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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:radixfexp:Z -> Zforall x y : Z, (x <= y)%Z \/ ~ (x <= y)%Z

Proof.beta:radixfexp:Z -> Zforall x y : Z, (x <= y)%Z \/ ~ (x <= y)%Z

intros.beta:radixfexp:Z -> Zx, y:Z(x <= y)%Z \/ ~ (x <= y)%Z

destruct (Z_le_dec x y).beta:radixfexp:Z -> Zx, y:Zl:(x <= y)%Z(x <= y)%Z \/ ~ (x <= y)%Z1 more goalbeta:radixfexp:Z -> Zx, y:Zn:~ (x <= y)%Z(x <= y)%Z \/ ~ (x <= y)%Z

now left.beta:radixfexp:Z -> Zx, y:Zn:~ (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:radixfexp:Z -> Znegligible_exp_prop negligible_exp

Proof.beta:radixfexp:Z -> Znegligible_exp_prop negligible_exp

unfold negligible_exp; destruct LPO_Z as [(n,Hn)|Hn].beta:radixfexp:Z -> Zn:ZHn:(n <= fexp n)%Znegligible_exp_prop
  (Some
     (proj1_sig
        (exist (fun n0 : Z => (n0 <= fexp n0)%Z) n Hn)))1 more goalbeta:radixfexp:Z -> ZHn:forall n : Z, ~ (n <= fexp n)%Znegligible_exp_prop None

now apply negligible_Some.beta:radixfexp:Z -> ZHn:forall n : Z, ~ (n <= fexp n)%Znegligible_exp_prop None

apply negligible_None.beta:radixfexp:Z -> ZHn:forall n : Z, ~ (n <= fexp n)%Zforall 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:radixfexp:Z -> Znegligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) \/
(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z)

Proof.beta:radixfexp:Z -> Znegligible_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:radixfexp:Z -> Zn:ZHn:(n <= fexp n)%ZSome
  (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)1 more goalbeta:radixfexp:Z -> ZHn:forall n : Z, ~ (n <= fexp n)%ZNone = 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:radixfexp:Z -> ZHn:forall n : Z, ~ (n <= fexp n)%ZNone = None /\ (forall n : Z, (fexp n < n)%Z) \/
(exists n : Z, None = Some n /\ (n <= fexp n)%Z)

left; split; trivial.beta:radixfexp:Z -> ZHn:forall n : Z, ~ (n <= fexp n)%Zforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall n m : Z,
(n <= fexp n)%Z -> (m <= fexp m)%Z -> fexp n = fexp m

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall n m : Z,
(n <= fexp n)%Z -> (m <= fexp m)%Z -> fexp n = fexp m

intros n m Hn Hm.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn, m:ZHn:(n <= fexp n)%ZHm:(m <= fexp m)%Zfexp n = fexp m

case (Zle_or_lt n m); intros H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn, m:ZHn:(n <= fexp n)%ZHm:(m <= fexp m)%ZH:(n <= m)%Zfexp n = fexp m1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn, m:ZHn:(n <= fexp n)%ZHm:(m <= fexp m)%ZH:(m < n)%Zfexp n = fexp m

apply valid_exp; omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn, m:ZHn:(n <= fexp n)%ZHm:(m <= fexp m)%ZH:(m < n)%Zfexp 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
x <> 0%R -> ulp x = bpow (cexp beta fexp x)

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
x <> 0%R -> ulp x = bpow (cexp beta fexp x)

intros  x Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:x <> 0%Rulp x = bpow (cexp beta fexp x)

unfold ulp; case (Req_bool_spec x); trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:x <> 0%Rx = 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, ulp (- x) = ulp x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, ulp (- x) = ulp x

intros x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Rulp (- x) = ulp x

unfold ulp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:(- x)%R = 0%Rmatch 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))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:(- x)%R <> 0%Rbpow (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:(- x)%R = 0%Rx = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:(- x)%R <> 0%Rbpow (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:(- x)%R <> 0%Rbpow (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:(- x)%R <> 0%Rbpow (cexp beta fexp (- x)) = bpow (cexp beta fexp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:(- x)%R <> 0%Rx <> 0%R

now rewrite cexp_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:(- x)%R <> 0%Rx <> 0%R

intros H2; apply H1; rewrite H2; ring.
Qed.

Theorem ulp_abs :
  forall x, ulp (Rabs x) = ulp x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, ulp (Rabs x) = ulp x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, ulp (Rabs x) = ulp x

intros x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Rulp (Rabs x) = ulp x

unfold ulp; case (Req_bool_spec x 0); intros H1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1: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
end1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:x = 0%RRabs x = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:x <> 0%Rbpow (cexp beta fexp (Rabs x)) =
bpow (cexp beta fexp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:x <> 0%RRabs x <> 0%R

now rewrite cexp_abs.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH1:x <> 0%RRabs x <> 0%R

now apply Rabs_no_R0.
Qed.

Theorem ulp_ge_0:
  forall x, (0 <= ulp x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 <= ulp x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 <= ulp x)%R

intros x; unfold ulp; case Req_bool_spec; intros.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH:x = 0%R(0 <=
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH:x <> 0%R(0 <= bpow (cexp beta fexp x))%R

case negligible_exp; intros.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH:x = 0%Rz:Z(0 <= bpow (fexp z))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH:x = 0%R(0 <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH:x <> 0%R(0 <= bpow (cexp beta fexp x))%R

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH:x = 0%R(0 <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH:x <> 0%R(0 <= bpow (cexp beta fexp x))%R

apply Rle_refl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RH: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 < x)%R -> F x -> (ulp x <= x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 < x)%R -> F x -> (ulp x <= x)%R

intros x Zx Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(ulp x <= x)%R

rewrite <- (Rmult_1_l (ulp x)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(1 * ulp x <= x)%R

pattern x at 2; rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(1 * ulp x <=
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(1 * bpow (cexp beta fexp x) <=
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xx <> 0%R

2: now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(0 <= bpow (cexp beta fexp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R

apply IZR_le, (Zlt_le_succ 0).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(0 < Ztrunc (scaled_mantissa beta fexp x))%Z

apply gt_0_F2R with beta (cexp beta fexp x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, x <> 0%R -> F x -> (ulp x <= Rabs x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, x <> 0%R -> F x -> (ulp x <= Rabs x)%R

intros x Zx Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RFx:F x(ulp x <= Rabs x)%R

rewrite <- ulp_abs.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RFx:F x(ulp (Rabs x) <= Rabs x)%R

apply ulp_le_id.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RFx:F x(0 < Rabs x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RFx:F xF (Rabs x)

now apply Rabs_pos_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RFx:F xF (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
~ F x ->
round beta fexp Zceil x =
(round beta fexp Zfloor x + ulp x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
~ F x ->
round beta fexp Zceil x =
(round beta fexp Zfloor x + ulp x)%R

intros x Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xround beta fexp Zceil x =
(round beta fexp Zfloor x + ulp x)%R

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xround beta fexp Zceil x =
(round beta fexp Zfloor x + bpow (cexp beta fexp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

unfold round.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xF2R
  {|
  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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xF2R
  {|
  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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

unfold F2R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ 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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ 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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

rewrite Zceil_floor_neq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ 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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xIZR (Zfloor (scaled_mantissa beta fexp x)) <>
scaled_mantissa beta fexp xbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

rewrite plus_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ 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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xIZR (Zfloor (scaled_mantissa beta fexp x)) <>
scaled_mantissa beta fexp xbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ 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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xIZR (Zfloor (scaled_mantissa beta fexp x)) <>
scaled_mantissa beta fexp xbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xIZR (Zfloor (scaled_mantissa beta fexp x)) <>
scaled_mantissa beta fexp x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

intros H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xH:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp xFalse1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

apply Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xH:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp xF x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

unfold generic_format, F2R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xH:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp xx =
(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 |}))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xH:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp xx =
(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

rewrite <- H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xH:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp xx =
(IZR
   (Ztrunc
      (IZR (Zfloor (scaled_mantissa beta fexp x)))) *
 bpow (cexp beta fexp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

rewrite Ztrunc_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xH:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp xx =
(IZR (Zfloor (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

rewrite H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xH:IZR (Zfloor (scaled_mantissa beta fexp x)) =
scaled_mantissa beta fexp xx =
(scaled_mantissa beta fexp x * bpow (cexp beta fexp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

now rewrite scaled_mantissa_mult_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xx <> 0%R

intros V; apply Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xV:x = 0%RF x

rewrite V.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xV:x = 0%RF 0

apply generic_format_0.
Qed.


Theorem ulp_bpow :
  forall e, ulp (bpow e) = bpow (fexp (e + 1)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall e : Z, ulp (bpow e) = bpow (fexp (e + 1))

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall e : Z, ulp (bpow e) = bpow (fexp (e + 1))

intros e.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zulp (bpow e) = bpow (fexp (e + 1))

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow (cexp beta fexp (bpow e)) = bpow (fexp (e + 1))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow e <> 0%R

apply f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zcexp beta fexp (bpow e) = fexp (e + 1)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow e <> 0%R

apply cexp_fexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(bpow (e + 1 - 1) <= Rabs (bpow e) < bpow (e + 1))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow e <> 0%R

rewrite Rabs_pos_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(bpow (e + 1 - 1) <= bpow e < bpow (e + 1))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(0 <= bpow e)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow e <> 0%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(bpow (e + 1 - 1) <= bpow e)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(bpow e < bpow (e + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(0 <= bpow e)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow e <> 0%R

ring_simplify (e + 1 - 1)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(bpow e <= bpow e)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(bpow e < bpow (e + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(0 <= bpow e)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow e <> 0%R

apply Rle_refl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(bpow e < bpow (e + 1))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(0 <= bpow e)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow e <> 0%R

apply bpow_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(e < e + 1)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(0 <= bpow e)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow e <> 0%R

apply Zlt_succ.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(0 <= bpow e)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow e <> 0%R

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zbpow e <> 0%R

apply Rgt_not_eq, Rlt_gt, bpow_gt_0.
Qed.


Lemma generic_format_ulp_0 :
  F (ulp 0).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpF (ulp 0)

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpF (ulp 0)

unfold ulp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpF
  (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpF
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end

case negligible_exp_spec.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexp(forall n : Z, (fexp n < n)%Z) -> F 01 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall n : Z, (n <= fexp n)%Z -> F (bpow (fexp n))

intros _; apply generic_format_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall n : Z, (n <= fexp n)%Z -> F (bpow (fexp n))

intros n H1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn:ZH1:(n <= fexp n)%ZF (bpow (fexp n))

apply generic_format_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn:ZH1:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall e : Z, (ulp 0 <= bpow e)%R -> F (bpow e)

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall e : Z, (ulp 0 <= bpow e)%R -> F (bpow e)

intros e; unfold ulp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= bpow e)%R -> F (bpow e)

case negligible_exp_spec.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Z(forall n : Z, (fexp n < n)%Z) ->
(0 <= bpow e)%R -> F (bpow e)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zforall n : Z,
(n <= fexp n)%Z ->
(bpow (fexp n) <= bpow e)%R -> F (bpow e)

intros H1 _.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:ZH1:forall n : Z, (fexp n < n)%ZF (bpow e)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zforall n : Z,
(n <= fexp n)%Z ->
(bpow (fexp n) <= bpow e)%R -> F (bpow e)

apply generic_format_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:ZH1:forall n : Z, (fexp n < n)%Z(fexp (e + 1) <= e)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zforall n : Z,
(n <= fexp n)%Z ->
(bpow (fexp n) <= bpow e)%R -> F (bpow e)

specialize (H1 (e+1)%Z); omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:Zforall n : Z,
(n <= fexp n)%Z ->
(bpow (fexp n) <= bpow e)%R -> F (bpow e)

intros n H1 H2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RF (bpow e)

apply generic_format_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%R(fexp (e + 1) <= e)%Z

case (Zle_or_lt (e+1) (fexp (e+1))); intros H4.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(e + 1 <= fexp (e + 1))%Z(fexp (e + 1) <= e)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(fexp (e + 1) < e + 1)%Z(fexp (e + 1) <= e)%Z

absurd (e+1 <= e)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(e + 1 <= fexp (e + 1))%Z~ (e + 1 <= e)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(e + 1 <= fexp (e + 1))%Z(e + 1 <= e)%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(fexp (e + 1) < e + 1)%Z(fexp (e + 1) <= e)%Z

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(e + 1 <= fexp (e + 1))%Z(e + 1 <= e)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(fexp (e + 1) < e + 1)%Z(fexp (e + 1) <= e)%Z

apply Z.le_trans with (1:=H4).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(e + 1 <= fexp (e + 1))%Z(fexp (e + 1) <= e)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(fexp (e + 1) < e + 1)%Z(fexp (e + 1) <= e)%Z

replace (fexp (e+1)) with (fexp n).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(e + 1 <= fexp (e + 1))%Z(fexp n <= e)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(e + 1 <= fexp (e + 1))%Zfexp n = fexp (e + 1)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(fexp (e + 1) < e + 1)%Z(fexp (e + 1) <= e)%Z

now apply le_bpow with beta.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(e + 1 <= fexp (e + 1))%Zfexp n = fexp (e + 1)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(fexp (e + 1) < e + 1)%Z(fexp (e + 1) <= e)%Z

now apply fexp_negligible_exp_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe, n:ZH1:(n <= fexp n)%ZH2:(bpow (fexp n) <= bpow e)%RH4:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpExp_not_FTZ fexp -> forall x : R, F (ulp x)

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpExp_not_FTZ fexp -> forall x : R, F (ulp x)

unfold Exp_not_FTZ; intros H x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RF (ulp x)

case (Req_dec x 0); intros Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x = 0%RF (ulp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%RF (ulp x)

rewrite Hx; apply generic_format_ulp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%RF (ulp x)

rewrite (ulp_neq_0 _ Hx).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%RF (bpow (cexp beta fexp x))

apply generic_format_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexp(forall x : R, F (ulp x)) -> Exp_not_FTZ fexp

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexp(forall x : R, F (ulp x)) -> Exp_not_FTZ fexp

intros H e.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall x : R, F (ulp x)e:Z(fexp (fexp e + 1) <= fexp e)%Z

specialize (H (bpow (e-1))).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:ZH:F (ulp (bpow (e - 1)))(fexp (fexp e + 1) <= fexp e)%Z

rewrite ulp_neq_0 in H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:ZH:F (bpow (cexp beta fexp (bpow (e - 1))))(fexp (fexp e + 1) <= fexp e)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:ZH:F (ulp (bpow (e - 1)))bpow (e - 1) <> 0%R

2: apply Rgt_not_eq, bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:ZH:F (bpow (cexp beta fexp (bpow (e - 1))))(fexp (fexp e + 1) <= fexp e)%Z

unfold cexp in H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:ZH:F (bpow (fexp (mag beta (bpow (e - 1)))))(fexp (fexp e + 1) <= fexp e)%Z

rewrite mag_bpow in H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:ZH:F (bpow (fexp (e - 1 + 1)))(fexp (fexp e + 1) <= fexp e)%Z

apply generic_format_bpow_inv' in H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpe:ZH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpExp_not_FTZ fexp -> forall x : R, (ulp 0 <= ulp x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpExp_not_FTZ fexp -> forall x : R, (ulp 0 <= ulp x)%R

unfold Exp_not_FTZ; intros H x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:R(ulp 0 <= ulp x)%R

case (Req_dec x 0); intros Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x = 0%R(ulp 0 <= ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%R(ulp 0 <= ulp x)%R

rewrite Hx; now right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%R(ulp 0 <= ulp x)%R

unfold ulp at 1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%R(match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= ulp x)%R

case negligible_exp_spec'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rnegligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) ->
(match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Z(bpow (fexp n) <= ulp x)%R

rewrite ulp_neq_0; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Z(bpow (fexp n) <= bpow (cexp beta fexp x))%R

apply bpow_le; unfold cexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Z(fexp n <= fexp (mag beta x))%Z

generalize (mag beta x); intros l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zl:mag_prop beta x(fexp n <= fexp l)%Z

case (Zle_or_lt l (fexp l)); intros Hl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zl:mag_prop beta xHl:(l <= fexp l)%Z(fexp n <= fexp l)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zl:mag_prop beta xHl:(fexp l < l)%Z(fexp n <= fexp l)%Z

rewrite (fexp_negligible_exp_eq n l); trivial; apply Z.le_refl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zl:mag_prop beta xHl:(fexp l < l)%Z(fexp n <= fexp l)%Z

case (Zle_or_lt (fexp n) (fexp l)); trivial; intros K.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zl:mag_prop beta xHl:(fexp l < l)%ZK:(fexp l < fexp n)%Z(fexp n <= fexp l)%Z

absurd (fexp n <= fexp l)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zl:mag_prop beta xHl:(fexp l < l)%ZK:(fexp l < fexp n)%Z~ (fexp n <= fexp l)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zl:mag_prop beta xHl:(fexp l < l)%ZK:(fexp l < fexp n)%Z(fexp n <= fexp l)%Z

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zl:mag_prop beta xHl:(fexp l < l)%ZK:(fexp l < fexp n)%Z(fexp n <= fexp l)%Z

apply Z.le_trans with (2:= H _).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zl:mag_prop beta xHl:(fexp l < l)%ZK:(fexp l < fexp n)%Z(fexp n <= fexp (fexp l + 1))%Z

apply Zeq_le, sym_eq, valid_exp; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall e : Z, (fexp (fexp e + 1) <= fexp e)%Zx:RHx:x <> 0%Rn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zl:mag_prop beta xHl:(fexp l < l)%ZK:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexp(forall x : R, (ulp 0 <= ulp x)%R) -> Exp_not_FTZ fexp

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexp(forall x : R, (ulp 0 <= ulp x)%R) -> Exp_not_FTZ fexp

intros H e.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall x : R, (ulp 0 <= ulp x)%Re:Z(fexp (fexp e + 1) <= fexp e)%Z

apply generic_format_bpow_inv' with beta.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall x : R, (ulp 0 <= ulp x)%Re:ZF (bpow (fexp e))

apply generic_format_bpow_ge_ulp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall x : R, (ulp 0 <= ulp x)%Re:Z(ulp 0 <= bpow (fexp e))%R

replace e with ((e-1)+1)%Z by ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall x : R, (ulp 0 <= ulp x)%Re:Z(ulp 0 <= bpow (fexp (e - 1 + 1)))%R

rewrite <- ulp_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:forall x : R, (ulp 0 <= ulp x)%Re: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_exp fexp ->
forall x y : R,
(0 <= x)%R -> (x <= y)%R -> (ulp x <= ulp y)%R

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_exp fexp ->
forall x y : R,
(0 <= x)%R -> (x <= y)%R -> (ulp x <= ulp y)%R

intros Hm x y Hx Hxy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 <= x)%RHxy:(x <= y)%R(ulp x <= ulp y)%R

destruct Hx as [Hx|Hx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%R(ulp x <= ulp y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%R(ulp x <= ulp y)%R

