -
 round beta fexp
   (Znearest
      (fun t : Z => negb (choice (- (t + 1))%Z)))
   (- v))%R
rewrite <- (Ropp_involutive u).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(- - u <=
 -
 round beta fexp
   (Znearest
      (fun t : Z => negb (choice (- (t + 1))%Z)))
   (- v))%R
apply Ropp_le_contravar.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(round beta fexp
   (Znearest
      (fun t : Z => negb (choice (- (t + 1))%Z)))
   (- v) <= - u)%R
apply round_N_le_midp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
F (- u)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(- v < (- u + succ (- u)) / 2)%R
now apply generic_format_opp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(- v < (- u + succ (- u)) / 2)%R
apply Ropp_lt_cancel.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(- ((- u + succ (- u)) / 2) < - - v)%R
rewrite Ropp_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(- ((- u + succ (- u)) / 2) < v)%R
apply Rle_lt_trans with (2:=H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(- ((- u + succ (- u)) / 2) <= (u + pred u) / 2)%R
unfold pred.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Fu:F u
H:((u + pred u) / 2 < v)%R
(- ((- u + succ (- u)) / 2) <= (u + - succ (- u)) / 2)%R
right; field.
Qed.

Lemma round_N_ge_ge_midp : forall choice u v,
       F u ->
       (u <= round beta fexp (Znearest choice) v)%R ->
       ((u + pred u) / 2 <= v)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (u v : R),
F u ->
(u <= round beta fexp (Znearest choice) v)%R ->
((u + pred u) / 2 <= v)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (u v : R),
F u ->
(u <= round beta fexp (Znearest choice) v)%R ->
((u + pred u) / 2 <= v)%R
intros choice u v Hu H2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
((u + pred u) / 2 <= v)%R
assert (K: ((u=0)%R /\ negligible_exp = None) \/ (pred u < u)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
case (Req_dec u 0); intros Zu.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u <> 0%R
u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
case_eq (negligible_exp).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
forall z : Z,
negligible_exp = Some z ->
u = 0%R /\ Some z = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
negligible_exp = None ->
u = 0%R /\ None = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u <> 0%R
u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
intros n Hn; right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
n:Z
Hn:negligible_exp = Some n
(pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
negligible_exp = None ->
u = 0%R /\ None = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u <> 0%R
u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
rewrite Zu, pred_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
n:Z
Hn:negligible_exp = Some n
(- ulp 0 < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
negligible_exp = None ->
u = 0%R /\ None = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u <> 0%R
u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
unfold ulp; rewrite Req_bool_true, Hn; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
n:Z
Hn:negligible_exp = Some n
(- bpow (fexp n) < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
negligible_exp = None ->
u = 0%R /\ None = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u <> 0%R
u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
rewrite <- Ropp_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
n:Z
Hn:negligible_exp = Some n
(- bpow (fexp n) < - 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
negligible_exp = None ->
u = 0%R /\ None = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u <> 0%R
u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
apply Ropp_lt_contravar, bpow_gt_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u = 0%R
negligible_exp = None ->
u = 0%R /\ None = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u <> 0%R
u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
intros _; left; split; easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u <> 0%R
u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
Zu:u <> 0%R
(pred u < u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
apply pred_lt_id...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
((u + pred u) / 2 <= v)%R
(* *)
case K.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
u = 0%R /\ negligible_exp = None ->
((u + pred u) / 2 <= v)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
intros (K1,K2).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
((u + pred u) / 2 <= v)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
(* . *)
rewrite K1, pred_0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