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%R(bpow (cexp beta fexp x) <= ulp y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%R(ulp x <= ulp y)%R

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%R(bpow (cexp beta fexp x) <= bpow (cexp beta fexp y))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Ry <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%R(ulp x <= ulp y)%R

apply bpow_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%R(cexp beta fexp x <= cexp beta fexp y)%Z3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Ry <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%R(ulp x <= ulp y)%R

apply Hm.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%R(mag beta x <= mag beta y)%Z3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Ry <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%R(ulp x <= ulp y)%R

now apply mag_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Ry <> 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%R(ulp x <= ulp y)%R

apply Rgt_not_eq, Rlt_gt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%R(0 < y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%R(ulp x <= ulp y)%R

now apply Rlt_le_trans with (1:=Hx).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:(0 < x)%RHxy:(x <= y)%Rx <> 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%R(ulp x <= ulp y)%R

now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%R(ulp x <= ulp y)%R

rewrite <- Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%R(ulp 0 <= ulp y)%R

apply ulp_ge_ulp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHx:0%R = xHxy:(x <= y)%RExp_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_exp fexp ->
forall x y : R,
(Rabs x <= Rabs y)%R -> (ulp x <= ulp y)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_exp fexp ->
forall x y : R,
(Rabs x <= Rabs y)%R -> (ulp x <= ulp y)%R

intros Hm x y Hxy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHxy:(Rabs x <= Rabs y)%R(ulp x <= ulp y)%R

rewrite <- ulp_abs.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHxy:(Rabs x <= Rabs y)%R(ulp (Rabs x) <= ulp y)%R

rewrite <- (ulp_abs y).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHxy:(Rabs x <= Rabs y)%R(ulp (Rabs x) <= ulp (Rabs y))%R

apply ulp_le_pos; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpx, y:RHxy:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpnegligible_exp = None ->
forall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, round beta fexp rnd x = 0%R -> x = 0%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpnegligible_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%Rx = 0%R

case (Req_dec x 0); try easy; intros Hx2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%Rx = 0%R

absurd (Rabs (round beta fexp rnd x) = 0%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%RRabs (round beta fexp rnd x) <> 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%RRabs (round beta fexp rnd x) = 0%R

2: rewrite Hx, Rabs_R0; easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%RRabs (round beta fexp rnd x) <> 0%R

apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(Rabs (round beta fexp rnd x) > 0)%R

apply Rlt_le_trans with (bpow (mag beta x - 1)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(0 < bpow (mag beta x - 1))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(bpow (mag beta x - 1) <= Rabs (round beta fexp rnd x))%R

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(bpow (mag beta x - 1) <= Rabs (round beta fexp rnd x))%R

apply abs_round_ge_generic; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%RF (bpow (mag beta x - 1))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(bpow (mag beta x - 1) <= Rabs x)%R

apply generic_format_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(bpow (mag beta x - 1) <= Rabs x)%R

case negligible_exp_spec'; [intros (K1,K2)|idtac].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%RK1:negligible_exp = NoneK2:forall n : Z, (fexp n < n)%Z(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(bpow (mag beta x - 1) <= Rabs x)%R

ring_simplify (mag beta x-1+1)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%RK1:negligible_exp = NoneK2:forall n : Z, (fexp n < n)%Z(fexp (mag beta x) <= mag beta x - 1)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(bpow (mag beta x - 1) <= Rabs x)%R

specialize (K2 (mag beta x)); now auto with zarith.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(bpow (mag beta x - 1) <= Rabs x)%R

intros (n,(Hn1,Hn2)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%Rn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(fexp (mag beta x - 1 + 1) <= mag beta x - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2:x <> 0%R(bpow (mag beta x - 1) <= Rabs x)%R

rewrite Hn1 in H; discriminate.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonernd:R -> ZVr:Valid_rnd rndx:RHx:round beta fexp rnd x = 0%RHx2: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 <= x)%R -> pred x = pred_pos x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 <= x)%R -> pred x = pred_pos x

intros x Hx; unfold pred, succ.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%R(- (- x + ulp (- x)))%R = pred_pos x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(- x < 0)%R(- - pred_pos (- - x))%R = pred_pos x

assert (K:(x = 0)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%Rx = 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%RK:x = 0%R(- (- x + ulp (- x)))%R = pred_pos xbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(- x < 0)%R(- - pred_pos (- - x))%R = pred_pos x

apply Rle_antisym; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%R(x <= 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%RK:x = 0%R(- (- x + ulp (- x)))%R = pred_pos xbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(- x < 0)%R(- - pred_pos (- - x))%R = pred_pos x

apply Ropp_le_cancel.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%R(- 0 <= - x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%RK:x = 0%R(- (- x + ulp (- x)))%R = pred_pos xbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(- x < 0)%R(- - pred_pos (- - x))%R = pred_pos x

now rewrite Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%RK:x = 0%R(- (- x + ulp (- x)))%R = pred_pos x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(- x < 0)%R(- - pred_pos (- - x))%R = pred_pos x

rewrite K; unfold pred_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%RK: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)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(- x < 0)%R(- - pred_pos (- - x))%R = pred_pos x

rewrite Req_bool_false.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%RK:x = 0%R(- (- 0 + ulp (- 0)))%R = (0 - ulp 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%RK:x = 0%R0%R <> bpow (mag beta 0 - 1)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(- x < 0)%R(- - pred_pos (- - x))%R = pred_pos x

2: apply Rlt_not_eq, bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(0 <= - x)%RK:x = 0%R(- (- 0 + ulp (- 0)))%R = (0 - ulp 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(- x < 0)%R(- - pred_pos (- - x))%R = pred_pos x

rewrite Ropp_0; ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 <= x)%RHx':(- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 <= x)%R -> succ x = (x + ulp x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 <= x)%R -> succ x = (x + ulp x)%R

intros x Hx; unfold succ.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, succ (- x) = (- pred x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, succ (- x) = (- pred x)%R

intros x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Rsucc (- x) = (- pred x)%R

now apply sym_eq, Ropp_involutive.
Qed.

Theorem pred_opp :
  forall x, pred (-x) = (- succ x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, pred (- x) = (- succ x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, pred (- x) = (- succ x)%R

intros x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Rpred (- x) = (- succ x)%R

unfold pred.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:R(- succ (- - x))%R = (- succ x)%R

now rewrite Ropp_involutive.
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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (x : R) (e : Z),
F x ->
x <> ulp x ->
(bpow e < x)%R -> (bpow e <= x - ulp x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%R(bpow e <= x - ulp x)%R

(* *)
assert (1 <= Ztrunc (scaled_mantissa beta fexp x))%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%R(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <= x - ulp x)%R

assert (0 <  Ztrunc (scaled_mantissa beta fexp x))%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%R(0 < Ztrunc (scaled_mantissa beta fexp x))%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 <= Ztrunc (scaled_mantissa beta fexp x))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%R(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 <= Ztrunc (scaled_mantissa beta fexp x))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <= x - ulp x)%R

rewrite <- Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%R(0 < x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 <= Ztrunc (scaled_mantissa beta fexp x))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <= x - ulp x)%R

apply Rle_lt_trans with (2:=Hx).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%R(0 <= bpow e)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 <= Ztrunc (scaled_mantissa beta fexp x))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <= x - ulp x)%R

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <= x - ulp x)%R

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <= x - ulp x)%R

case (Zle_lt_or_eq _ _ H); intros Hm.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <= x - ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

(* *)
pattern x at 1 ; rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(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)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Zx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

unfold F2R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Zx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Zx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Zx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

rewrite <- Rmult_minus_distr_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <=
 (IZR (Ztrunc (scaled_mantissa beta fexp x)) - 1) *
 bpow (cexp beta fexp x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Zx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

rewrite <- minus_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <=
 IZR (Ztrunc (scaled_mantissa beta fexp x) - 1) *
 bpow (cexp beta fexp x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Zx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

apply bpow_le_F2R_m1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 < Ztrunc (scaled_mantissa beta fexp x))%Z3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |})%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Zx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z(bpow e <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |})%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Zx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

now rewrite <- Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Zx <> 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

apply Rgt_not_eq, Rlt_gt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:(1 < Ztrunc (scaled_mantissa beta fexp x))%Z(0 < x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

apply Rlt_trans with (2:=Hx), bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx':x <> ulp xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(bpow e <= x - ulp x)%R

(* *)
contradict Hx'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)x = ulp x

pattern x at 1; rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)F2R {| Fnum := 1; Fexp := cexp beta fexp x |} = ulp x

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)F2R {| Fnum := 1; Fexp := cexp beta fexp x |} =
bpow (cexp beta fexp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)x <> 0%R

unfold F2R; simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)(1 * bpow (cexp beta fexp x))%R =
bpow (cexp beta fexp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)x <> 0%R

now rewrite Rmult_1_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm:1%Z = Ztrunc (scaled_mantissa beta fexp x)x <> 0%R

apply Rgt_not_eq, Rlt_gt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZFx:F xHx:(bpow e < x)%RH:(1 <= Ztrunc (scaled_mantissa beta fexp x))%ZHm: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (x : R) (e : Z),
(0 < x)%R ->
F x -> (x < bpow e)%R -> (x + ulp x <= bpow e)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R(x + ulp x <= bpow e)%R

pattern x at 1 ; rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} +
 bpow (cexp beta fexp x) <= bpow e)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%Rx <> 0%R

unfold F2R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(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)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%Rx <> 0%R
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) + bpow (cexp beta fexp x) <=
 bpow e)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%Rx <> 0%R

pattern (bpow (cexp beta fexp x)) at 2 ; rewrite <- Rmult_1_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) + 1 * bpow (cexp beta fexp x) <=
 bpow e)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%Rx <> 0%R

rewrite <- Rmult_plus_distr_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R((IZR (Ztrunc (scaled_mantissa beta fexp x)) + 1) *
 bpow (cexp beta fexp x) <= bpow e)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%Rx <> 0%R

rewrite <- plus_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R(IZR (Ztrunc (scaled_mantissa beta fexp x) + 1) *
 bpow (cexp beta fexp x) <= bpow e)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%Rx <> 0%R

apply F2R_p1_le_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R(0 < Ztrunc (scaled_mantissa beta fexp x))%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < 
 bpow e)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%Rx <> 0%R

apply gt_0_F2R with beta (cexp beta fexp x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < 
 bpow e)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%Rx <> 0%R

now rewrite <- Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%R(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < bpow e)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%Rx <> 0%R

now rewrite <- Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Re:ZZx:(0 < x)%RFx:F xHx:(x < bpow e)%Rx <> 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 < x)%R ->
F x -> x <> bpow (mag beta x - 1) -> F (x - ulp x)

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 < x)%R ->
F x -> x <> bpow (mag beta x - 1) -> F (x - ulp x)

intros x Zx Fx Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)F (x - ulp x)

destruct (mag beta x) as (ex, Ex).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (Build_mag_prop beta x ex Ex - 1)F (x - ulp x)

simpl in Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)F (x - ulp x)

specialize (Ex (Rgt_not_eq _ _ Zx)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)F (x - ulp x)

assert (Ex' : (bpow (ex - 1) < x < bpow ex)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)(bpow (ex - 1) < x < bpow ex)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RF (x - ulp x)

rewrite Rabs_pos_eq in Ex.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= x < bpow ex)%RHx:x <> bpow (ex - 1)(bpow (ex - 1) < x < bpow ex)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RF (x - ulp x)

destruct Ex as (H,H'); destruct H; split; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZH:bpow (ex - 1) = xH':(x < bpow ex)%RHx:x <> bpow (ex - 1)(bpow (ex - 1) < x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RF (x - ulp x)

contradict Hx; easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)(0 <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RF (x - ulp x)

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RF (x - ulp x)

unfold generic_format, scaled_mantissa, cexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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 |}1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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 |}1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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 |}2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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 |}2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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 |}2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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 |}))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) <= Rabs (x - ulp x) < bpow ex)%R
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) <= x - ulp x < bpow ex)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) <= x - ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> ulp x3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) < x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> bpow (cexp beta fexp x)4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) < x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

intro H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RH:x = bpow (cexp beta fexp x)False4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) < x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RH:x = bpow (cexp beta fexp x)(ex - 1 < cexp beta fexp x < ex)%Z5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RH:x = bpow (cexp beta fexp x)H0:(ex - 1 < cexp beta fexp x < ex)%ZFalsebeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) < x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RH:x = bpow (cexp beta fexp x)H0:(ex - 1 < cexp beta fexp x < ex)%ZFalse4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) < x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

clear -H0.beta:radixfexp:Z -> Zx:Rex:ZH0:(ex - 1 < cexp beta fexp x < ex)%ZFalse4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) < x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R
 omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) < x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(bpow (ex - 1) < x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

apply Ex'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x < bpow ex)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(x - ulp x <= x + 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

apply Rplus_le_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(- ulp x <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

rewrite <-Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(- ulp x <= - 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

apply Ropp_le_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

apply ulp_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= x - ulp x)%R

apply Rle_0_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(ulp x <= x)%R

pattern x at 2; rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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 |})%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

unfold F2R; simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

pattern (bpow (cexp beta fexp x)) at 1; rewrite <- Rmult_1_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

apply Rmult_le_compat_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= bpow (cexp beta fexp x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

apply IZR_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

assert (0 <  Ztrunc (scaled_mantissa beta fexp x))%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 < Ztrunc (scaled_mantissa beta fexp x))%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RH:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 <= Ztrunc (scaled_mantissa beta fexp x))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

apply gt_0_F2R with beta (cexp beta fexp x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx: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 |})%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RH:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 <= Ztrunc (scaled_mantissa beta fexp x))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

rewrite <- Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 < x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RH:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 <= Ztrunc (scaled_mantissa beta fexp x))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

apply Rle_lt_trans with (2:=proj1 Ex').beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%R(0 <= bpow (ex - 1))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RH:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 <= Ztrunc (scaled_mantissa beta fexp x))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%RH:(0 < Ztrunc (scaled_mantissa beta fexp x))%Z(1 <= Ztrunc (scaled_mantissa beta fexp x))%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 0%R

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RHx:x <> bpow (ex - 1)Ex':(bpow (ex - 1) < x < bpow ex)%Rx <> 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)F (x - bpow (fexp (e - 1)))

pose (f:=(x - bpow (fexp (e - 1)))%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RF (x - bpow (fexp (e - 1)))

fold f.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RF f

assert (He:(fexp (e-1) <= e-1)%Z).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:R(fexp (e - 1) <= e - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) <= e - 1)%ZF f

apply generic_format_bpow_inv with beta; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RF (bpow (e - 1))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) <= e - 1)%ZF f

rewrite <- Hx; assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) <= e - 1)%ZF f

case (Zle_lt_or_eq _ _ He); clear He; intros He.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZF f1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

assert (f = F2R (Float beta (Zpower beta (e-1-(fexp (e-1))) -1) (fexp (e-1))))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%Zf =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}F fbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

unfold f; rewrite Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(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) |}2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}F fbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

unfold F2R; simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(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)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}F fbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

rewrite minus_IZR, IZR_Zpower.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(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)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%Z(0 <= e - 1 - fexp (e - 1))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}F fbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

rewrite Rmult_minus_distr_r, Rmult_1_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(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)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%Z(0 <= e - 1 - fexp (e - 1))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}F fbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

rewrite <- bpow_plus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(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)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%Z(0 <= e - 1 - fexp (e - 1))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}F fbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

now replace (e - 1 - fexp (e - 1) + fexp (e - 1))%Z with (e-1)%Z by ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%Z(0 <= e - 1 - fexp (e - 1))%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}F fbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}F f1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

rewrite H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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) |})1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply generic_format_F2R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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))%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

intros _.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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))%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply Zeq_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply cexp_fexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

rewrite <- H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1 - 1) <= Rabs f < bpow (e - 1))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

unfold f; rewrite Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

rewrite Rabs_right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply Rplus_le_reg_l with (bpow (fexp (e-1))).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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))))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

ring_simplify.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply Rle_trans with (bpow (e - 2) + bpow (e - 2))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 2) + bpow (e - 2) <= bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply Rplus_le_compat ; apply bpow_le ; omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 2) + bpow (e - 2) <= bpow (e - 1))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply Rle_trans with (2*bpow (e - 2))%R;[right; ring|idtac].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(2 * bpow (e - 2) <= bpow (e - 1))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply Rle_trans with (bpow 1*bpow (e - 2))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(2 * bpow (e - 2) <= bpow 1 * bpow (e - 2))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow 1 * bpow (e - 2) <= bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply Rmult_le_compat_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(0 <= bpow (e - 2))%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(2 <= bpow 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow 1 * bpow (e - 2) <= bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(2 <= bpow 1)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow 1 * bpow (e - 2) <= bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

replace (bpow 1) with (IZR beta).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(2 <= IZR beta)%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}IZR beta = bpow 1beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow 1 * bpow (e - 2) <= bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply IZR_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(2 <= beta)%Z5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}IZR beta = bpow 1beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow 1 * bpow (e - 2) <= bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply <- Zle_is_le_bool.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(2 <=? beta)%Z = true5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}IZR beta = bpow 1beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow 1 * bpow (e - 2) <= bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

now destruct beta.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}IZR beta = bpow 14 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow 1 * bpow (e - 2) <= bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}IZR beta = IZR (Z.pow_pos beta 1)4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow 1 * bpow (e - 2) <= bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

unfold Zpower_pos; simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}IZR beta = IZR (beta * 1)4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow 1 * bpow (e - 2) <= bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

now rewrite Zmult_1_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow 1 * bpow (e - 2) <= bpow (e - 1))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

rewrite <- bpow_plus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (1 + (e - 2)) <= bpow (e - 1))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

replace (1+(e-2))%Z with (e-1)%Z by ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) <= bpow (e - 1))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

now right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

rewrite <- Rplus_0_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH: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)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply Rplus_lt_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(- bpow (fexp (e - 1)) < 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

rewrite <- Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(- bpow (fexp (e - 1)) < - 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply Ropp_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(0 < bpow (fexp (e - 1)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (e - 1) - bpow (fexp (e - 1)) >= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply Rle_ge; apply Rle_0_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(bpow (fexp (e - 1)) <= bpow (e - 1))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

apply bpow_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:(fexp (e - 1) < e - 1)%ZH:f =
F2R
  {|
  Fnum := beta ^ (e - 1 - fexp (e - 1)) - 1;
  Fexp := fexp (e - 1) |}(fexp (e - 1) <= e - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF f

replace f with 0%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%ZF 01 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%Z0%R = f

apply generic_format_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%Z0%R = f

unfold f.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%Z0%R = (x - bpow (fexp (e - 1)))%R

rewrite Hx, He.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHx:x = bpow (e - 1)f:=(x - bpow (fexp (e - 1)))%R:RHe:fexp (e - 1) = (e - 1)%Z0%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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 < x)%R -> F x -> F (x + ulp x)

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 < x)%R -> F x -> F (x + ulp x)

intros x Zx Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xF (x + ulp x)

destruct (mag beta x) as (ex, Ex).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%RF (x + ulp x)

specialize (Ex (Rgt_not_eq _ _ Zx)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%RF (x + ulp x)

assert (Ex' := Ex).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%RF (x + ulp x)

rewrite Rabs_pos_eq in Ex'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RF (x + ulp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

destruct (id_p_ulp_le_bpow x ex) ; try easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%RF (x + ulp x)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

unfold generic_format, scaled_mantissa, cexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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)) |}2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite mag_unique with beta (x + ulp x)%R ex.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(x + ulp x)%R =
F2R
  {|
  Fnum := Ztrunc ((x + ulp x) * bpow (- fexp ex));
  Fexp := fexp ex |}3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

pattern x at 1 3 ; rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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 |}3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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 |}4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

unfold scaled_mantissa.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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 |}4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite cexp_fexp with (1 := Ex).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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 |}4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

unfold F2R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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 |}))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite Rmult_plus_distr_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite Rmult_assoc.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite <- bpow_plus, Zplus_opp_r, Rmult_1_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

change (bpow 0) with 1%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite <- plus_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite Ztrunc_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite plus_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite Rmult_plus_distr_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(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))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

now rewrite Rmult_1_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%Rx <> 0%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + ulp x) < bpow ex)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite Rabs_pos_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= x + ulp x < bpow ex)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(bpow (ex - 1) <= x + ulp x)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(x + ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

apply Rle_trans with (1 := proj1 Ex').beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(x <= x + ulp x)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(x + ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

pattern x at 1 ; rewrite <- Rplus_0_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(x + 0 <= x + ulp x)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(x + ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

apply Rplus_le_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= ulp x)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(x + ulp x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

apply ulp_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(x + ulp x < bpow ex)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

exact H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= x + ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

apply Rplus_le_le_0_compat.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x < bpow ex)%R(0 <= ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

apply ulp_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (x + ulp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exF (bpow ex)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

apply generic_format_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow ex(fexp (ex + 1) <= ex)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

apply valid_exp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow ex(fexp ex < ex)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

destruct (Zle_or_lt ex (fexp ex)) ; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%Z(fexp ex < ex)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

elim Rlt_not_le with (1 := Zx).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%Z(x <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%Z(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

replace (Ztrunc (scaled_mantissa beta fexp x)) with Z0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%Z(F2R {| Fnum := 0; Fexp := cexp beta fexp x |} <= 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%Z0%Z = Ztrunc (scaled_mantissa beta fexp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite F2R_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%Z(0 <= 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%Z0%Z = Ztrunc (scaled_mantissa beta fexp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

apply Rle_refl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%Z0%Z = Ztrunc (scaled_mantissa beta fexp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

unfold scaled_mantissa.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%Z0%Z = Ztrunc (x * bpow (- cexp beta fexp x))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite cexp_fexp with (1 := Ex).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%Z0%Z = Ztrunc (x * bpow (- fexp ex))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

destruct (mantissa_small_pos beta fexp x ex) ; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R0%Z = Ztrunc (x * bpow (- fexp ex))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

rewrite Ztrunc_floor.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R0%Z = Zfloor (x * bpow (- fexp ex))2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R(0 <= x * bpow (- fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

apply sym_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%RZfloor (x * bpow (- fexp ex)) = 0%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R(0 <= x * bpow (- fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

apply Zfloor_imp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R(0 <= x * bpow (- fexp ex) < IZR (0 + 1))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R(0 <= x * bpow (- fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R(0 <= x * bpow (- fexp ex))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R(x * bpow (- fexp ex) < IZR (0 + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R(0 <= x * bpow (- fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R(x * bpow (- fexp ex) < IZR (0 + 1))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R(0 <= x * bpow (- fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

exact H2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:(bpow (ex - 1) <= Rabs x < bpow ex)%REx':(bpow (ex - 1) <= x < bpow ex)%RH:(x + ulp x)%R = bpow exH0:(ex <= fexp ex)%ZH1:(0 < x * bpow (- fexp ex))%RH2:(x * bpow (- fexp ex) < 1)%R(0 <= x * bpow (- fexp ex))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, Ex':(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx, 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> (0 < x)%R -> F (pred_pos x)

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> (0 < x)%R -> F (pred_pos x)

intros x Fx Zx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(0 < x)%RF (pred_pos x)

unfold pred_pos; case Req_bool_spec; intros H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(0 < x)%RH:x = bpow (mag beta x - 1)F (x - bpow (fexp (mag beta x - 1)))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(0 < x)%RH:x <> bpow (mag beta x - 1)F (x - ulp x)

now apply generic_format_pred_aux2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(0 < x)%RH: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> F (succ x)

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> F (succ x)

intros x Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xF (succ x)

unfold succ; case Rle_bool_spec; intros Zx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(0 <= x)%RF (x + ulp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(x < 0)%RF (- pred_pos (- x))

destruct Zx as [Zx|Zx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(0 < x)%RF (x + ulp x)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:0%R = xF (x + ulp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(x < 0)%RF (- pred_pos (- x))

now apply generic_format_succ_aux1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:0%R = xF (x + ulp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(x < 0)%RF (- pred_pos (- x))

rewrite <- Zx, Rplus_0_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:0%R = xF (ulp 0)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(x < 0)%RF (- pred_pos (- x))

apply generic_format_ulp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(x < 0)%RF (- pred_pos (- x))

apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(x < 0)%RF (pred_pos (- x))

apply generic_format_pred_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(x < 0)%RF (- x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(x < 0)%R(0 < - x)%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xZx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> F (pred x)

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> F (pred x)

intros x Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xF (pred x)

unfold pred.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xF (- succ (- x))

apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xF (succ (- x))

apply generic_format_succ.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xF (- x)

now apply generic_format_opp.
Qed.

Lemma pred_pos_lt_id :
  forall x, (x <> 0)%R ->
  (pred_pos x < x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, x <> 0%R -> (pred_pos x < x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, x <> 0%R -> (pred_pos x < x)%R

intros x Zx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%R(pred_pos x < x)%R

unfold pred_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x = bpow (mag beta x - 1)(x - bpow (fexp (mag beta x - 1)) < x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x <> bpow (mag beta x - 1)(x - ulp x < x)%R

(* *)
rewrite <- Rplus_0_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x = bpow (mag beta x - 1)(x - bpow (fexp (mag beta x - 1)) < x + 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x <> bpow (mag beta x - 1)(x - ulp x < x)%R

apply Rplus_lt_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x = bpow (mag beta x - 1)(- bpow (fexp (mag beta x - 1)) < 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x <> bpow (mag beta x - 1)(x - ulp x < x)%R

rewrite <- Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x = bpow (mag beta x - 1)(- bpow (fexp (mag beta x - 1)) < - 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x <> bpow (mag beta x - 1)(x - ulp x < x)%R

apply Ropp_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x = bpow (mag beta x - 1)(0 < bpow (fexp (mag beta x - 1)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x <> bpow (mag beta x - 1)(x - ulp x < x)%R

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x <> bpow (mag beta x - 1)(x - ulp x < x)%R

(* *)
rewrite <- Rplus_0_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x <> bpow (mag beta x - 1)(x - ulp x < x + 0)%R

apply Rplus_lt_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x <> bpow (mag beta x - 1)(- ulp x < 0)%R

rewrite <- Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x <> bpow (mag beta x - 1)(- ulp x < - 0)%R

apply Ropp_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH:x <> bpow (mag beta x - 1)(0 < ulp x)%R

rewrite ulp_neq_0; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RH: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, x <> 0%R -> (x < succ x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, x <> 0%R -> (x < succ x)%R

intros x Zx; unfold succ.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(0 <= x)%R(x < x + ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(x < 0)%R(x < - pred_pos (- x))%R

pattern x at 1; rewrite <- (Rplus_0_r x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(0 <= x)%R(x + 0 < x + ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(x < 0)%R(x < - pred_pos (- x))%R

apply Rplus_lt_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(0 <= x)%R(0 < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(x < 0)%R(x < - pred_pos (- x))%R

rewrite ulp_neq_0; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(0 <= x)%R(0 < bpow (cexp beta fexp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(x < 0)%R(x < - pred_pos (- x))%R

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(x < 0)%R(x < - pred_pos (- x))%R

pattern x at 1; rewrite <- (Ropp_involutive x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(x < 0)%R(- - x < - pred_pos (- x))%R

apply Ropp_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(x < 0)%R(pred_pos (- x) < - x)%R

apply pred_pos_lt_id.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%RHx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, x <> 0%R -> (pred x < x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, x <> 0%R -> (pred x < x)%R

intros x Zx; unfold pred.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%R(- succ (- x) < x)%R

pattern x at 2; rewrite <- (Ropp_involutive x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%R(- succ (- x) < - - x)%R

apply Ropp_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%R(- x < succ (- x))%R

apply succ_gt_id.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:x <> 0%R(- x)%R <> 0%R

auto with real.
Qed.

Theorem succ_ge_id :
  forall x, (x <= succ x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (x <= succ x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (x <= succ x)%R

intros x; case (Req_dec x 0).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Rx = 0%R -> (x <= succ x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Rx <> 0%R -> (x <= succ x)%R

intros V; rewrite V.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RV:x = 0%R(0 <= succ 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Rx <> 0%R -> (x <= succ x)%R

unfold succ; rewrite Rle_bool_true;[idtac|now right].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RV:x = 0%R(0 <= 0 + ulp 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Rx <> 0%R -> (x <= succ x)%R

rewrite Rplus_0_l; apply ulp_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:Rx <> 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (pred x <= x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (pred x <= x)%R

intros x; unfold pred.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:R(- succ (- x) <= x)%R

pattern x at 2; rewrite <- (Ropp_involutive x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:R(- succ (- x) <= - - x)%R

apply Ropp_le_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 < x)%R -> F x -> (0 <= pred_pos x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 < x)%R -> F x -> (0 <= pred_pos x)%R

intros x Zx Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(0 <= pred_pos x)%R

unfold pred_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x = bpow (mag beta x - 1)(0 <= x - bpow (fexp (mag beta x - 1)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(0 <= x - ulp x)%R

(* *)
apply Rle_0_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x = bpow (mag beta x - 1)(bpow (fexp (mag beta x - 1)) <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(0 <= x - ulp x)%R

rewrite H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x = bpow (mag beta x - 1)(bpow (fexp (mag beta (bpow (mag beta x - 1)) - 1)) <=
 bpow (mag beta x - 1))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(0 <= x - ulp x)%R

apply bpow_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x = bpow (mag beta x - 1)(fexp (mag beta (bpow (mag beta x - 1)) - 1) <=
 mag beta x - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(0 <= x - ulp x)%R

destruct (mag beta x) as (ex,Ex) ; simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%RH:x = bpow (Build_mag_prop beta x ex Ex - 1)(fexp (mag beta (bpow (ex - 1)) - 1) <= ex - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(0 <= x - ulp x)%R

rewrite mag_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%RH:x = bpow (Build_mag_prop beta x ex Ex - 1)(fexp (ex - 1 + 1 - 1) <= ex - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(0 <= x - ulp x)%R

ring_simplify (ex - 1 + 1 - 1)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%RH:x = bpow (Build_mag_prop beta x ex Ex - 1)(fexp (ex - 1) <= ex - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(0 <= x - ulp x)%R

apply generic_format_bpow_inv with beta; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%RH:x = bpow (Build_mag_prop beta x ex Ex - 1)F (bpow (ex - 1))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(0 <= x - ulp x)%R

simpl in H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZEx:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%RH:x = bpow (ex - 1)F (bpow (ex - 1))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(0 <= x - ulp x)%R

rewrite <- H; assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(0 <= x - ulp x)%R

apply Rle_0_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 < x)%R -> F x -> (0 <= pred x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, (0 < x)%R -> F x -> (0 <= pred x)%R

intros x Zx Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(0 <= pred x)%R

rewrite pred_eq_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(0 <= pred_pos x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(0 <= x)%R

now apply pred_pos_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 < x)%R ->
F x ->
x <> bpow (mag beta x - 1) ->
(x - ulp x + ulp (x - ulp x))%R = x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)(x - ulp x + ulp (x - ulp x))%R = x

replace (ulp (x - ulp x)) with (ulp x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)(x - ulp x + ulp x)%R = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)ulp x = ulp (x - ulp x)

ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)ulp x = ulp (x - ulp x)

assert (H : x <> 0%R) by now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%Rulp x = ulp (x - ulp x)

assert (H' : x <> bpow (cexp beta fexp x)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%Rx <> bpow (cexp beta fexp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

unfold cexp ; intros M.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RM:x = bpow (fexp (mag beta x))False1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

case_eq (mag beta x); intros ex Hex T.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RM:x = bpow (fexp (mag beta x))ex:ZHex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%RT:mag beta x = Build_mag_prop beta x ex HexFalse1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

assert (Lex:(mag_val beta x (mag beta x) = ex)%Z).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RM:x = bpow (fexp (mag beta x))ex:ZHex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%RT:mag beta x = Build_mag_prop beta x ex Hexmag beta x = ex2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RM:x = bpow (fexp (mag beta x))ex:ZHex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%RT:mag beta x = Build_mag_prop beta x ex HexLex:mag beta x = exFalsebeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

rewrite T; reflexivity.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RM:x = bpow (fexp (mag beta x))ex:ZHex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%RT:mag beta x = Build_mag_prop beta x ex HexLex:mag beta x = exFalse1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

rewrite Lex in *.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%RT:mag beta x = Build_mag_prop beta x ex HexLex:mag beta x = exFalse1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

clear T; simpl in *; specialize (Hex H).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = exFalse1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

rewrite Rabs_pos_eq in Hex by now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = exFalse1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

assert (ex - 1 < fexp ex < ex)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(ex - 1 < fexp ex < ex)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = exH0:(ex - 1 < fexp ex < ex)%ZFalsebeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

  split ; apply (lt_bpow beta) ; rewrite <- M by easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) < x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = exH0:(ex - 1 < fexp ex < ex)%ZFalsebeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

  lra.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x < bpow ex)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = exH0:(ex - 1 < fexp ex < ex)%ZFalsebeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

  apply Hex.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RM:x = bpow (fexp ex)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = exH0:(ex - 1 < fexp ex < ex)%ZFalse1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ulp x = ulp (x - ulp x)

rewrite 2!ulp_neq_0 by lra.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ex:ZHex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%RT:mag beta x = Build_mag_prop beta x ex HexBuild_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ex:ZHex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%RT:mag beta x = Build_mag_prop beta x ex Hexmag beta x = ex1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ex:ZHex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%RT:mag beta x = Build_mag_prop beta x ex HexLex:mag beta x = exBuild_mag_prop beta x ex Hex =
mag beta
  (x - bpow (fexp (Build_mag_prop beta x ex Hex)))

rewrite T; reflexivity.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xHx:x <> bpow (mag beta x - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)ex:ZHex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%RT:mag beta x = Build_mag_prop beta x ex HexLex:mag beta x = exBuild_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = exex = mag beta (x - bpow (fexp ex))

specialize (Hex H).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = exex = mag beta (x - bpow (fexp ex))

apply sym_eq, mag_unique.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= Rabs (x - bpow (fexp ex)) < bpow ex)%R

rewrite Rabs_right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex) < bpow ex)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

rewrite Rabs_right in Hex.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex) < bpow ex)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

2: apply Rle_ge; apply Rlt_le; easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex) < bpow ex)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

destruct Hex as ([H1|H1],H2).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:(bpow (ex - 1) < x)%RH2:(x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:bpow (ex - 1) = xH2:(x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

apply Rle_trans with (x-ulp x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:(bpow (ex - 1) < x)%RH2:(x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - ulp x)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:(bpow (ex - 1) < x)%RH2:(x < bpow ex)%RLex:mag beta x = ex(x - ulp x <= x - bpow (fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:bpow (ex - 1) = xH2:(x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

apply id_m_ulp_ge_bpow; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:(bpow (ex - 1) < x)%RH2:(x < bpow ex)%RLex:mag beta x = exx <> ulp x4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:(bpow (ex - 1) < x)%RH2:(x < bpow ex)%RLex:mag beta x = ex(x - ulp x <= x - bpow (fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:bpow (ex - 1) = xH2:(x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

rewrite ulp_neq_0; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:(bpow (ex - 1) < x)%RH2:(x < bpow ex)%RLex:mag beta x = ex(x - ulp x <= x - bpow (fexp ex))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:bpow (ex - 1) = xH2:(x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

rewrite ulp_neq_0; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:(bpow (ex - 1) < x)%RH2:(x < bpow ex)%RLex:mag beta x = ex(x - bpow (cexp beta fexp x) <= x - bpow (fexp ex))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:bpow (ex - 1) = xH2:(x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

right; unfold cexp; now rewrite Lex.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)H1:bpow (ex - 1) = xH2:(x < bpow ex)%RLex:mag beta x = ex(bpow (ex - 1) <= x - bpow (fexp ex))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

lra.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) < bpow ex)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

apply Rle_lt_trans with (2:=proj2 Hex).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

rewrite <- Rplus_0_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) <= x + 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

apply Rplus_le_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(- bpow (fexp ex) <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

rewrite <- Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(- bpow (fexp ex) <= - 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

apply Ropp_le_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= x < bpow ex)%RLex:mag beta x = ex(0 <= bpow (fexp ex))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(x - bpow (fexp ex) >= 0)%R

apply Rle_ge.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(0 <= x - bpow (fexp ex))%R

apply Rle_0_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(bpow (fexp ex) <= x)%R

rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(bpow (fexp ex) <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (fexp (mag beta x)))%R

rewrite Lex.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(0 <= bpow (fexp ex))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(1 <= IZR (Ztrunc (scaled_mantissa beta fexp x)))%R

apply IZR_le, (Zlt_le_succ 0).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex:mag beta x = ex(0 < Ztrunc (scaled_mantissa beta fexp x))%Z

apply gt_0_F2R with beta (cexp beta fexp x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xex:ZHx:x <> bpow (ex - 1)H:x <> 0%RH':x <> bpow (cexp beta fexp x)Hex:(bpow (ex - 1) <= Rabs x < bpow ex)%RLex: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%R(x - bpow (fexp (e - 1)) + bpow (fexp (e - 1)))%R = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%Rbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%Rbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

assert (He:(fexp (e-1) <= e-1)%Z).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%R(fexp (e - 1) <= e - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) <= e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply generic_format_bpow_inv with beta; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RF (bpow (e - 1))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) <= e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

rewrite <- Hxe; assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) <= e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

case (Zle_lt_or_eq _ _ He); clear He; intros He.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