((0 + - ulp 0) / 2 <= v)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
unfold ulp; rewrite Req_bool_true, K2; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
((0 + - 0) / 2 <= v)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
replace ((0+-0)/2)%R with 0%R by field.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
(0 <= v)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
case (Rle_or_lt 0 v); try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
(v < 0)%R -> (0 <= v)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
intros H3; contradict H2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
~ (u <= round beta fexp (Znearest choice) v)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
rewrite K1; apply Rlt_not_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
(round beta fexp (Znearest choice) v < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
assert (H4: (round beta fexp (Znearest choice) v <= 0)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
(round beta fexp (Znearest choice) v <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
H4:(round beta fexp (Znearest choice) v <= 0)%R
(round beta fexp (Znearest choice) v < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
apply round_le_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
F 0
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
(v <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
H4:(round beta fexp (Znearest choice) v <= 0)%R
(round beta fexp (Znearest choice) v < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
apply generic_format_0...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
(v <= 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
H4:(round beta fexp (Znearest choice) v <= 0)%R
(round beta fexp (Znearest choice) v < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
now left.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
H4:(round beta fexp (Znearest choice) v <= 0)%R
(round beta fexp (Znearest choice) v < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
case H4; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
H4:(round beta fexp (Znearest choice) v <= 0)%R
round beta fexp (Znearest choice) v = 0%R ->
(round beta fexp (Znearest choice) v < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
intros H5.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
H4:(round beta fexp (Znearest choice) v <= 0)%R
H5:round beta fexp (Znearest choice) v = 0%R
(round beta fexp (Znearest choice) v < 0)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
absurd (v=0)%R; try auto with real.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:u = 0%R
K2:negligible_exp = None
H3:(v < 0)%R
H4:(round beta fexp (Znearest choice) v <= 0)%R
H5:round beta fexp (Znearest choice) v = 0%R
v = 0%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
apply eq_0_round_0_negligible_exp with (Znearest choice)...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
(pred u < u)%R -> ((u + pred u) / 2 <= v)%R
(* . *)
intros K1.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
((u + pred u) / 2 <= v)%R
case (Rle_or_lt ((u + pred u) / 2) v); try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
(v < (u + pred u) / 2)%R -> ((u + pred u) / 2 <= v)%R
intros H3.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
H3:(v < (u + pred u) / 2)%R
((u + pred u) / 2 <= v)%R
absurd (u <= round beta fexp (Znearest choice) v)%R; try easy.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
H3:(v < (u + pred u) / 2)%R
~ (u <= round beta fexp (Znearest choice) v)%R
apply Rlt_not_le.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
H3:(v < (u + pred u) / 2)%R
(round beta fexp (Znearest choice) v < u)%R
apply Rle_lt_trans with (2:=K1).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
H3:(v < (u + pred u) / 2)%R
(round beta fexp (Znearest choice) v <= pred u)%R
apply round_N_le_midp...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
H3:(v < (u + pred u) / 2)%R
F (pred u)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
H3:(v < (u + pred u) / 2)%R
(v < (pred u + succ (pred u)) / 2)%R
apply generic_format_pred...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
H3:(v < (u + pred u) / 2)%R
(v < (pred u + succ (pred u)) / 2)%R
rewrite succ_pred...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
H3:(v < (u + pred u) / 2)%R
(v < (pred u + u) / 2)%R
apply Rlt_le_trans with (1:=H3).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(u <= round beta fexp (Znearest choice) v)%R
K:u = 0%R /\ negligible_exp = None \/ (pred u < u)%R
K1:(pred u < u)%R
H3:(v < (u + pred u) / 2)%R
((u + pred u) / 2 <= (pred u + u) / 2)%R
right; f_equal; ring.
Qed.