(* *)
rewrite ulp_neq_0; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Zbpow (fexp (e - 1)) =
bpow (cexp beta fexp (x - bpow (fexp (e - 1))))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Zfexp (e - 1) =
cexp beta fexp (x - bpow (fexp (e - 1)))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

unfold cexp ; apply f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(e - 1)%Z = mag beta (x - bpow (fexp (e - 1)))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply sym_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Zmag beta (x - bpow (fexp (e - 1))) = (e - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply mag_unique.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (e - 1 - 1) <= Rabs (x - bpow (fexp (e - 1))) <
 bpow (e - 1))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

rewrite Rabs_right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (e - 1 - 1) <= x - bpow (fexp (e - 1)) <
 bpow (e - 1))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (e - 1 - 1) <= x - bpow (fexp (e - 1)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply Rplus_le_reg_l with (bpow (fexp (e-1))).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (fexp (e - 1)) + bpow (e - 1 - 1) <=
 bpow (fexp (e - 1)) + (x - bpow (fexp (e - 1))))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

ring_simplify.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (fexp (e - 1)) + bpow (e - 1 - 1) <= x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply Rle_trans with (bpow (e - 2) + bpow (e - 2))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (fexp (e - 1)) + bpow (e - 1 - 1) <=
 bpow (e - 2) + bpow (e - 2))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (e - 2) + bpow (e - 2) <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply Rplus_le_compat; apply bpow_le; omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (e - 2) + bpow (e - 2) <= x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply Rle_trans with (2*bpow (e - 2))%R;[right; ring|idtac].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(2 * bpow (e - 2) <= x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply Rle_trans with (bpow 1*bpow (e - 2))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(2 * bpow (e - 2) <= bpow 1 * bpow (e - 2))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow 1 * bpow (e - 2) <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply Rmult_le_compat_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(0 <= bpow (e - 2))%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(2 <= bpow 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow 1 * bpow (e - 2) <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(2 <= bpow 1)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow 1 * bpow (e - 2) <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

replace (bpow 1) with (IZR beta).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(2 <= IZR beta)%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%ZIZR beta = bpow 1beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow 1 * bpow (e - 2) <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply IZR_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(2 <= beta)%Z5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%ZIZR beta = bpow 1beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow 1 * bpow (e - 2) <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply <- Zle_is_le_bool.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(2 <=? beta)%Z = true5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%ZIZR beta = bpow 1beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow 1 * bpow (e - 2) <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

now destruct beta.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%ZIZR beta = bpow 14 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow 1 * bpow (e - 2) <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%ZIZR beta = IZR (Z.pow_pos beta 1)4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow 1 * bpow (e - 2) <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

unfold Zpower_pos; simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%ZIZR beta = IZR (beta * 1)4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow 1 * bpow (e - 2) <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

now rewrite Zmult_1_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow 1 * bpow (e - 2) <= x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

rewrite <- bpow_plus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (1 + (e - 2)) <= x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

replace (1+(e-2))%Z with (e-1)%Z by ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (e - 1) <= x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

now right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) < bpow (e - 1))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

rewrite <- Rplus_0_r, Hxe.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (e - 1) - bpow (fexp (e - 1)) < bpow (e - 1) + 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply Rplus_lt_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(- bpow (fexp (e - 1)) < 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

rewrite <- Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(- bpow (fexp (e - 1)) < - 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply Ropp_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(0 < bpow (fexp (e - 1)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(x - bpow (fexp (e - 1)) >= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply Rle_ge; apply Rle_0_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (fexp (e - 1)) <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

rewrite Hxe.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(bpow (fexp (e - 1)) <= bpow (e - 1))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

apply bpow_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:(fexp (e - 1) < e - 1)%Z(fexp (e - 1) <= e - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe:x = bpow (e - 1)Zp:(x - bpow (fexp (e - 1)))%R <> 0%RHe:fexp (e - 1) = (e - 1)%Zbpow (fexp (e - 1)) = ulp (x - bpow (fexp (e - 1)))

(* *)
contradict Zp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZHxe: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%Rulp 0 = x

assert (H3:(x = bpow (fexp (e - 1)))).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%Rx = bpow (fexp (e - 1))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))ulp 0 = x

now apply Rminus_diag_uniq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))ulp 0 = x

assert (H4: (fexp (e-1) = e-1)%Z).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))fexp (e - 1) = (e - 1)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zulp 0 = x

apply bpow_inj with beta.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))bpow (fexp (e - 1)) = bpow (e - 1)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zulp 0 = x

now rewrite <- H1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zulp 0 = x

unfold ulp; rewrite Req_bool_true; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zmatch negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end = x

case negligible_exp_spec.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Z(forall n : Z, (fexp n < n)%Z) -> 0%R = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zforall n : Z, (n <= fexp n)%Z -> bpow (fexp n) = x

intros K.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%ZK:forall n : Z, (fexp n < n)%Z0%R = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zforall n : Z, (n <= fexp n)%Z -> bpow (fexp n) = x

specialize (K (e-1)%Z).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%ZK:(fexp (e - 1) < e - 1)%Z0%R = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zforall n : Z, (n <= fexp n)%Z -> bpow (fexp n) = x

contradict K; omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zforall n : Z, (n <= fexp n)%Z -> bpow (fexp n) = x

intros n Hn.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zn:ZHn:(n <= fexp n)%Zbpow (fexp n) = x

rewrite H3; apply f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zn:ZHn:(n <= fexp n)%Zfexp n = fexp (e - 1)

case (Zle_or_lt n (e-1)); intros H6.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zn:ZHn:(n <= fexp n)%ZH6:(n <= e - 1)%Zfexp n = fexp (e - 1)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zn:ZHn:(n <= fexp n)%ZH6:(e - 1 < n)%Zfexp n = fexp (e - 1)

apply valid_exp; omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xe:=mag_val beta x (mag beta x):ZH1:x = bpow (e - 1)H2:(x - bpow (fexp (e - 1)))%R = 0%RH3:x = bpow (fexp (e - 1))H4:fexp (e - 1) = (e - 1)%Zn:ZHn:(n <= fexp n)%ZH6:(e - 1 < n)%Zfexp 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 < x)%R ->
F x -> (pred_pos x + ulp (pred_pos x))%R = x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 < x)%R ->
F x -> (pred_pos x + ulp (pred_pos x))%R = x

intros x Zx Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F x(pred_pos x + ulp (pred_pos x))%R = x

unfold pred_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x = bpow (mag beta x - 1)(x - bpow (fexp (mag beta x - 1)) +
 ulp (x - bpow (fexp (mag beta x - 1))))%R = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH: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 = x2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH: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 = xbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(x - ulp x + ulp (x - ulp x))%R = x

rewrite H1, Rplus_0_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x = bpow (mag beta x - 1)H1:(x - bpow (fexp (mag beta x - 1)))%R = 0%Rulp 0 = x2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH: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 = xbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(x - ulp x + ulp (x - ulp x))%R = x

now apply pred_pos_plus_ulp_aux3.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH: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 = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH:x <> bpow (mag beta x - 1)(x - ulp x + ulp (x - ulp x))%R = x

now apply pred_pos_plus_ulp_aux2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xH: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 < x)%R -> F x -> (pred x + ulp (pred x))%R = x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 < x)%R -> F x -> (pred x + ulp (pred x))%R = x

intros x Hx Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F x(pred x + ulp (pred x))%R = x

rewrite pred_eq_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F x(pred_pos x + ulp (pred_pos x))%R = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F x(0 <= x)%R

now apply pred_pos_plus_ulp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 < x)%R ->
F x ->
forall eps : R,
(0 <= eps < ulp x)%R ->
mag beta (x + eps) = mag beta x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rmag beta (x + eps) = mag beta x

destruct (mag beta x) as (ex, He).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%Rmag beta (x + eps) = Build_mag_prop beta x ex He

simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:x <> 0%R -> (bpow (ex - 1) <= Rabs x < bpow ex)%Rmag beta (x + eps) = ex

specialize (He (Rgt_not_eq _ _ Zx)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%Rmag beta (x + eps) = ex

apply mag_unique.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(bpow (ex - 1) <= Rabs (x + eps) < bpow ex)%R

rewrite Rabs_pos_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(bpow (ex - 1) <= x + eps < bpow ex)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

rewrite Rabs_pos_eq in He.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(bpow (ex - 1) <= x + eps < bpow ex)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(bpow (ex - 1) <= x + eps)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(x + eps < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

apply Rle_trans with (1 := proj1 He).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(x <= x + eps)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(x + eps < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

pattern x at 1 ; rewrite <- Rplus_0_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(x + 0 <= x + eps)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(x + eps < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

now apply Rplus_le_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(x + eps < bpow ex)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

apply Rlt_le_trans with (x + ulp x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(x + eps < x + ulp x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(x + ulp x <= bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

now apply Rplus_lt_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(x + ulp x <= bpow ex)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

pattern x at 1 ; rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |} + ulp x <= bpow ex)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(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)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

unfold F2R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(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)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(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)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(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)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

rewrite <- Rmult_plus_distr_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R((IZR (Ztrunc (scaled_mantissa beta fexp x)) + 1) *
 bpow (cexp beta fexp x) <= bpow ex)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

rewrite <- plus_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(IZR (Ztrunc (scaled_mantissa beta fexp x) + 1) *
 bpow (cexp beta fexp x) <= bpow ex)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

apply F2R_p1_le_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(0 < Ztrunc (scaled_mantissa beta fexp x))%Z4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < 
 bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

apply gt_0_F2R with beta (cexp beta fexp x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < 
 bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

now rewrite <- Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%R(F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := fexp (mag beta x) |} < bpow ex)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

now rewrite <- Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= x < bpow ex)%Rx <> 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x + eps)%R

apply Rplus_le_le_0_compat.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 <= eps)%R

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rex:ZHe:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 <= x)%R ->
F x ->
forall eps : R,
(0 <= eps < ulp x)%R ->
round beta fexp Zfloor (x + eps) = x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

destruct Zx as [Zx|Zx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

(* . 0 < x *)
pattern x at 2 ; rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) =
F2R
  {|
  Fnum := Ztrunc (scaled_mantissa beta fexp x);
  Fexp := cexp beta fexp x |}1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

unfold round.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%RF2R
  {|
  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 |}1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

unfold scaled_mantissa.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%RF2R
  {|
  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 |}1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%RF2R
  {|
  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 |}1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

unfold cexp at 1 2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%RF2R
  {|
  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 |}1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite mag_plus_eps ; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%RF2R
  {|
  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 |}1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply (f_equal (fun m => F2R (Float beta m _))).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%RZfloor ((x + eps) * bpow (- fexp (mag beta x))) =
Ztrunc (x * bpow (- cexp beta fexp x))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite Ztrunc_floor.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%RZfloor ((x + eps) * bpow (- fexp (mag beta x))) =
Zfloor (x * bpow (- cexp beta fexp x))2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply Zfloor_imp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(IZR (Zfloor (x * bpow (- cexp beta fexp x))) <=
 (x + eps) * bpow (- fexp (mag beta x)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply (Rle_trans _ _ _ (Zfloor_lb _)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(x * bpow (- cexp beta fexp x) <=
 (x + eps) * bpow (- fexp (mag beta x)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply Rmult_le_compat_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= bpow (- fexp (mag beta x)))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(x <= x + eps)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(x <= x + eps)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

pattern x at 1 ; rewrite <- Rplus_0_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(x + 0 <= x + eps)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

now apply Rplus_le_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + eps) * bpow (- fexp (mag beta x)) <
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply Rlt_le_trans with ((x + ulp x) * bpow (- cexp beta fexp x))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + eps) * bpow (- fexp (mag beta x)) <
 (x + ulp x) * bpow (- cexp beta fexp x))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + ulp x) * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply Rmult_lt_compat_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 < bpow (- cexp beta fexp x))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(x + eps < x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + ulp x) * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(x + eps < x + ulp x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + ulp x) * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

now apply Rplus_lt_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R((x + ulp x) * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x)) + 1))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite Rmult_plus_distr_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite plus_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(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)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply Rplus_le_compat.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(x * bpow (- cexp beta fexp x) <=
 IZR (Zfloor (x * bpow (- cexp beta fexp x))))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(ulp x * bpow (- cexp beta fexp x) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

pattern x at 1 3 ; rewrite Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(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))))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(ulp x * bpow (- cexp beta fexp x) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

unfold F2R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(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))))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(ulp x * bpow (- cexp beta fexp x) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x
 simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(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))))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(ulp x * bpow (- cexp beta fexp x) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite Rmult_assoc.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(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)))))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(ulp x * bpow (- cexp beta fexp x) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite <- bpow_plus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(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))))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(ulp x * bpow (- cexp beta fexp x) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite Zplus_opp_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(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)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(ulp x * bpow (- cexp beta fexp x) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite Rmult_1_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(IZR (Ztrunc (scaled_mantissa beta fexp x)) <=
 IZR
   (Zfloor
      (IZR (Ztrunc (scaled_mantissa beta fexp x)))))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(ulp x * bpow (- cexp beta fexp x) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite Zfloor_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(IZR (Ztrunc (scaled_mantissa beta fexp x)) <=
 IZR (Ztrunc (scaled_mantissa beta fexp x)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(ulp x * bpow (- cexp beta fexp x) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply Rle_refl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(ulp x * bpow (- cexp beta fexp x) <= 1)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(bpow (cexp beta fexp x) * bpow (- cexp beta fexp x) <=
 1)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%Rx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

2: now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(bpow (cexp beta fexp x) * bpow (- cexp beta fexp x) <=
 1)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite <- bpow_plus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(bpow (cexp beta fexp x + - cexp beta fexp x) <= 1)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

rewrite Zplus_opp_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(bpow 0 <= 1)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply Rle_refl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x * bpow (- cexp beta fexp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply Rmult_le_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= bpow (- cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 < x)%RFx:F xeps:RHeps:(0 <= eps < ulp x)%R(0 <= bpow (- cexp beta fexp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp x)%Rround beta fexp Zfloor (x + eps) = x

(* . x=0 *)
rewrite <- Zx, Rplus_0_l; rewrite <- Zx in Heps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%Rround beta fexp Zfloor eps = 0%R

case (proj1 Heps); intros P.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:(0 < eps)%Rround beta fexp Zfloor eps = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

unfold round, scaled_mantissa, cexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:(0 < eps)%RF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

revert Heps; unfold ulp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(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%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

rewrite Req_bool_true; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(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%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

case negligible_exp_spec.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(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%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rforall 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%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

intros _ (H1,H2).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%RH1:(0 <= eps)%RH2:(eps < 0)%RF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rforall 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%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

exfalso ; lra.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rforall 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%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

intros n Hn H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

assert (fexp (mag beta eps) = fexp n).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%Rfexp (mag beta eps) = fexp n2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

apply valid_exp; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%R(mag beta eps <= fexp n)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

assert(mag beta eps-1 < fexp n)%Z;[idtac|omega].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%R(mag beta eps - 1 < fexp n)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

apply lt_bpow with beta.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%R(bpow (mag beta eps - 1) < bpow (fexp n))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

apply Rle_lt_trans with (2:=proj2 H).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%R(bpow (mag beta eps - 1) <= eps)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

destruct (mag beta eps) as (e,He).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%Re:ZHe:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R(bpow (Build_mag_prop beta eps e He - 1) <= eps)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

simpl; rewrite Rabs_pos_eq in He.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%Re:ZHe:eps <> 0%R -> (bpow (e - 1) <= eps < bpow e)%R(bpow (e - 1) <= eps)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%Re:ZHe:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R(0 <= eps)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

now apply He, Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%Re:ZHe:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R(0 <= eps)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

now left.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zfloor (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

replace (Zfloor (eps * bpow (- fexp (mag beta eps)))) with 0%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R {| Fnum := 0; Fexp := fexp (mag beta eps) |} = 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n0%Z = Zfloor (eps * bpow (- fexp (mag beta eps)))beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

unfold F2R; simpl; ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n0%Z = Zfloor (eps * bpow (- fexp (mag beta eps)))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

apply sym_eq, Zfloor_imp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(0 <= eps * bpow (- fexp (mag beta eps)) < IZR (0 + 1))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(0 <= eps * bpow (- fexp (mag beta eps)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) < IZR (0 + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

apply Rmult_le_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(0 <= eps)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(0 <= bpow (- fexp (mag beta eps)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) < IZR (0 + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

now left.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(0 <= bpow (- fexp (mag beta eps)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) < IZR (0 + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) < IZR (0 + 1))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

apply Rmult_lt_reg_r with (bpow (fexp n)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(0 < bpow (fexp n))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) * bpow (fexp n) <
 IZR (0 + 1) * bpow (fexp n))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) * bpow (fexp n) <
 IZR (0 + 1) * bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

rewrite Rmult_assoc, <- bpow_plus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps) + fexp n) <
 IZR (0 + 1) * bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

rewrite H0; ring_simplify (-fexp n + fexp n)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow 0 < IZR (0 + 1) * bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

simpl; rewrite Rmult_1_l, Rmult_1_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RP:(0 < eps)%Rn:ZHn:(n <= fexp n)%ZH:(0 <= eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps < bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

apply H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsround beta fexp Zfloor eps = 0%R

rewrite <- P, round_0; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:0%R = xFx:F xeps:RHeps:(0 <= eps < ulp 0)%RP:0%R = epsValid_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:R(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

case Zx; intros Zx1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%R(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

(* . 0 < x *)
intros (Heps1,[Heps2|Heps2]).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%Rround beta fexp Zceil (x + eps) = (x + ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%R(0 <= eps < ulp x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%Rround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%R(0 <= eps)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%R(eps < ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%Rround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%R(eps < ulp x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%Rround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

exact Heps2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%Rround beta fexp Zceil (x + eps) = (x + ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = xround beta fexp Zceil (x + eps) = (x + ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

rewrite round_UP_DN_ulp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x(round beta fexp Zfloor (x + eps) + ulp (x + eps))%R =
(x + ulp x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x~ F (x + eps)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

rewrite Hd.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x(x + ulp (x + eps))%R = (x + ulp x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x~ F (x + eps)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

rewrite 2!ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x(x + bpow (cexp beta fexp (x + eps)))%R =
(x + bpow (cexp beta fexp x))%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = xx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x(x + eps)%R <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x~ F (x + eps)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

unfold cexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x(x + bpow (fexp (mag beta (x + eps))))%R =
(x + bpow (fexp (mag beta x)))%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = xx <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x(x + eps)%R <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x~ F (x + eps)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

now rewrite mag_plus_eps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = xx <> 0%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x(x + eps)%R <> 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x~ F (x + eps)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x(x + eps)%R <> 0%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x~ F (x + eps)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = x~ F (x + eps)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

intros Fs.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:round beta fexp Zfloor (x + eps) = xFs:F (x + eps)False2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

rewrite round_generic in Hd...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:(x + eps)%R = xFs:F (x + eps)False2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:(x + eps)%R = xFs:F (x + eps)(x + eps > x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:(eps < ulp x)%RHeps:(0 <= eps < ulp x)%RHd:(x + eps)%R = xFs:F (x + eps)(x + eps > x + 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

now apply Rplus_lt_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + eps) = (x + ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

rewrite Heps2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xround beta fexp Zceil (x + ulp x) = (x + ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

apply round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:(0 < x)%RHeps1:(0 < eps)%RHeps2:eps = ulp xF (x + ulp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp x)%R ->
round beta fexp Zceil (x + eps) = (x + ulp x)%R

now apply generic_format_succ_aux1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = x(0 < eps <= ulp 0)%R ->
round beta fexp Zceil eps = ulp 0

intros Heps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rround beta fexp Zceil eps = ulp 0

case (proj2 Heps).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%R(eps < ulp 0)%R -> round beta fexp Zceil eps = ulp 01 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

unfold round, scaled_mantissa, cexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%R(eps < ulp 0)%R ->
F2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = ulp 01 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

unfold ulp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(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))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

rewrite Req_bool_true; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(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
end1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

case negligible_exp_spec.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(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%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

lra.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rforall 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)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

intros n Hn H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RF2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = bpow (fexp n)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

assert (fexp (mag beta eps) = fexp n).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%Rfexp (mag beta eps) = fexp n2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

apply valid_exp; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%R(mag beta eps <= fexp n)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

assert(mag beta eps-1 < fexp n)%Z;[idtac|omega].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%R(mag beta eps - 1 < fexp n)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

apply lt_bpow with beta.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%R(bpow (mag beta eps - 1) < bpow (fexp n))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

apply Rle_lt_trans with (2:=H).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%R(bpow (mag beta eps - 1) <= eps)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

destruct (mag beta eps) as (e,He).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%Re:ZHe:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R(bpow (Build_mag_prop beta eps e He - 1) <= eps)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

simpl; rewrite Rabs_pos_eq in He.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%Re:ZHe:eps <> 0%R -> (bpow (e - 1) <= eps < bpow e)%R(bpow (e - 1) <= eps)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%Re:ZHe:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R(0 <= eps)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

now apply He, Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%Re:ZHe:eps <> 0%R ->
(bpow (e - 1) <= Rabs eps < bpow e)%R(0 <= eps)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = 
bpow (fexp n)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

now left.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R
  {|
  Fnum := Zceil (eps * bpow (- fexp (mag beta eps)));
  Fexp := fexp (mag beta eps) |} = bpow (fexp n)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

replace (Zceil (eps * bpow (- fexp (mag beta eps)))) with 1%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp nF2R {| Fnum := 1; Fexp := fexp (mag beta eps) |} =
bpow (fexp n)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n1%Z = Zceil (eps * bpow (- fexp (mag beta eps)))beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

unfold F2R; simpl; rewrite H0; ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n1%Z = Zceil (eps * bpow (- fexp (mag beta eps)))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

apply sym_eq, Zceil_imp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(IZR (1 - 1) < eps * bpow (- fexp (mag beta eps)) <= 1)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(IZR (1 - 1) < eps * bpow (- fexp (mag beta eps)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

simpl; apply Rmult_lt_0_compat.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(0 < eps)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(0 < bpow (- fexp (mag beta eps)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

apply Heps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(0 < bpow (- fexp (mag beta eps)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) <= 1)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) <= 1)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

apply Rmult_le_reg_r with (bpow (fexp n)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(0 < bpow (fexp n))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) * bpow (fexp n) <=
 1 * bpow (fexp n))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps)) * bpow (fexp n) <=
 1 * bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

rewrite Rmult_assoc, <- bpow_plus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow (- fexp (mag beta eps) + fexp n) <=
 1 * bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

rewrite H0; ring_simplify (-fexp n + fexp n)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps * bpow 0 <= 1 * bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

simpl; rewrite Rmult_1_l, Rmult_1_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Rn:ZHn:(n <= fexp n)%ZH:(eps < bpow (fexp n))%RH0:fexp (mag beta eps) = fexp n(eps <= bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

now left.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%Reps = ulp 0 -> round beta fexp Zceil eps = ulp 0

intros P; rewrite P.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%RP:eps = ulp 0round beta fexp Zceil (ulp 0) = ulp 0

apply round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RZx:(0 <= x)%RFx:F xeps:RZx1:0%R = xHeps:(0 < eps <= ulp 0)%RP:eps = ulp 0F (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%Rround beta fexp Zceil (pred x + eps) = x

rewrite round_UP_plus_eps_pos; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%R(pred x + ulp (pred x))%R = x2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%R(0 <= pred x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%RF (pred x)

rewrite pred_eq_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%R(pred_pos x + ulp (pred_pos x))%R = x3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%R(0 <= pred x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%RF (pred x)

apply pred_pos_plus_ulp; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%R(0 <= x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%R(0 <= pred x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%RF (pred x)

now left.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%R(0 <= pred x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%RF (pred x)

now apply pred_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred x))%RF (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:R(0 < eps <= ulp (pred x))%R ->
round beta fexp Zfloor (x - eps) = pred x

rewrite pred_eq_pos;[intros Heps|now left].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%Rround beta fexp Zfloor (x - eps) = pred_pos x

replace (x-eps)%R with (pred_pos x + (ulp (pred_pos x)-eps))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%Rround beta fexp Zfloor
  (pred_pos x + (ulp (pred_pos x) - eps)) = pred_pos x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%Rround beta fexp Zfloor
  (pred_pos x + (ulp (pred_pos x) - eps)) = pred_pos x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%Rround beta fexp Zfloor
  (pred_pos x + (ulp (pred_pos x) - eps)) = pred_pos x

rewrite round_DN_plus_eps_pos; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%R(0 <= pred_pos x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%RF (pred_pos x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%RF (pred_pos x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%R(0 <= ulp (pred_pos x) - eps < ulp (pred_pos x))%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%R(0 <= ulp (pred_pos x) - eps)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%R(ulp (pred_pos x) - eps < ulp (pred_pos x))%R

apply Rle_0_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%R(eps <= ulp (pred_pos x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%R(ulp (pred_pos x) - eps < ulp (pred_pos x))%R

now apply Heps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%R(ulp (pred_pos x) - eps < ulp (pred_pos x))%R

rewrite <- Rplus_0_r.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%R(ulp (pred_pos x) - eps < ulp (pred_pos x) + 0)%R

apply Rplus_lt_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%R(- eps < 0)%R

rewrite <- Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(0 < eps <= ulp (pred_pos x))%R(- eps < - 0)%R

apply Ropp_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHpx:(0 < x)%RFx:F xeps:RHeps:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%Rround beta fexp Zfloor (x + eps) = x

case (Rle_or_lt 0 x); intros Zx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(0 <= x)%Rround beta fexp Zfloor (x + eps) = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(x < 0)%Rround beta fexp Zfloor (x + eps) = x

apply round_DN_plus_eps_pos; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(0 <= x)%R(0 <= eps < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(x < 0)%Rround beta fexp Zfloor (x + eps) = x

split; try apply Heps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(0 <= x)%R(eps < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(x < 0)%Rround beta fexp Zfloor (x + eps) = x

rewrite Rle_bool_true in Heps; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp x)%RZx:(0 <= x)%R(eps < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(x < 0)%Rround beta fexp Zfloor (x + eps) = x

now apply Heps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps <
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(x < 0)%Rround beta fexp Zfloor (x + eps) = x

(* *)
rewrite Rle_bool_false in Heps; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%Rround beta fexp Zfloor (x + eps) = x

rewrite <- (Ropp_involutive (x+eps)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%Rround beta fexp Zfloor (- - (x + eps)) = x

pattern x at 2; rewrite <- (Ropp_involutive x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%Rround beta fexp Zfloor (- - (x + eps)) = (- - x)%R

rewrite round_DN_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(- round beta fexp Zceil (- (x + eps)))%R = (- - x)%R

apply f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%Rround beta fexp Zceil (- (x + eps)) = (- x)%R

replace (-(x+eps))%R with (pred (-x) + (ulp (pred (-x)) - eps))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%Rround beta fexp Zceil
  (pred (- x) + (ulp (pred (- x)) - eps)) = (- x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R

rewrite round_UP_pred_plus_eps_pos; try reflexivity.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(0 < - x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(0 < ulp (pred (- x)) - eps <= ulp (pred (- x)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R

now apply Ropp_0_gt_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%RF (- x)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(0 < ulp (pred (- x)) - eps <= ulp (pred (- x)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(0 < ulp (pred (- x)) - eps <= ulp (pred (- x)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(0 < ulp (pred (- x)) - eps)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(ulp (pred (- x)) - eps <= ulp (pred (- x)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R

apply Rplus_lt_reg_l with eps; ring_simplify.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(eps < ulp (pred (- x)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(ulp (pred (- x)) - eps <= ulp (pred (- x)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R

apply Heps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(ulp (pred (- x)) - eps <= ulp (pred (- x)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(0 <= eps)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R

apply Heps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(pred (- x) + (ulp (pred (- x)) - eps))%R =
(- (x + eps))%R

unfold pred.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(- succ (- - x) + (ulp (- succ (- - x)) - eps))%R =
(- (x + eps))%R

rewrite Ropp_involutive.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (ulp (- succ x) - eps))%R =
(- (x + eps))%R

unfold succ; rewrite Rle_bool_false; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(- - pred_pos (- x) + (ulp (- - pred_pos (- x)) - eps))%R =
(- (x + eps))%R

rewrite Ropp_involutive; unfold Rminus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(pred_pos (- x) + (ulp (pred_pos (- x)) + - eps))%R =
(- (x + eps))%R

rewrite <- Rplus_assoc, pred_pos_plus_ulp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(- x + - eps)%R = (- (x + eps))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(0 < - x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%RF (- x)

ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%R(0 < - x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%RF (- x)

now apply Ropp_0_gt_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 <= eps < ulp (pred (- x)))%RZx:(x < 0)%RF (- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%Rround beta fexp Zceil (x + eps) = succ x

case (Rle_or_lt 0 x); intros Zx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(0 <= x)%Rround beta fexp Zceil (x + eps) = succ x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(x < 0)%Rround beta fexp Zceil (x + eps) = succ x

rewrite succ_eq_pos; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(0 <= x)%Rround beta fexp Zceil (x + eps) = (x + ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(x < 0)%Rround beta fexp Zceil (x + eps) = succ x

rewrite Rle_bool_true in Heps; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp x)%RZx:(0 <= x)%Rround beta fexp Zceil (x + eps) = (x + ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(x < 0)%Rround beta fexp Zceil (x + eps) = succ x

apply round_UP_plus_eps_pos; assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool 0 x
  then ulp x
  else ulp (pred (- x))))%RZx:(x < 0)%Rround beta fexp Zceil (x + eps) = succ x

(* *)
rewrite Rle_bool_false in Heps; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%Rround beta fexp Zceil (x + eps) = succ x

rewrite <- (Ropp_involutive (x+eps)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%Rround beta fexp Zceil (- - (x + eps)) = succ x

rewrite <- (Ropp_involutive (succ x)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%Rround beta fexp Zceil (- - (x + eps)) = (- - succ x)%R

rewrite round_UP_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- round beta fexp Zfloor (- (x + eps)))%R =
(- - succ x)%R

apply f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%Rround beta fexp Zfloor (- (x + eps)) = (- succ x)%R

replace (-(x+eps))%R with (-succ x + (-eps + ulp (pred (-x))))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%Rround beta fexp Zfloor
  (- succ x + (- eps + ulp (pred (- x)))) =
(- succ x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

apply round_DN_plus_eps_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= - succ x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- succ x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

rewrite <- pred_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= pred (- x))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- succ x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

apply pred_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 < - x)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- succ x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

now apply Ropp_0_gt_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- x)3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- succ x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- succ x)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

now apply generic_format_opp, generic_format_succ.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= - eps + ulp (pred (- x)) < ulp (- succ x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= - eps + ulp (pred (- x)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- eps + ulp (pred (- x)) < ulp (- succ x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

apply Rplus_le_reg_l with eps; ring_simplify.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(eps <= ulp (pred (- x)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- eps + ulp (pred (- x)) < ulp (- succ x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

apply Heps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- eps + ulp (pred (- x)) < ulp (- succ x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

unfold pred; rewrite Ropp_involutive.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- eps + ulp (- succ x) < ulp (- succ x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 < eps)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

apply Heps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- succ x + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

unfold succ; rewrite Rle_bool_false; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- - pred_pos (- x) + (- eps + ulp (pred (- x))))%R =
(- (x + eps))%R

apply trans_eq with (-x +-eps)%R;[idtac|ring].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- - pred_pos (- x) + (- eps + ulp (pred (- x))))%R =
(pred_pos (- x) + ulp (pred_pos (- x)) + - eps)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 < - x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- x)

rewrite pred_eq_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(- - pred_pos (- x) + (- eps + ulp (pred_pos (- x))))%R =
(pred_pos (- x) + ulp (pred_pos (- x)) + - eps)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= - x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 < - x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- x)

ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 <= - x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 < - x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- x)

left; now apply Ropp_0_gt_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%R(0 < - x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- x)

now apply Ropp_0_gt_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <= ulp (pred (- x)))%RZx:(x < 0)%RF (- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (0 <= x < y)%R -> (x <= pred_pos y)%R

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (0 <= x < y)%R -> (x <= pred_pos y)%R

intros x y Fx Fy H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%R(x <= pred_pos y)%R

case (proj1 H); intros V.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%R(x <= pred_pos y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

assert (Zy:(0 < y)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%R(0 < y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Rle_lt_trans with (1:=proj1 H).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%R(x < y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%R(x <= pred_pos y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

(* *)
assert (Zp: (0 < pred y)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%R(0 < pred y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

assert (Zp:(0 <= pred y)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%R(0 <= pred y)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 <= pred y)%R(0 < pred y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply pred_ge_0 ; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 <= pred y)%R(0 < pred y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

destruct Zp; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred y(0 < pred y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

generalize H0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred y0%R = pred y -> (0 < pred y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite pred_eq_pos;[idtac|now left].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred y0%R = pred_pos y -> (0 < pred_pos y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

unfold pred_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred y0%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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

destruct (mag beta y) as (ey,Hey); simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%R0%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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

case Req_bool_spec; intros Hy2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)0%R = (y - bpow (fexp (ey - 1)))%R ->
(0 < y - bpow (fexp (ey - 1)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

(* . *)
intros Hy3.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%R(0 < y - bpow (fexp (ey - 1)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

assert (ey-1 = fexp (ey -1))%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%R(ey - 1)%Z = fexp (ey - 1)4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply bpow_inj with beta.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%Rbpow (ey - 1) = bpow (fexp (ey - 1))4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite <- Hy2, <- Rplus_0_l, Hy3.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%Ry =
(y - bpow (fexp (ey - 1)) + bpow (fexp (ey - 1)))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)(0 < y - bpow (fexp (ey - 1)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

assert (Zx: (x <> 0)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)x <> 0%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%R(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%R(0 < y - bpow (fexp (ey - 1)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

destruct (mag beta x) as (ex,Hex).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:x <> 0%R ->
(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 < y - bpow (fexp (ey - 1)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

specialize (Hex Zx).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(0 < y - bpow (fexp (ey - 1)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

assert (ex <= ey)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(ex <= ey)%Z4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%Z(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply bpow_lt_bpow with beta.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(bpow (ex - 1) < bpow ey)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%Z(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Rle_lt_trans with (1:=proj1 Hex).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(Rabs x < bpow ey)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%Z(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Rlt_trans with (Rabs y).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(Rabs x < Rabs y)%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(Rabs y < bpow ey)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%Z(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite 2!Rabs_right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(x < y)%R7 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(y >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(x >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(Rabs y < bpow ey)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%Z(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(y >= 0)%R6 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(x >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(Rabs y < bpow ey)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%Z(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rgt_ge.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(x >= 0)%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(Rabs y < bpow ey)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%Z(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rgt_ge.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%R(Rabs y < bpow ey)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%Z(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Hey.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%Ry <> 0%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%Z(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%Z(0 < y - bpow (fexp (ey - 1)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

case (Zle_lt_or_eq _ _ H2); intros Hexy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%Z(0 < y - bpow (fexp (ey - 1)))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

assert (fexp ex = fexp (ey-1))%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%Zfexp ex = fexp (ey - 1)5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply valid_exp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%Z(ey - 1 <= fexp (ey - 1))%Z6 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%Z(ex <= fexp (ey - 1))%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%Z(ex <= fexp (ey - 1))%Z5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite <- H1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%Z(ex <= ey - 1)%Z5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(0 < y - bpow (fexp (ey - 1)))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

absurd (0 < Ztrunc (scaled_mantissa beta fexp x) < 1)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)~ (0 < Ztrunc (scaled_mantissa beta fexp x) < 1)%Z5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(0 < Ztrunc (scaled_mantissa beta fexp x) < 1)%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(0 < Ztrunc (scaled_mantissa beta fexp x) < 1)%Z4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(0 < Ztrunc (scaled_mantissa beta fexp x))%Z5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(Ztrunc (scaled_mantissa beta fexp x) < 1)%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply gt_0_F2R with beta (cexp beta fexp x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(0 <
 F2R
   {|
   Fnum := Ztrunc (scaled_mantissa beta fexp x);
   Fexp := cexp beta fexp x |})%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(Ztrunc (scaled_mantissa beta fexp x) < 1)%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now rewrite <- Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(Ztrunc (scaled_mantissa beta fexp x) < 1)%Z4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply lt_IZR.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(IZR (Ztrunc (scaled_mantissa beta fexp x)) < 1)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Rmult_lt_reg_r with (bpow (cexp beta fexp x)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(0 < bpow (cexp beta fexp x))%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) < 
 1 * bpow (cexp beta fexp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(IZR (Ztrunc (scaled_mantissa beta fexp x)) *
 bpow (cexp beta fexp x) < 1 * bpow (cexp beta fexp x))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(x < 1 * bpow (cexp beta fexp x))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite Rmult_1_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(x < bpow (cexp beta fexp x))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

unfold cexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(x < bpow (fexp (mag beta x)))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite mag_unique with beta x ex.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(x < bpow (fexp ex))%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(bpow (ex - 1) <= Rabs x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite H3,<-H1, <- Hy2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(x < y)%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(bpow (ex - 1) <= Rabs x < bpow ex)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:(ex < ey)%ZH3:fexp ex = fexp (ey - 1)(bpow (ex - 1) <= Rabs x < bpow ex)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

exact Hex.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(0 < y - bpow (fexp (ey - 1)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

absurd (y <= x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey~ (y <= x)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(y <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rlt_not_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(y <= x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite Rabs_right in Hex.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(y <= x)%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(x >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Rle_trans with (2:=proj1 Hex).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(y <= bpow (ex - 1))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(x >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite Hexy, Hy2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(bpow (ey - 1) <= bpow (ey - 1))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(x >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rle_refl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y = bpow (ey - 1)Hy3:0%R = (y - bpow (fexp (ey - 1)))%RH1:(ey - 1)%Z = fexp (ey - 1)Zx:x <> 0%Rex:ZHex:(bpow (ex - 1) <= Rabs x < bpow ex)%RH2:(ex <= ey)%ZHexy:ex = ey(x >= 0)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rgt_ge.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)0%R = (y - ulp y)%R -> (0 < y - ulp y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