Lemma round_N_le_le_midp : forall choice u v,
       F u ->
       (round beta fexp (Znearest choice) v <= u)%R ->
       (v <= (u + succ u) / 2)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (u v : R),
F u ->
(round beta fexp (Znearest choice) v <= u)%R ->
(v <= (u + succ u) / 2)%R
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (u v : R),
F u ->
(round beta fexp (Znearest choice) v <= u)%R ->
(v <= (u + succ u) / 2)%R
intros choice u v Hu H2.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(round beta fexp (Znearest choice) v <= u)%R
(v <= (u + succ u) / 2)%R
apply Ropp_le_cancel.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(round beta fexp (Znearest choice) v <= u)%R
(- ((u + succ u) / 2) <= - v)%R
apply Rle_trans with (((-u)+pred (-u))/2)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(round beta fexp (Znearest choice) v <= u)%R
(- ((u + succ u) / 2) <= (- u + pred (- u)) / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(round beta fexp (Znearest choice) v <= u)%R
((- u + pred (- u)) / 2 <= - v)%R
rewrite pred_opp; right; field.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(round beta fexp (Znearest choice) v <= u)%R
((- u + pred (- u)) / 2 <= - v)%R
apply round_N_ge_ge_midp with
   (choice := fun t:Z => negb (choice (- (t + 1))%Z))...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(round beta fexp (Znearest choice) v <= u)%R
F (- u)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(round beta fexp (Znearest choice) v <= u)%R
(- u <=
 round beta fexp
   (Znearest
      (fun t : Z => negb (choice (- (t + 1))%Z)))
   (- v))%R
apply generic_format_opp...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(round beta fexp (Znearest choice) v <= u)%R
(- u <=
 round beta fexp
   (Znearest
      (fun t : Z => negb (choice (- (t + 1))%Z)))
   (- v))%R
rewrite <- (Ropp_involutive (round _ _ _ _)).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(round beta fexp (Znearest choice) v <= u)%R
(- u <=
 -
 -
 round beta fexp
   (Znearest
      (fun t : Z => negb (choice (- (t + 1))%Z)))
   (- v))%R
rewrite <- round_N_opp, Ropp_involutive.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
u, v:R
Hu:F u
H2:(round beta fexp (Znearest choice) v <= u)%R
(- u <= - round beta fexp (Znearest choice) v)%R
apply Ropp_le_contravar; easy.
Qed.


Lemma round_N_eq_DN: forall choice x,
       let d:=round beta fexp Zfloor x in
       let u:=round beta fexp Zceil x in
      (x<(d+u)/2)%R ->
     round beta fexp (Znearest choice) x = d.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (x : R),
let d := round beta fexp Zfloor x in
let u := round beta fexp Zceil x in
(x < (d + u) / 2)%R ->
round beta fexp (Znearest choice) x = d
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (x : R),
let d := round beta fexp Zfloor x in
let u := round beta fexp Zceil x in
(x < (d + u) / 2)%R ->
round beta fexp (Znearest choice) x = d
intros choice x d u H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
round beta fexp (Znearest choice) x = d
apply Rle_antisym.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
(round beta fexp (Znearest choice) x <= d)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
(d <= round beta fexp (Znearest choice) x)%R
destruct (generic_format_EM beta fexp x) as [Fx|Fx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
Fx:F x
(round beta fexp (Znearest choice) x <= d)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
Fx:~ F x
(round beta fexp (Znearest choice) x <= d)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
(d <= round beta fexp (Znearest choice) x)%R
rewrite round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
Fx:F x
(x <= d)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
Fx:~ F x
(round beta fexp (Znearest choice) x <= d)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
(d <= round beta fexp (Znearest choice) x)%R
apply round_DN_pt; trivial; now right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
Fx:~ F x
(round beta fexp (Znearest choice) x <= d)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
(d <= round beta fexp (Znearest choice) x)%R
apply round_N_le_midp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
Fx:~ F x
F d
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
Fx:~ F x
(x < (d + succ d) / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
(d <= round beta fexp (Znearest choice) x)%R
apply round_DN_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
Fx:~ F x
(x < (d + succ d) / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
(d <= round beta fexp (Znearest choice) x)%R
apply Rlt_le_trans with (1:=H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
Fx:~ F x
((d + u) / 2 <= (d + succ d) / 2)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
(d <= round beta fexp (Znearest choice) x)%R
right; apply f_equal2; trivial; apply f_equal.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
Fx:~ F x
u = succ d
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
(d <= round beta fexp (Znearest choice) x)%R
now apply sym_eq, succ_DN_eq_UP.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x < (d + u) / 2)%R
(d <= round beta fexp (Znearest choice) x)%R
apply round_ge_generic; try apply round_DN_pt...
Qed.