(* . *)
intros Hy3.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%R(0 < y - ulp y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

assert (y = bpow (fexp ey))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%Ry = bpow (fexp ey)3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Rminus_diag_uniq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%R(y - bpow (fexp ey))%R = 0%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite Hy3.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%R(y - bpow (fexp ey))%R = (y - ulp y)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite ulp_neq_0;[idtac|now apply Rgt_not_eq].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%R(y - bpow (fexp ey))%R =
(y - bpow (cexp beta fexp y))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

unfold cexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%R(y - bpow (fexp ey))%R =
(y - bpow (fexp (mag beta y)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite (mag_unique beta y ey); trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%R(bpow (ey - 1) <= Rabs y < bpow ey)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Hey.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%Ry <> 0%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(0 < y - ulp y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy2:y <> bpow (ey - 1)Hy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(0 < y - ulp y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

contradict Hy2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)y = bpow (ey - 1)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite H1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)bpow (fexp ey) = bpow (ey - 1)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)fexp ey = (ey - 1)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Zplus_reg_l with 1%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(1 + fexp ey)%Z = (1 + (ey - 1))%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

ring_simplify.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(fexp ey + 1)%Z = ey2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply trans_eq with (mag beta y).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(fexp ey + 1)%Z = mag beta y3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)mag beta y = eybeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply sym_eq; apply mag_unique.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey + 1 - 1) <= Rabs y < bpow (fexp ey + 1))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)mag beta y = eybeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite H1, Rabs_right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey + 1 - 1) <= bpow (fexp ey) <
 bpow (fexp ey + 1))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)mag beta y = eybeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey + 1 - 1) <= bpow (fexp ey))%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey) < bpow (fexp ey + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)mag beta y = eybeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply bpow_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(fexp ey + 1 - 1 <= fexp ey)%Z5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey) < bpow (fexp ey + 1))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)mag beta y = eybeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey) < bpow (fexp ey + 1))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)mag beta y = eybeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply bpow_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(fexp ey < fexp ey + 1)%Z4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey) >= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)mag beta y = eybeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

omega.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (fexp ey) >= 0)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)mag beta y = eybeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Rle_ge; apply bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)mag beta y = ey2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply mag_unique.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)(bpow (ey - 1) <= Rabs y < bpow ey)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Hey.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RH0:0%R = pred yey:ZHey:y <> 0%R ->
(bpow (ey - 1) <= Rabs y < bpow ey)%RHy3:0%R = (y - ulp y)%RH1:y = bpow (fexp ey)y <> 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%R(x <= pred_pos y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

(* *)
case (Rle_or_lt (ulp (pred_pos y)) (y-x)); intros H1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(ulp (pred_pos y) <= y - x)%R(x <= pred_pos y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

(* . *)
apply Rplus_le_reg_r with (-x + ulp (pred_pos y))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(ulp (pred_pos y) <= y - x)%R(x + (- x + ulp (pred_pos y)) <=
 pred_pos y + (- x + ulp (pred_pos y)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

ring_simplify (x+(-x+ulp (pred_pos y)))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(ulp (pred_pos y) <= y - x)%R(ulp (pred_pos y) <=
 pred_pos y + (- x + ulp (pred_pos y)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Rle_trans with (1:=H1).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(ulp (pred_pos y) <= y - x)%R(y - x <= pred_pos y + (- x + ulp (pred_pos y)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite <- (pred_pos_plus_ulp y) at 1; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(ulp (pred_pos y) <= y - x)%R(pred_pos y + ulp (pred_pos y) - x <=
 pred_pos y + (- x + ulp (pred_pos y)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(x <= pred_pos y)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Req_le; ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(x <= pred_pos y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

(* . *)
replace x with (y-(y-x))%R by ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(y - (y - x) <= pred_pos y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite <- pred_eq_pos;[idtac|now left].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(y - (y - x) <= pred y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite <- round_DN_minus_eps_pos with (eps:=(y-x)%R); try easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(y - (y - x) <= round beta fexp Zfloor (y - (y - x)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(0 < y - x <= ulp (pred y))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

ring_simplify (y-(y-x))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(x <= round beta fexp Zfloor x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(0 < y - x <= ulp (pred y))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply Req_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%Rx = round beta fexp Zfloor x2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(0 < y - x <= ulp (pred y))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply sym_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%Rround beta fexp Zfloor x = x2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(0 < y - x <= ulp (pred y))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

apply round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(0 < y - x <= ulp (pred y))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

split; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(0 < y - x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(y - x <= ulp (pred y))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rlt_Rminus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(y - x <= ulp (pred y))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite pred_eq_pos;[idtac|now left].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:(0 < x)%RZy:(0 < y)%RZp:(0 < pred y)%RH1:(y - x < ulp (pred_pos y))%R(y - x <= ulp (pred_pos y))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV:0%R = x(x <= pred_pos y)%R

rewrite <- V; apply pred_pos_ge_0; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(0 <= x < y)%RV: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x ->
F y -> (0 <= x)%R -> (x < y)%R -> (succ x <= y)%R

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%R(succ x <= y)%R

rewrite succ_eq_pos; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%R(x + ulp x <= y)%R

case (Rle_or_lt (ulp x) (y-x)); intros H1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(ulp x <= y - x)%R(x + ulp x <= y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R(x + ulp x <= y)%R

apply Rplus_le_reg_r with (-x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(ulp x <= y - x)%R(x + ulp x + - x <= y + - x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R(x + ulp x <= y)%R

now ring_simplify (x+ulp x + -x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R(x + ulp x <= y)%R

replace y with (x+(y-x))%R by ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R(x + ulp x <= x + (y - x))%R

absurd (x < y)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R~ (x < y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R(x < y)%R

2: apply H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R~ (x < y)%R

apply Rle_not_lt; apply Req_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%Ry = x

rewrite <- round_DN_plus_eps_pos with (eps:=(y-x)%R); try easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%Ry = round beta fexp Zfloor (x + (y - x))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R(0 <= y - x < ulp x)%R

ring_simplify (x+(y-x))%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%Ry = round beta fexp Zfloor y1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R(0 <= y - x < ulp x)%R

apply sym_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%Rround beta fexp Zfloor y = y1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R(0 <= y - x < ulp x)%R

apply round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(y - x < ulp x)%R(0 <= y - x < ulp x)%R

split; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RHx:F xHy:F yZx:(0 <= x)%RH:(x < y)%RH1:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x < y)%R -> (succ x <= y)%R

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x < y)%R -> (succ x <= y)%R

intros x y Fx Fy H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%R(succ x <= y)%R

destruct (Rle_or_lt 0 x) as [Hx|Hx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(0 <= x)%R(succ x <= y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%R(succ x <= y)%R

now apply succ_le_lt_aux.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%R(succ x <= y)%R

unfold succ; rewrite Rle_bool_false; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%R(- pred_pos (- x) <= y)%R

case (Rle_or_lt y 0); intros Hy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%R(- pred_pos (- x) <= y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- pred_pos (- x) <= y)%R

rewrite <- (Ropp_involutive y).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%R(- pred_pos (- x) <= - - y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- pred_pos (- x) <= y)%R

apply Ropp_le_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%R(- y <= pred_pos (- x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- pred_pos (- x) <= y)%R

apply le_pred_pos_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%RF (- y)3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%R(0 <= - y < - x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- pred_pos (- x) <= y)%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%RF (- x)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%R(0 <= - y < - x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- pred_pos (- x) <= y)%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%R(0 <= - y < - x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- pred_pos (- x) <= y)%R

split.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%R(0 <= - y)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%R(- y < - x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- pred_pos (- x) <= y)%R

rewrite <- Ropp_0; now apply Ropp_le_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(y <= 0)%R(- y < - x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- pred_pos (- x) <= y)%R

now apply Ropp_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- pred_pos (- x) <= y)%R

apply Rle_trans with (-0)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- pred_pos (- x) <= - 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- 0 <= y)%R

apply Ropp_le_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(0 <= pred_pos (- x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- 0 <= y)%R

apply pred_pos_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(0 < - x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- 0 <= y)%R

rewrite <- Ropp_0; now apply Ropp_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%RF (- x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(0 < y)%R(- 0 <= y)%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yH:(x < y)%RHx:(x < 0)%RHy:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x < y)%R -> (x <= pred y)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x < y)%R -> (x <= pred y)%R

intros x y Fx Fy Hxy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(x <= pred y)%R

rewrite <- (Ropp_involutive x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(- - x <= pred y)%R

unfold pred; apply Ropp_le_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(succ (- y) <= - x)%R

apply succ_le_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%RF (- y)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(- y < - x)%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%RF (- x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(- y < - x)%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
y <> 0%R -> (x <= y)%R -> (x < succ y)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
y <> 0%R -> (x <= y)%R -> (x < succ y)%R

intros x y Zy Hxy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RZy:y <> 0%RHxy:(x <= y)%R(x < succ y)%R

apply Rle_lt_trans with (1 := Hxy).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RZy:y <> 0%RHxy:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
x <> 0%R -> (x <= y)%R -> (pred x < y)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
x <> 0%R -> (x <= y)%R -> (pred x < y)%R

intros x y Zy Hxy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RZy:x <> 0%RHxy:(x <= y)%R(pred x < y)%R

apply Rlt_le_trans with (2 := Hxy).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RZy:x <> 0%RHxy:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> (0 < x)%R -> succ (pred x) = x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> (0 < x)%R -> succ (pred x) = x

intros x Fx Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%Rsucc (pred x) = x

rewrite pred_eq_pos by now left.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%Rsucc (pred_pos x) = x

rewrite succ_eq_pos by now apply pred_pos_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexppred (ulp 0) = 0%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexppred (ulp 0) = 0%R

rewrite pred_eq_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexppred_pos (ulp 0) = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexp(0 <= ulp 0)%R

2: apply ulp_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexppred_pos (ulp 0) = 0%R

unfold ulp; rewrite Req_bool_true; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexppred_pos
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end = 0%R

case negligible_exp_spec'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpnegligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) ->
pred_pos
  match negligible_exp with
  | Some n => bpow (fexp n)
  | None => 0
  end = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH1:negligible_exp = Nonepred_pos 0 = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH1:negligible_exp = None(0 - ulp 0)%R = 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH1:negligible_exp = None0%R <> bpow (mag beta 0 - 1)beta:radixfexp:Z -> Zvalid_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH1:negligible_exp = None(0 - ulp 0)%R = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH1:negligible_exp = None(0 -
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_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:radixfexp:Z -> Zvalid_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zpred_pos (bpow (fexp n)) = 0%R

unfold pred_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn:ZH1:negligible_exp = Some nH2:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn:ZH1:negligible_exp = Some nH2:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn:ZH1:negligible_exp = Some nH2:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Z(bpow (fexp n) - bpow (fexp (fexp n)))%R = 0%R

apply Rminus_diag_eq, f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpn:ZH1:negligible_exp = Some nH2:(n <= fexp n)%Zfexp n = fexp (fexp n)

apply sym_eq, valid_exp; omega.
Qed.

Theorem succ_0 :
  succ 0 = ulp 0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpsucc 0 = ulp 0

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpsucc 0 = ulp 0

unfold succ.beta:radixfexp:Z -> Zvalid_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexp(0 + ulp 0)%R = ulp 01 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexp(0 <= 0)%R

apply Rplus_0_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexp(0 <= 0)%R

apply Rle_refl.
Qed.

Theorem pred_0 :
  pred 0 = Ropp (ulp 0).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexppred 0 = (- ulp 0)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexppred 0 = (- ulp 0)%R

rewrite <- succ_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexppred 0 = (- succ 0)%R

rewrite <- Ropp_0 at 1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexppred (- 0) = (- succ 0)%R

apply pred_opp.
Qed.

Lemma pred_succ_pos :
  forall x, F x -> (0 < x)%R ->
  pred (succ x) = x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> (0 < x)%R -> pred (succ x) = x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> (0 < x)%R -> pred (succ x) = x

intros x Fx Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%Rpred (succ x) = x

apply Rle_antisym.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R(pred (succ x) <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R(x <= pred (succ x))%R

-beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R(pred (succ x) <= x)%R
 apply Rnot_lt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R~ (x < pred (succ x))%R

  intros H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RH:(x < pred (succ x))%RFalse

  apply succ_le_lt with (1 := Fx) in H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RH:(succ x <= pred (succ x))%RFalse1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RH:(x < pred (succ x))%RF (pred (succ x))

  revert H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R(succ x <= pred (succ x))%R -> False1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RH:(x < pred (succ x))%RF (pred (succ x))

  apply Rlt_not_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R(pred (succ x) < succ x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RH:(x < pred (succ x))%RF (pred (succ x))

  apply pred_lt_id.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%Rsucc x <> 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RH:(x < pred (succ x))%RF (pred (succ x))

  apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R(succ x > 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RH:(x < pred (succ x))%RF (pred (succ x))

  apply Rlt_le_trans with (1 := Hx).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R(x <= succ x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RH:(x < pred (succ x))%RF (pred (succ x))

  apply succ_ge_id.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RH:(x < pred (succ x))%RF (pred (succ x))

  now apply generic_format_pred, generic_format_succ.
-beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R(x <= pred (succ x))%R
 apply pred_ge_gt with (1 := Fx).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RF (succ x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R(x < succ x)%R

  now apply generic_format_succ.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%R(x < succ x)%R

  apply succ_gt_id.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%Rx <> 0%R

  now apply Rgt_not_eq.
Qed.

Theorem succ_pred :
  forall x, F x ->
  succ (pred x) = x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> succ (pred x) = x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> succ (pred x) = x

intros x Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xsucc (pred x) = x

destruct (Rle_or_lt 0 x) as [[Hx|Hx]|Hx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%Rsucc (pred x) = x2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:0%R = xsucc (pred x) = xbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%Rsucc (pred x) = x

now apply succ_pred_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:0%R = xsucc (pred x) = x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%Rsucc (pred x) = x

rewrite <- Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:0%R = xsucc (pred 0) = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%Rsucc (pred x) = x

rewrite pred_0, succ_opp, pred_ulp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:0%R = x(- 0)%R = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%Rsucc (pred x) = x

apply Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%Rsucc (pred x) = x

unfold pred.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%Rsucc (- succ (- x)) = x

rewrite succ_opp, pred_succ_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%R(- - x)%R = x2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%R(0 < - x)%R

apply Ropp_involutive.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%RF (- x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(x < 0)%R(0 < - x)%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> pred (succ x) = x

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R, F x -> pred (succ x) = x

intros x Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xpred (succ x) = x

rewrite <- (Ropp_involutive x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xpred (succ (- - x)) = (- - x)%R

rewrite succ_opp, pred_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F x(- succ (pred (- x)))%R = (- - x)%R

apply f_equal, succ_pred.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xF (- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%Rround beta fexp Zceil (pred x + eps) = x

rewrite round_UP_plus_eps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%Rsucc (pred x) = x2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (pred x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (pred x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite ulp_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

generalize Heps; case (Rle_bool_spec x 0); intros H1 H2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite Rle_bool_false; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%R(pred x < 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

case H1; intros H1'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':(x < 0)%R(pred x < 0)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(pred x < 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

apply Rlt_le_trans with (2:=H1).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':(x < 0)%R(pred x < x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(pred x < 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

apply pred_lt_id.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':(x < 0)%Rx <> 0%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(pred x < 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

now apply Rlt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(pred x < 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite H1'; unfold pred, succ.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(-
 (if Rle_bool 0 (- 0)
  then - 0 + ulp (- 0)
  else - pred_pos (- - 0)) < 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite Ropp_0; rewrite Rle_bool_true;[idtac|now right].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(- (0 + ulp 0) < 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite Rplus_0_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(- ulp 0 < 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite <- Ropp_0; apply Ropp_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(0 < ulp (- 0))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

apply Rlt_le_trans with (1:=proj1 H2).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(eps <= ulp (- 0))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

apply Rle_trans with (1:=proj2 H2).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(ulp x <= ulp (- 0))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite Ropp_0, H1'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(ulp 0 <= ulp 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

now right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool 0 (pred x) then ulp (pred x) else ulp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite Rle_bool_true; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 <= pred x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

now apply pred_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%Rround beta fexp Zfloor (x - eps) = pred x

replace (x-eps)%R with (-(-x+eps))%R by ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%Rround beta fexp Zfloor (- (- x + eps)) = pred x

rewrite round_DN_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%Rround beta fexp Zceil (- x + eps) = succ (- x)

pattern (-x)%R at 1; rewrite <- (pred_succ (-x)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%Rround beta fexp Zceil (pred (succ (- x)) + eps) =
succ (- x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

apply round_UP_pred_plus_eps.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (succ (- x))2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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)))))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

now apply generic_format_succ, generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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)))))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite pred_succ.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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)))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite ulp_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

generalize Heps; case (Rle_bool_spec x 0); intros H1 H2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite Rle_bool_false; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%R(0 < succ (- x))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

case H1; intros H1'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':(x < 0)%R(0 < succ (- x))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(0 < succ (- x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

apply Rlt_le_trans with (-x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':(x < 0)%R(0 < - x)%R5 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':(x < 0)%R(- x <= succ (- x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(0 < succ (- x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

now apply Ropp_0_gt_lt_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':(x < 0)%R(- x <= succ (- x))%R4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(0 < succ (- x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

apply succ_ge_id.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(0 < succ (- x))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite H1', Ropp_0, succ_eq_pos;[idtac|now right].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(0 < 0 + ulp 0)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite Rplus_0_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(0 < ulp 0)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

apply Rlt_le_trans with (1:=proj1 H2).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(x <= 0)%RH2:(0 < eps <= ulp x)%RH1':x = 0%R(eps <= ulp 0)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite H1' in H2; apply H2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <=
 (if Rle_bool (succ (- x)) 0
  then ulp (succ (- x))
  else ulp x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite Rle_bool_true.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 < eps <= ulp (succ (- x)))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(succ (- x) <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

now rewrite succ_opp, ulp_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(succ (- x) <= 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite succ_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(- pred x <= 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

rewrite <- Ropp_0; apply Ropp_le_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RH1:(0 < x)%RH2:(0 < eps <= ulp (pred x))%R(0 <= pred x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

now apply pred_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- x)

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xeps:RHeps:(0 < eps <=
 (if Rle_bool x 0 then ulp x else ulp (pred x)))%RF (- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%R(Rabs (round beta fexp rnd x - x) < ulp x)%R

destruct (generic_format_EM beta fexp x) as [Hx|Hx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:F x(Rabs (round beta fexp rnd x - x) < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp rnd x - x) < ulp x)%R

(* x = rnd x *)
rewrite round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:F x(Rabs (x - x) < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp rnd x - x) < ulp x)%R

unfold Rminus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:F x(Rabs (x + - x) < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp rnd x - x) < ulp x)%R

rewrite Rplus_opp_r, Rabs_R0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:F x(0 < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp rnd x - x) < ulp x)%R

rewrite ulp_neq_0; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:F x(0 < bpow (cexp beta fexp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp rnd x - x) < ulp x)%R

apply bpow_gt_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zfloor x - x) < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

(* . *)
rewrite Rabs_left1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(- (round beta fexp Zfloor x - x) < ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

rewrite Ropp_minus_distr.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(x - round beta fexp Zfloor x < ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

apply Rplus_lt_reg_l with (round beta fexp Zfloor x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x +
 (x - round beta fexp Zfloor x) <
 round beta fexp Zfloor x + ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

rewrite <- round_UP_DN_ulp with (1 := Hx).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x +
 (x - round beta fexp Zfloor x) <
 round beta fexp Zceil x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

ring_simplify.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(x < round beta fexp Zceil x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

assert (Hu: (x <= round beta fexp Zceil x)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(x <= round beta fexp Zceil x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHu:(x <= round beta fexp Zceil x)%R(x < round beta fexp Zceil x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

apply round_UP_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHu:(x <= round beta fexp Zceil x)%R(x < round beta fexp Zceil x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

destruct Hu as [Hu|Hu].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHu:(x < round beta fexp Zceil x)%R(x < round beta fexp Zceil x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHu:x = round beta fexp Zceil x(x < round beta fexp Zceil x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

exact Hu.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHu:x = round beta fexp Zceil x(x < round beta fexp Zceil x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

elim Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHu:x = round beta fexp Zceil xF x2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

rewrite Hu.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHu:x = round beta fexp Zceil xF (round beta fexp Zceil x)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

apply generic_format_round...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x - x <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

apply Rle_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

apply round_DN_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(Rabs (round beta fexp Zceil x - x) < ulp x)%R

(* . *)
rewrite Rabs_pos_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zceil x - x < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

rewrite round_UP_DN_ulp with (1 := Hx).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x + ulp x - x < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

apply Rplus_lt_reg_r with (x - ulp x)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x + ulp x - x + (x - ulp x) <
 ulp x + (x - ulp x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

ring_simplify.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x < x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

assert (Hd: (round beta fexp Zfloor x <= x)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(round beta fexp Zfloor x <= x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHd:(round beta fexp Zfloor x <= x)%R(round beta fexp Zfloor x < x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

apply round_DN_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHd:(round beta fexp Zfloor x <= x)%R(round beta fexp Zfloor x < x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

destruct Hd as [Hd|Hd].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHd:(round beta fexp Zfloor x < x)%R(round beta fexp Zfloor x < x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHd:round beta fexp Zfloor x = x(round beta fexp Zfloor x < x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

exact Hd.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHd:round beta fexp Zfloor x = x(round beta fexp Zfloor x < x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

elim Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHd:round beta fexp Zfloor x = xF x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

rewrite <- Hd.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F xHd:round beta fexp Zfloor x = xF (round beta fexp Zfloor x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

apply generic_format_round...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ F x(0 <= round beta fexp Zceil x - x)%R

apply Rle_0_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x <> 0%RHx:~ 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R,
(Rabs (round beta fexp rnd x - x) <= ulp x)%R

intros  rnd Zrnd x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:R(Rabs (round beta fexp rnd x - x) <= ulp x)%R

case (Req_dec x 0).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:Rx = 0%R ->
(Rabs (round beta fexp rnd x - x) <= ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:Rx <> 0%R ->
(Rabs (round beta fexp rnd x - x) <= ulp x)%R

intros Zx; rewrite Zx, round_0...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R(Rabs (0 - 0) <= ulp 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:Rx <> 0%R ->
(Rabs (round beta fexp rnd x - x) <= ulp x)%R

unfold Rminus; rewrite Rplus_0_l, Ropp_0, Rabs_R0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R(0 <= ulp 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:Rx <> 0%R ->
(Rabs (round beta fexp rnd x - x) <= ulp x)%R

apply ulp_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:Rx <> 0%R ->
(Rabs (round beta fexp rnd x - x) <= ulp x)%R

intros Zx; left.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (choice : Z -> bool) (x : R),
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (choice : Z -> bool) (x : R),
(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

intros choice x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

destruct (generic_format_EM beta fexp x) as [Hx|Hx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F x(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

(* x = rnd x *)
rewrite round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(Rabs (x - x) <= / 2 * ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F x(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

unfold Rminus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(Rabs (x + - x) <= / 2 * ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F x(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

rewrite Rplus_opp_r, Rabs_R0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(0 <= / 2 * ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F x(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

apply Rmult_le_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(0 <= / 2)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(0 <= ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F x(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(0 < / 2)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(0 <= ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F x(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

apply Rinv_0_lt_compat.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(0 < 2)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(0 <= ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F x(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

now apply IZR_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:F x(0 <= ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F x(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

apply ulp_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (d - x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(Rabs (d - x) <= / 2 * ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

apply Hr2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%RF d2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(Rabs (d - x) <= / 2 * ulp x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(Rabs (d - x) <= / 2 * ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

rewrite Rabs_left1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(- (d - x) <= / 2 * ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

rewrite Ropp_minus_distr.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(x - d <= / 2 * ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(0 < 2)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R((x - d) * 2 <= / 2 * ulp x * 2)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

now apply IZR_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R((x - d) * 2 <= / 2 * ulp x * 2)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R((x - d) * 2 + (d - x) <= / 2 * ulp x * 2 + (d - x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

ring_simplify.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(x - d <= - x + d + 2 * / 2 * ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d + ulp x - x <= - x + d + 2 * / 2 * ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d + ulp x - x)%R = (- x + d + 2 * / 2 * ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d - x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R
 field.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d - x <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

apply Rle_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(x - d <= d + ulp x - x)%R(d <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%R(d + ulp x)%R = round beta fexp Zceil x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R

unfold d.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%R(round beta fexp Zfloor x + ulp x)%R =
round beta fexp Zceil x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (d + ulp x - x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(Rabs (d + ulp x - x) <= / 2 * ulp x)%R

apply Hr2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil xF (d + ulp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(Rabs (d + ulp x - x) <= / 2 * ulp x)%R

rewrite Hu.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil xF (round beta fexp Zceil x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(Rabs (d + ulp x - x) <= / 2 * ulp x)%R

rewrite Rabs_pos_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(d + ulp x - x <= / 2 * ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(0 <= d + ulp x - x)%R

apply Rmult_le_reg_r with 2%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(0 < 2)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x((d + ulp x - x) * 2 <= / 2 * ulp x * 2)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(0 <= d + ulp x - x)%R

now apply IZR_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x((d + ulp x - x) * 2 <= / 2 * ulp x * 2)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(0 <= d + ulp x - x)%R

ring_simplify.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(d + ulp x - x <= - d + 2 * ulp x * / 2 - ulp x + x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(0 <= d + ulp x - x)%R

apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(d + ulp x - x < - d + 2 * ulp x * / 2 - ulp x + x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(0 <= d + ulp x - x)%R

apply Rlt_le_trans with (1 := H).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(x - d <= - d + 2 * ulp x * / 2 - ulp x + x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(0 <= d + ulp x - x)%R

right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(x - d)%R = (- d + 2 * ulp x * / 2 - ulp x + x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(0 <= d + ulp x - x)%R
 field.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(0 <= d + ulp x - x)%R

apply Rle_0_minus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(d + ulp x)%R = round beta fexp Zceil x(x <= d + ulp x)%R

rewrite Hu.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:RHx:~ F xd:=round beta fexp Zfloor x:RHr1:F (round beta fexp (Znearest choice) x)Hr2:forall g : R,
F g ->
(Rabs (round beta fexp (Znearest choice) x - x) <=
 Rabs (g - x))%RH:(d + ulp x - x < x - d)%RHu:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 <= x)%R -> ulp (round beta fexp Zfloor x) = ulp x

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(0 <= x)%R -> ulp (round beta fexp Zfloor x) = ulp x

intros x [Hx|Hx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%Rulp (round beta fexp Zfloor x) = ulp x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:0%R = xulp (round beta fexp Zfloor x) = ulp x

-beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%Rulp (round beta fexp Zfloor x) = ulp x
 rewrite (ulp_neq_0 x) by now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%Rulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)

  destruct (round_ge_generic beta fexp Zfloor 0 x) as [Hd|Hd].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RF 03 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:(0 < round beta fexp Zfloor x)%Rulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:0%R = round beta fexp Zfloor xulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)

    apply generic_format_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%R(0 <= x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:(0 < round beta fexp Zfloor x)%Rulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:0%R = round beta fexp Zfloor xulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)

    now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:(0 < round beta fexp Zfloor x)%Rulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:0%R = round beta fexp Zfloor xulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)

  +beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:(0 < round beta fexp Zfloor x)%Rulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)
 rewrite ulp_neq_0 by now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:(0 < round beta fexp Zfloor x)%Rbpow (cexp beta fexp (round beta fexp Zfloor x)) =
bpow (cexp beta fexp x)

    now rewrite cexp_DN with (2 := Hd).
  +beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:0%R = round beta fexp Zfloor xulp (round beta fexp Zfloor x) =
bpow (cexp beta fexp x)
 rewrite <- Hd.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:0%R = round beta fexp Zfloor xulp 0 = bpow (cexp beta fexp x)

    unfold cexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:0%R = round beta fexp Zfloor xulp 0 = bpow (fexp (mag beta x))

    destruct (mag beta x) as [e He].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:0%R = round beta fexp Zfloor xe:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%Rulp 0 = bpow (fexp (Build_mag_prop beta x e He))

    simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:0%R = round beta fexp Zfloor xe:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%Rulp 0 = bpow (fexp e)

    specialize (He (Rgt_not_eq _ _ Hx)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:0%R = round beta fexp Zfloor xe:ZHe:(bpow (e - 1) <= Rabs x < bpow e)%Rulp 0 = bpow (fexp e)

    apply sym_eq in Hd.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:round beta fexp Zfloor x = 0%Re:ZHe:(bpow (e - 1) <= Rabs x < bpow e)%Rulp 0 = bpow (fexp e)

    assert (H := exp_small_round_0 _ _ _ _ _ He Hd).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:round beta fexp Zfloor x = 0%Re:ZHe:(bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%Zulp 0 = bpow (fexp e)

    unfold ulp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:round beta fexp Zfloor x = 0%Re:ZHe:(bpow (e - 1) <= Rabs x < bpow e)%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:round beta fexp Zfloor x = 0%Re:ZHe:(bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%Zmatch negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end = bpow (fexp e)

    destruct negligible_exp_spec as [H0|k Hk].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:round beta fexp Zfloor x = 0%Re:ZHe:(bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%ZH0:forall n : Z, (fexp n < n)%Z0%R = bpow (fexp e)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:round beta fexp Zfloor x = 0%Re:ZHe:(bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%Zk:ZHk:(k <= fexp k)%Zbpow (fexp k) = bpow (fexp e)

    now elim Zlt_not_le with (1 := H0 e).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:(0 < x)%RHd:round beta fexp Zfloor x = 0%Re:ZHe:(bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%Zk:ZHk:(k <= fexp k)%Zbpow (fexp k) = bpow (fexp e)

    now apply f_equal, fexp_negligible_exp_eq.
-beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RHx:0%R = xulp (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpnegligible_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpnegligible_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%RFalse

case negligible_exp_spec'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%Rnegligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) -> False1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%R(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False

intros (_,Hn).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%RHn:forall n : Z, (fexp n < n)%ZFalse1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%RHn:forall n : Z, (fexp n < n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%RFalse1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%RHn:forall n : Z, (fexp n < n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R~ (fexp e < e)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%RHn:forall n : Z, (fexp n < n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(fexp e < e)%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%R(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False

apply Zle_not_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%RHn:forall n : Z, (fexp n < n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(e <= fexp e)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%RHn:forall n : Z, (fexp n < n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(fexp e < e)%Zbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%RHn:forall n : Z, (fexp n < n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(fexp e < e)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%R(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False

apply (Hn e).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%R(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
False

intros (n,(H1,_)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:negligible_exp = Nonerndn:R -> ZHrnd:Valid_rnd rndnx:RHx:x <> 0%RK:round beta fexp rndn x = 0%Rn:ZH1:negligible_exp = Some nFalse

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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm: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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%R(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%R(0 <= x)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%R(x < 0)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:(0 < x)%R(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:0%R = x(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%R(x < 0)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:0%R = x(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%R(x < 0)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%R(x < 0)%R ->
(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:(x < 0)%R(Rabs (round beta fexp rnd x - x) <
 ulp (round beta fexp rnd x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:(x < 0)%R(Rabs (round beta fexp rnd (- - x) - - - x) <
 ulp (round beta fexp rnd (- - x)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:(x < 0)%R(Rabs (- round beta fexp (Zrnd_opp rnd) (- x) - - - x) <
 ulp (round beta fexp (Zrnd_opp rnd) (- x)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:(x < 0)%R(Rabs (- (round beta fexp (Zrnd_opp rnd) (- x) - - x)) <
 ulp (round beta fexp (Zrnd_opp rnd) (- x)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:(x < 0)%R(Rabs (round beta fexp (Zrnd_opp rnd) (- x) - - x) <
 ulp (round beta fexp (Zrnd_opp rnd) (- x)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:(x < 0)%RValid_rnd (Zrnd_opp rnd)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:(x < 0)%R(0 < - x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpM: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))%Rrnd:R -> ZHrnd:Valid_rnd rndx:RZx:x <> 0%RH:(x < 0)%R(0 < - x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(Rabs (round beta fexp rnd x - x) < ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(ulp x <= ulp (round beta fexp rnd x))%R

apply error_lt_ulp...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%Rx <> 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(ulp x <= ulp (round beta fexp rnd x))%R

now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(ulp x <= ulp (round beta fexp rnd x))%R

rewrite <- ulp_DN; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(ulp (round beta fexp Zfloor x) <=
 ulp (round beta fexp rnd x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%R

apply ulp_le_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= round beta fexp Zfloor x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(round beta fexp Zfloor x <= round beta fexp rnd x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%R

apply round_ge_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%RF 03 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(round beta fexp Zfloor x <= round beta fexp rnd x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%R

apply generic_format_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(round beta fexp Zfloor x <= round beta fexp rnd x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%R

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(round beta fexp Zfloor x <= round beta fexp rnd x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%R

case (round_DN_or_UP beta fexp rnd x); intros V; rewrite V.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%RV:round beta fexp rnd x = round beta fexp Zfloor x(round beta fexp Zfloor x <= round beta fexp Zfloor x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%RV:round beta fexp rnd x = round beta fexp Zceil x(round beta fexp Zfloor x <= round beta fexp Zceil x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%R

apply Rle_refl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%RV:round beta fexp rnd x = round beta fexp Zceil x(round beta fexp Zfloor x <= round beta fexp Zceil x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%R

apply Rle_trans with x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%RV:round beta fexp rnd x = round beta fexp Zceil x(round beta fexp Zfloor x <= x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%RV:round beta fexp rnd x = round beta fexp Zceil x(x <= round beta fexp Zceil x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%R

apply round_DN_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%RV:round beta fexp rnd x = round beta fexp Zceil x(x <= round beta fexp Zceil x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(0 < x)%R(0 <= x)%R

apply round_UP_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexprnd:R -> ZHrnd:Valid_rnd rndx:RHx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:R(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R

destruct (Req_dec x 0) as [Zx|Nzx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:RZx:x = 0%R(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:RNzx:x <> 0%R(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R

{beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:RZx:x = 0%R(Rabs (round beta fexp rnd x - x) <=
 ulp (round beta fexp rnd x))%R
 rewrite Zx, round_0; [|exact Vrnd].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:RZx:x = 0%R(Rabs (0 - 0) <= ulp 0)%R

  unfold Rminus; rewrite Ropp_0, Rplus_0_l, Rabs_R0; apply ulp_ge_0. }beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:RNzx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x = 0%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x = 0%R(/ 2 * ulp x <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x = 0%R(/ 2 * ulp 0 <= / 2 * ulp 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx: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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%R(Rabs (0 - x) <= / 2 * ulp x)%R ->
(Rabs (0 - x) <= / 2 * ulp 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%R(Rabs x <= / 2 * ulp x)%R -> (Rabs x <= / 2 * ulp 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%R(Rabs x <= / 2 * ulp 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%R(Rabs x <=
 / 2 *
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rnegligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) ->
(Rabs x <=
 / 2 *
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(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)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%RH1:negligible_exp = NoneH2:forall n : Z, (fexp n < n)%Z(Rabs x <=
 / 2 *
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(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)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%RH1:negligible_exp = NoneH2:forall n : Z, (fexp n < n)%Zround beta fexp (Znearest choice) x <> 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(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)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(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)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Z(Rabs x <= / 2 * bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Z(/ 2 * ulp x <= / 2 * bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Zulp x = bpow (fexp n)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Zbpow (cexp beta fexp x) = bpow (fexp n)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Zcexp beta fexp x = fexp n1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Zfexp (mag beta x) = fexp n1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Z(mag beta x <= fexp n)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Z(mag beta x - 1 < fexp n)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Z(bpow (mag beta x - 1) < bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(bpow (Build_mag_prop beta x e He - 1) < bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(bpow (e - 1) < bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(bpow (e - 1) <= Rabs x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(Rabs x < bpow (fexp n))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(Rabs x < bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(Rabs x <=
 Rabs (round beta fexp (Znearest choice) x - x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(Rabs (round beta fexp (Znearest choice) x - x) <
 bpow (fexp n))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%RRabs x = Rabs (- x)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(Rabs (round beta fexp (Znearest choice) x - x) <
 bpow (fexp n))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(Rabs (round beta fexp (Znearest choice) x - x) <
 bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(Rabs (round beta fexp (Znearest choice) x - x) <
 ulp 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(ulp 0 <= bpow (fexp n))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe: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))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(ulp 0 <= bpow (fexp n))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(ulp 0 <= bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x = 0%RHx:x <> 0%RN:(Rabs x <= / 2 * ulp x)%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(bpow (fexp n) <= bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(0 <= x)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(0 <= x)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp Zfloor x))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(0 <= x)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(/ 2 * ulp x <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(0 <= / 2)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(ulp x <= ulp (round beta fexp (Znearest choice) x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(ulp x <= ulp (round beta fexp (Znearest choice) x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(Rabs x <= Rabs (round beta fexp (Znearest choice) x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(- x <= Rabs (round beta fexp (Znearest choice) x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(- x <= Rabs (round beta fexp Zfloor x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(- x <= - round beta fexp Zfloor x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(round beta fexp Zfloor x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(round beta fexp Zfloor x <= x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(round beta fexp Zfloor x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(round beta fexp Zfloor x <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%RF 02 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(x <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zfloor xH:(x < 0)%R(x <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(0 <= x)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(0 <= x)%R(/ 2 * ulp x <=
 / 2 * ulp (round beta fexp (Znearest choice) x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(0 <= x)%R(0 <= / 2)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(0 <= x)%R(ulp x <= ulp (round beta fexp (Znearest choice) x))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(0 <= x)%R(ulp x <= ulp (round beta fexp (Znearest choice) x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(0 <= x)%R(x <= round beta fexp (Znearest choice) x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (- round beta fexp Zceil x))%R

rewrite <- round_DN_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (round beta fexp Zfloor (- x)))%R

rewrite ulp_DN; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(Rabs (round beta fexp (Znearest choice) x - x) <=
 / 2 * ulp (- x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(0 <= - x)%R

pattern x at 1 2; rewrite <- Ropp_involutive.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(Rabs
   (round beta fexp (Znearest choice) (- - x) - - - x) <=
 / 2 * ulp (- x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(0 <= - x)%R

rewrite round_N_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(Rabs
   (-
    round beta fexp
      (Znearest
         (fun t : Z => negb (choice (- (t + 1))%Z)))
      (- x) - - - x) <= / 2 * ulp (- x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(0 <= - x)%R

unfold Rminus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(Rabs
   (-
    round beta fexp
      (Znearest
         (fun t : Z => negb (choice (- (t + 1))%Z)))
      (- x) + - - - x) <= / 2 * ulp (- x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(0 <= - x)%R

rewrite <- Ropp_plus_distr, Rabs_Ropp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(Rabs
   (round beta fexp
      (Znearest
         (fun t : Z => negb (choice (- (t + 1))%Z)))
      (- x) + - - x) <= / 2 * ulp (- x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(0 <= - x)%R

apply error_le_half_ulp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(0 <= - x)%R

rewrite <- Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(x < 0)%R(- 0 <= - x)%R

apply Ropp_le_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice:Z -> boolx:RHfx:round beta fexp (Znearest choice) x <> 0%RHx:round beta fexp (Znearest choice) x =
round beta fexp Zceil xH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x <= y)%R -> (pred x <= pred y)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x <= y)%R -> (pred x <= pred y)%R

intros x y Fx Fy [Hxy| ->].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(pred x <= pred y)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpy:RFx, Fy:F y(pred y <= pred y)%R