Lemma round_N_eq_DN_pt: forall choice x d u,
      Rnd_DN_pt F x d -> Rnd_UP_pt F x u ->
      (x<(d+u)/2)%R ->
     round beta fexp (Znearest choice) x = d.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (x d u : R),
Rnd_DN_pt F x d ->
Rnd_UP_pt F x u ->
(x < (d + u) / 2)%R ->
round beta fexp (Znearest choice) x = d
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (x d u : R),
Rnd_DN_pt F x d ->
Rnd_UP_pt F x u ->
(x < (d + u) / 2)%R ->
round beta fexp (Znearest choice) x = d
intros choice x d u Hd Hu H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:(x < (d + u) / 2)%R
round beta fexp (Znearest choice) x = d
assert (H0:(d = round beta fexp Zfloor x)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:(x < (d + u) / 2)%R
d = round beta fexp Zfloor x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:(x < (d + u) / 2)%R
H0:d = round beta fexp Zfloor x
round beta fexp (Znearest choice) x = d
apply Rnd_DN_pt_unique with (1:=Hd).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:(x < (d + u) / 2)%R
Rnd_DN_pt F x (round beta fexp Zfloor x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:(x < (d + u) / 2)%R
H0:d = round beta fexp Zfloor x
round beta fexp (Znearest choice) x = d
apply round_DN_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:(x < (d + u) / 2)%R
H0:d = round beta fexp Zfloor x
round beta fexp (Znearest choice) x = d
rewrite H0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:(x < (d + u) / 2)%R
H0:d = round beta fexp Zfloor x
round beta fexp (Znearest choice) x =
round beta fexp Zfloor x
apply round_N_eq_DN.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:(x < (d + u) / 2)%R
H0:d = round beta fexp Zfloor x
(x <
 (round beta fexp Zfloor x + round beta fexp Zceil x) /
 2)%R
rewrite <- H0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:(x < (d + u) / 2)%R
H0:d = round beta fexp Zfloor x
(x < (d + round beta fexp Zceil x) / 2)%R
rewrite Rnd_UP_pt_unique with F x (round beta fexp Zceil x) u; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:(x < (d + u) / 2)%R
H0:d = round beta fexp Zfloor x
Rnd_UP_pt F x (round beta fexp Zceil x)
apply round_UP_pt...
Qed.

Lemma round_N_eq_UP: forall choice x,
      let d:=round beta fexp Zfloor x in
      let u:=round beta fexp Zceil x in
     ((d+u)/2 < x)%R ->
     round beta fexp (Znearest choice) x = u.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (x : R),
let d := round beta fexp Zfloor x in
let u := round beta fexp Zceil x in
((d + u) / 2 < x)%R ->
round beta fexp (Znearest choice) x = u
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (x : R),
let d := round beta fexp Zfloor x in
let u := round beta fexp Zceil x in
((d + u) / 2 < x)%R ->
round beta fexp (Znearest choice) x = u
intros choice x d u H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
round beta fexp (Znearest choice) x = u
apply Rle_antisym.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
(round beta fexp (Znearest choice) x <= u)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
(u <= round beta fexp (Znearest choice) x)%R
apply round_le_generic; try apply round_UP_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
(u <= round beta fexp (Znearest choice) x)%R
destruct (generic_format_EM beta fexp x) as [Fx|Fx].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
Fx:F x
(u <= round beta fexp (Znearest choice) x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
Fx:~ F x
(u <= round beta fexp (Znearest choice) x)%R
rewrite round_generic...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
Fx:F x
(u <= x)%R
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
Fx:~ F x
(u <= round beta fexp (Znearest choice) x)%R
apply round_UP_pt; trivial; now right.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
Fx:~ F x
(u <= round beta fexp (Znearest choice) x)%R
apply round_N_ge_midp.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
Fx:~ F x
F u
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
Fx:~ F x
((u + pred u) / 2 < x)%R
apply round_UP_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
Fx:~ F x
((u + pred u) / 2 < x)%R
apply Rle_lt_trans with (2:=H).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
Fx:~ F x
((u + pred u) / 2 <= (d + u) / 2)%R
right; apply f_equal2; trivial; rewrite Rplus_comm; apply f_equal2; trivial.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:((d + u) / 2 < x)%R
Fx:~ F x
pred u = d
now apply pred_UP_eq_DN.
Qed.

Lemma round_N_eq_UP_pt: forall choice x d u,
      Rnd_DN_pt F x d -> Rnd_UP_pt F x u ->
      ((d+u)/2 < x)%R ->
     round beta fexp (Znearest choice) x = u.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (x d u : R),
Rnd_DN_pt F x d ->
Rnd_UP_pt F x u ->
((d + u) / 2 < x)%R ->
round beta fexp (Znearest choice) x = u
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (choice : Z -> bool) (x d u : R),
Rnd_DN_pt F x d ->
Rnd_UP_pt F x u ->
((d + u) / 2 < x)%R ->
round beta fexp (Znearest choice) x = u
intros choice x d u Hd Hu H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:((d + u) / 2 < x)%R
round beta fexp (Znearest choice) x = u
assert (H0:(u = round beta fexp Zceil x)%R).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:((d + u) / 2 < x)%R
u = round beta fexp Zceil x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:((d + u) / 2 < x)%R
H0:u = round beta fexp Zceil x
round beta fexp (Znearest choice) x = u
apply Rnd_UP_pt_unique with (1:=Hu).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:((d + u) / 2 < x)%R
Rnd_UP_pt F x (round beta fexp Zceil x)
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:((d + u) / 2 < x)%R
H0:u = round beta fexp Zceil x
round beta fexp (Znearest choice) x = u
apply round_UP_pt...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:((d + u) / 2 < x)%R
H0:u = round beta fexp Zceil x
round beta fexp (Znearest choice) x = u
rewrite H0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:((d + u) / 2 < x)%R
H0:u = round beta fexp Zceil x
round beta fexp (Znearest choice) x =
round beta fexp Zceil x
apply round_N_eq_UP.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:((d + u) / 2 < x)%R
H0:u = round beta fexp Zceil x
((round beta fexp Zfloor x + round beta fexp Zceil x) /
 2 < x)%R
rewrite <- H0.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:((d + u) / 2 < x)%R
H0:u = round beta fexp Zceil x
((round beta fexp Zfloor x + u) / 2 < x)%R
rewrite Rnd_DN_pt_unique with F x (round beta fexp Zfloor x) d; try assumption.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
choice:Z -> bool
x, d, u:R
Hd:Rnd_DN_pt F x d
Hu:Rnd_UP_pt F x u
H:((d + u) / 2 < x)%R
H0:u = round beta fexp Zceil x
Rnd_DN_pt F x (round beta fexp Zfloor x)
apply round_DN_pt...
Qed.

Lemma round_N_plus_ulp_ge :
  forall { Hm : Monotone_exp fexp } choice1 choice2 x,
  let rx := round beta fexp (Znearest choice2) x in
  (x <= round beta fexp (Znearest choice1) (rx + ulp rx))%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall (choice1 choice2 : Z -> bool) (x : R),
let rx := round beta fexp (Znearest choice2) x in
(x <= round beta fexp (Znearest choice1) (rx + ulp rx))%R
Proof.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Monotone_exp fexp ->
forall (choice1 choice2 : Z -> bool) (x : R),
let rx := round beta fexp (Znearest choice2) x in
(x <= round beta fexp (Znearest choice1) (rx + ulp rx))%R
intros Hm choice1 choice2 x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice1, choice2:Z -> bool
x:R
let rx := round beta fexp (Znearest choice2) x in
(x <= round beta fexp (Znearest choice1) (rx + ulp rx))%R
simpl.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice1, choice2:Z -> bool
x:R
(x <=
 round beta fexp (Znearest choice1)
   (round beta fexp (Znearest choice2) x +
    ulp (round beta fexp (Znearest choice2) x)))%R
set (rx := round _ _ _ x).beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice1, choice2:Z -> bool
x:R
rx:=round beta fexp (Znearest choice2) x:R
(x <= round beta fexp (Znearest choice1) (rx + ulp rx))%R
assert (Vrnd1 : Valid_rnd (Znearest choice1)) by now apply valid_rnd_N.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice1, choice2:Z -> bool
x:R
rx:=round beta fexp (Znearest choice2) x:R
Vrnd1:Valid_rnd (Znearest choice1)
(x <= round beta fexp (Znearest choice1) (rx + ulp rx))%R
assert (Vrnd2 : Valid_rnd (Znearest choice2)) by now apply valid_rnd_N.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice1, choice2:Z -> bool
x:R
rx:=round beta fexp (Znearest choice2) x:R
Vrnd1:Valid_rnd (Znearest choice1)
Vrnd2:Valid_rnd (Znearest choice2)
(x <= round beta fexp (Znearest choice1) (rx + ulp rx))%R
apply (Rle_trans _ (succ rx)); [now apply succ_round_ge_id|].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice1, choice2:Z -> bool
x:R
rx:=round beta fexp (Znearest choice2) x:R
Vrnd1:Valid_rnd (Znearest choice1)
Vrnd2:Valid_rnd (Znearest choice2)
(succ rx <=
 round beta fexp (Znearest choice1) (rx + ulp rx))%R
rewrite round_generic; [now apply succ_le_plus_ulp|now simpl|].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
Hm:Monotone_exp fexp
choice1, choice2:Z -> bool
x:R
rx:=round beta fexp (Znearest choice2) x:R
Vrnd1:Valid_rnd (Znearest choice1)
Vrnd2:Valid_rnd (Znearest choice2)
F (rx + ulp rx)
now apply generic_format_plus_ulp, generic_format_round.
Qed.


Lemma round_N_eq_ties: forall c1 c2 x,
   (x - round beta fexp Zfloor x <> round beta fexp Zceil x - x)%R ->
   (round beta fexp (Znearest c1) x = round beta fexp (Znearest c2) x)%R.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (c1 c2 : Z -> bool) (x : R),
(x - round beta fexp Zfloor x)%R <>
(round beta fexp Zceil x - x)%R ->
round beta fexp (Znearest c1) x =
round beta fexp (Znearest c2) x
Proof with auto with typeclass_instances.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
forall (c1 c2 : Z -> bool) (x : R),
(x - round beta fexp Zfloor x)%R <>
(round beta fexp Zceil x - x)%R ->
round beta fexp (Znearest c1) x =
round beta fexp (Znearest c2) x
intros c1 c2 x.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
c1, c2:Z -> bool
x:R
(x - round beta fexp Zfloor x)%R <>
(round beta fexp Zceil x - x)%R ->
round beta fexp (Znearest c1) x =
round beta fexp (Znearest c2) x
pose (d:=round beta fexp Zfloor x); pose (u:=round beta fexp Zceil x); fold d; fold u; intros H.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
c1, c2:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x - d)%R <> (u - x)%R
round beta fexp (Znearest c1) x =
round beta fexp (Znearest c2) x
case (Rle_or_lt ((d+u)/2) x); intros L.beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
c1, c2:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x - d)%R <> (u - x)%R
L:((d + u) / 2 <= x)%R
round beta fexp (Znearest c1) x =
round beta fexp (Znearest c2) x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
c1, c2:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x - d)%R <> (u - x)%R
L:(x < (d + u) / 2)%R
round beta fexp (Znearest c1) x =
round beta fexp (Znearest c2) x
2:rewrite 2!round_N_eq_DN...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
c1, c2:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x - d)%R <> (u - x)%R
L:((d + u) / 2 <= x)%R
round beta fexp (Znearest c1) x =
round beta fexp (Znearest c2) x
destruct L as [L|L].beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
c1, c2:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x - d)%R <> (u - x)%R
L:((d + u) / 2 < x)%R
round beta fexp (Znearest c1) x =
round beta fexp (Znearest c2) x
beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
c1, c2:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x - d)%R <> (u - x)%R
L:((d + u) / 2)%R = x
round beta fexp (Znearest c1) x =
round beta fexp (Znearest c2) x
rewrite 2!round_N_eq_UP...beta:radix
fexp:Z -> Z
valid_exp:Valid_exp fexp
c1, c2:Z -> bool
x:R
d:=round beta fexp Zfloor x:R
u:=round beta fexp Zceil x:R
H:(x - d)%R <> (u - x)%R
L:((d + u) / 2)%R = x
round beta fexp (Znearest c1) x =
round beta fexp (Znearest c2) x
contradict H; rewrite <- L; field.
Qed.

End Fcore_ulp.