2: apply Rle_refl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(pred x <= pred y)%R

apply pred_ge_gt with (2 := Fy).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%RF (pred x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(pred x < y)%R

now apply generic_format_pred.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(pred x < y)%R

apply Rle_lt_trans with (2 := Hxy).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x <= y)%R -> (succ x <= succ y)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x <= y)%R -> (succ x <= succ y)%R

intros x y Fx Fy Hxy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x <= y)%R(succ x <= succ y)%R

apply Ropp_le_cancel.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x <= y)%R(- succ y <= - succ x)%R

rewrite <- 2!pred_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x <= y)%R(pred (- y) <= pred (- x))%R

apply pred_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x <= y)%RF (- y)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x <= y)%RF (- x)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x <= y)%R(- y <= - x)%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x <= y)%RF (- x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x <= y)%R(- y <= - x)%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (pred x <= pred y)%R -> (x <= y)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (pred x <= pred y)%R -> (x <= y)%R

intros x y Fx Fy Hxy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(pred x <= pred y)%R(x <= y)%R

rewrite <- (succ_pred x), <- (succ_pred y); try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (succ x <= succ y)%R -> (x <= y)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (succ x <= succ y)%R -> (x <= y)%R

intros x y Fx Fy Hxy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(succ x <= succ y)%R(x <= y)%R

rewrite <- (pred_succ x), <- (pred_succ y); try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x < y)%R -> (pred x < pred y)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x < y)%R -> (pred x < pred y)%R

intros x y Fx Fy Hxy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(pred x < pred y)%R

apply Rnot_le_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R~ (pred y <= pred x)%R

intros H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%RH:(pred y <= pred x)%RFalse

apply Rgt_not_le with (1 := Hxy).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x < y)%R -> (succ x < succ y)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x y : R,
F x -> F y -> (x < y)%R -> (succ x < succ y)%R

intros x y Fx Fy Hxy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R(succ x < succ y)%R

apply Rnot_le_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%R~ (succ y <= succ x)%R

intros H.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%RH:(succ y <= succ x)%RFalse

apply Rgt_not_le with (1 := Hxy).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFx:F xFy:F yHxy:(x < y)%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_exp fexp ->
forall x : R, (succ x <= x + ulp x)%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_exp fexp ->
forall x : R, (succ x <= x + ulp x)%R

intros Mexp x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:R(succ x <= x + ulp x)%R

destruct (Rle_or_lt 0 x) as [Px|Nx]; [now right; apply succ_eq_pos|].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(x < 0)%R(succ x <= x + ulp x)%R

replace (_ + _)%R with (- (-x - ulp x))%R by ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(x < 0)%R(succ x <= - (- x - ulp x))%R

unfold succ; rewrite (Rle_bool_false _ _ Nx), <-ulp_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(x < 0)%R(- pred_pos (- x) <= - (- x - ulp (- x)))%R

apply Ropp_le_contravar; unfold pred_pos.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(x < 0)%RHx:(- 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)))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(x < 0)%RHx:(- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(x < 0)%RHx:(- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(x < 0)%RHx:(- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)(bpow (fexp (mag beta (- x) - 1)) <= ulp (- x))%R

   unfold ulp; rewrite Req_bool_false; [|lra].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(x < 0)%RHx:(- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RNx:(x < 0)%RHx:(- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_exp fexp ->
forall x : R, F x -> F (x + ulp x)

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_exp fexp ->
forall x : R, F x -> F (x + ulp x)

intros Mexp x Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F xF (x + ulp x)

destruct (Rle_or_lt 0 x) as [Px|Nx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F xPx:(0 <= x)%RF (x + ulp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F xNx:(x < 0)%RF (x + ulp x)

{beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F xPx:(0 <= x)%RF (x + ulp x)
 now rewrite <-(succ_eq_pos _ Px); apply generic_format_succ. }beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F xNx:(x < 0)%RF (x + ulp x)

apply generic_format_opp in Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RF (x + ulp x)

replace (_ + _)%R with (- (-x - ulp x))%R by ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RF (- (- x - ulp x))

apply generic_format_opp; rewrite <-ulp_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RF (- x - ulp (- x))

destruct (Req_dec (-x) (bpow (mag beta (-x) - 1))) as [Hx|Hx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)F (- x - ulp (- x))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R <> bpow (mag beta (- x) - 1)F (- x - ulp (- x))

{beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)F (- x - ulp (- x))
 unfold ulp; rewrite Req_bool_false; [|lra].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)F (- x - bpow (cexp beta fexp (- x)))

  rewrite Hx at 1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)F
  (bpow (mag beta (- x) - 1) -
   bpow (cexp beta fexp (- x)))

  unfold cexp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)F
  (bpow (mag beta (- x) - 1) -
   bpow (fexp (mag beta (- x))))

  set (e := mag _ _).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)(fexp e < e)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%ZF (bpow (e - 1) - bpow (fexp e))

  {beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%ZF (bpow (e - 1) - bpow (fexp e))

  replace (e - 1)%Z with (e - 1 - fexp e + fexp e)%Z by ring.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%ZF (bpow (e - 1 - fexp e + fexp e) - bpow (fexp e))

  rewrite bpow_plus.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%ZF
  (bpow (e - 1 - fexp e) * bpow (fexp e) -
   bpow (fexp e))

  set (m := bpow (_ - _)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):RF (m * bpow (fexp e) - bpow (fexp e))

  replace (_ - _)%R with ((m - 1) * bpow (fexp e))%R; [|unfold m; ring].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):RF ((m - 1) * bpow (fexp e))

  case_eq (e - 1 - fexp e)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):R(e - 1 - fexp e)%Z = 0%Z ->
F ((m - 1) * bpow (fexp e))2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rforall p : positive,
(e - 1 - fexp e)%Z = Z.pos p ->
F ((m - 1) * bpow (fexp e))beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rforall p : positive,
(e - 1 - fexp e)%Z = Z.neg p ->
F ((m - 1) * bpow (fexp e))

  {beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):RHe:(e - 1 - fexp e)%Z = 0%ZF (0 * bpow (fexp e))

    rewrite Rmult_0_l; apply generic_format_0. }beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rforall p : positive,
(e - 1 - fexp e)%Z = Z.pos p ->
F ((m - 1) * bpow (fexp e))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rforall p : positive,
(e - 1 - fexp e)%Z = Z.neg p ->
F ((m - 1) * bpow (fexp e))

  {beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pF ((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaF ((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))

    apply (generic_format_F2R' _ _ _ f); [|intro Hm'; unfold f; simpl].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaF2R f =
((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaF2R f =
((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R
 now unfold Defs.F2R; simpl; rewrite minus_IZR. }beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rp:positiveHp:(e - 1 - fexp e)%Z = Z.pos pf:={| Fnum := Z.pow_pos beta p - 1; Fexp := fexp e |}
:
float beta:float betaHm':((IZR (Z.pow_pos beta p) - 1) * bpow (fexp e))%R <>
0%RHe:(1 <= e - 1 - fexp e)%Ze':=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R = bpow (mag beta (- x) - 1)e:=mag beta (- x):mag_prop beta (- x)Hfe:(fexp e < e)%Zm:=bpow (e - 1 - fexp e):Rforall p : positive,
(e - 1 - fexp e)%Z = Z.neg p ->
F ((m - 1) * bpow (fexp e))

  intros p Hp; exfalso; lia. }beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R <> bpow (mag beta (- x) - 1)F (- x - ulp (- x))

replace (_ - _)%R with (pred_pos (-x)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R <> bpow (mag beta (- x) - 1)F (pred_pos (- x))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R <> bpow (mag beta (- x) - 1)pred_pos (- x) = (- x - ulp (- x))%R

{beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- x)%R <> bpow (mag beta (- x) - 1)F (pred_pos (- x))
 now apply generic_format_pred_pos; [|lra]. }beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMexp:Monotone_exp fexpx:RFx:F (- x)Nx:(x < 0)%RHx:(- 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(y < round beta fexp Zceil x)%R(y <= round beta fexp Zfloor x)%R

apply round_DN_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(y < round beta fexp Zceil x)%R(y <= x)%R

apply Rnot_lt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(y < round beta fexp Zceil x)%R~ (x < y)%R

contradict Hlt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(x < y)%R~ (y < round beta fexp Zceil x)%R

apply RIneq.Rle_not_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(x < y)%R(round beta fexp Zceil x <= y)%R

apply round_UP_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(round beta fexp Zfloor x < y)%R(round beta fexp Zceil x <= y)%R

apply round_UP_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(round beta fexp Zfloor x < y)%R(x <= y)%R

apply Rnot_lt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(round beta fexp Zfloor x < y)%R~ (y < x)%R

contradict Hlt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(y < x)%R~ (round beta fexp Zfloor x < y)%R

apply RIneq.Rle_not_lt.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(y < x)%R(y <= round beta fexp Zfloor x)%R

apply round_DN_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, y:RFy:F yHlt:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R

intros x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:R(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R

destruct (generic_format_EM beta fexp x) as [Fx|Fx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F x(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F x(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R

rewrite !round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F x(pred x <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F x(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R

apply pred_le_id.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%R(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%R(- succ 0 <= round beta fexp Zfloor x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%R(- (0 + ulp 0) <= round beta fexp Zfloor x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%R(-
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= round beta fexp Zfloor x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x <> 0%R(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R

case negligible_exp_spec'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rnegligible_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)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x <> 0%R(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R

intros (H1,H2).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%RH1:negligible_exp = NoneH2:forall n : Z, (fexp n < n)%Z(-
 match negligible_exp with
 | Some n => bpow (fexp n)
 | None => 0
 end <= round beta fexp Zfloor x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xH1:negligible_exp = NoneH2:forall n : Z, (fexp n < n)%Zx <> 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%Z(- bpow (fexp n) <= round beta fexp Zfloor x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%ZK:(round beta fexp Zfloor x < - bpow (fexp n))%R(- bpow (fexp n) <= round beta fexp Zfloor x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%ZK:(round beta fexp Zfloor x < - bpow (fexp n))%R~ (round beta fexp Zceil x <= - bpow (fexp n))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%ZK:(round beta fexp Zfloor x < - bpow (fexp n))%R(round beta fexp Zceil x <= - bpow (fexp n))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x <> 0%R(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R

apply Rlt_not_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%ZK:(round beta fexp Zfloor x < - bpow (fexp n))%R(- bpow (fexp n) < round beta fexp Zceil x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%ZK:(round beta fexp Zfloor x < - bpow (fexp n))%R(round beta fexp Zceil x <= - bpow (fexp n))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x <> 0%R(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R

rewrite Zx, <- Ropp_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%ZK:(round beta fexp Zfloor x < - bpow (fexp n))%R(- bpow (fexp n) < - 0)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%ZK:(round beta fexp Zfloor x < - bpow (fexp n))%R(round beta fexp Zceil x <= - bpow (fexp n))%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%ZK:(round beta fexp Zfloor x < - bpow (fexp n))%R(round beta fexp Zceil x <= - bpow (fexp n))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%ZK:(round beta fexp Zfloor x < - bpow (fexp n))%RF (- bpow (fexp n))1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x = 0%Rn:ZH1:negligible_exp = Some nHn:(n <= fexp n)%ZK:(round beta fexp Zfloor x < - bpow (fexp n))%R(fexp (fexp n + 1) <= fexp n)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x <> 0%R(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R

now apply valid_exp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x <> 0%Rlet u := round beta fexp Zceil x in (pred u < u)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x <> 0%RHup: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x <> 0%RHup: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x <> 0%RHup:let u := round beta fexp Zceil x in
(pred u < u)%RF (pred (round beta fexp Zceil x))

apply generic_format_pred...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xZx:round beta fexp Zceil x <> 0%RHup:let u := round beta fexp Zceil x in
(pred u < u)%RF (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(round beta fexp Zceil x <=
 succ (round beta fexp Zfloor x))%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
(round beta fexp Zceil x <=
 succ (round beta fexp Zfloor x))%R

intros x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:R(round beta fexp Zceil x <=
 succ (round beta fexp Zfloor x))%R

rewrite <- (Ropp_involutive x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:R(round beta fexp Zceil (- - x) <=
 succ (round beta fexp Zfloor (- - x)))%R

rewrite round_DN_opp, round_UP_opp, succ_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:R(- round beta fexp Zfloor (- x) <=
 - pred (round beta fexp Zceil (- x)))%R

apply Ropp_le_contravar.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
~ F x ->
pred (round beta fexp Zceil x) =
round beta fexp Zfloor x

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
~ F x ->
pred (round beta fexp Zceil x) =
round beta fexp Zfloor x

intros x Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xpred (round beta fexp Zceil x) =
round beta fexp Zfloor x

apply Rle_antisym.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F x(pred (round beta fexp Zceil x) <=
 round beta fexp Zfloor x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F x(round beta fexp Zfloor x <=
 pred (round beta fexp Zceil x))%R

now apply pred_UP_le_DN.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F x(round beta fexp Zfloor x <=
 pred (round beta fexp Zceil x))%R

apply pred_ge_gt; try apply generic_format_round...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F x(round beta fexp Zfloor x < round beta fexp Zceil x)%R

pose proof round_DN_UP_lt _ _ _ Fx as HE.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xHE:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
~ F x ->
succ (round beta fexp Zfloor x) =
round beta fexp Zceil x

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
~ F x ->
succ (round beta fexp Zfloor x) =
round beta fexp Zceil x

intros x Fx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xsucc (round beta fexp Zfloor x) =
round beta fexp Zceil x

rewrite <- pred_UP_eq_DN; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xsucc (pred (round beta fexp Zceil x)) =
round beta fexp Zceil x

rewrite succ_pred; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:~ F xF (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x d : R,
F d ->
(d <= x < succ d)%R -> round beta fexp Zfloor x = d

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x d : R,
F d ->
(d <= x < succ d)%R -> round beta fexp Zfloor x = d

intros x d Fd (Hxd1,Hxd2).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%Rround beta fexp Zfloor x = d

generalize (round_DN_pt beta fexp x); intros (T1,(T2,T3)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%Rround beta fexp Zfloor x = d

apply sym_eq, Rle_antisym.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%R(d <= round beta fexp Zfloor x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RFx:F x(round beta fexp Zfloor x <= d)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RNFx:~ F x(round beta fexp Zfloor x <= d)%R

rewrite round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RFx:F x(x <= d)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RNFx:~ F x(round beta fexp Zfloor x <= d)%R

apply succ_le_inv; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RFx:F x(succ x <= succ d)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RNFx:~ F x(round beta fexp Zfloor x <= d)%R

apply succ_le_lt; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RFx:F xF (succ d)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RNFx:~ F x(round beta fexp Zfloor x <= d)%R

apply generic_format_succ...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RNFx:~ F x(round beta fexp Zfloor x <= d)%R

apply succ_le_inv; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RNFx:~ F x(succ (round beta fexp Zfloor x) <= succ d)%R

rewrite succ_DN_eq_UP; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RNFx:~ F x(round beta fexp Zceil x <= succ d)%R

apply round_UP_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RNFx:~ F xF (succ d)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RNFx:~ F x(x <= succ d)%R

apply generic_format_succ...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, d:RFd:F dHxd1:(d <= x)%RHxd2:(x < succ d)%RT1:F (round beta fexp Zfloor x)T2:(round beta fexp Zfloor x <= x)%RT3:forall g : R,
F g ->
(g <= x)%R -> (g <= round beta fexp Zfloor x)%RNFx:~ 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x u : R,
F u ->
(pred u < x <= u)%R -> round beta fexp Zceil x = u

Proof with auto with typeclass_instances.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x u : R,
F u ->
(pred u < x <= u)%R -> round beta fexp Zceil x = u

intros x u Fu Hux.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, u:RFu:F uHux:(pred u < x <= u)%Rround beta fexp Zceil x = u

rewrite <- (Ropp_involutive (round beta fexp Zceil x)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, u:RFu:F uHux:(pred u < x <= u)%R(- - round beta fexp Zceil x)%R = u

rewrite <- round_DN_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, u:RFu:F uHux:(pred u < x <= u)%R(- round beta fexp Zfloor (- x))%R = u

rewrite <- (Ropp_involutive u).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, u:RFu:F uHux:(pred u < x <= u)%R(- round beta fexp Zfloor (- x))%R = (- - u)%R

apply f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, u:RFu:F uHux:(pred u < x <= u)%Rround beta fexp Zfloor (- x) = (- u)%R

apply round_DN_eq; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, u:RFu:F uHux:(pred u < x <= u)%RF (- u)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, u:RFu:F uHux:(pred u < x <= u)%R(- u <= - x < succ (- u))%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, u:RFu:F uHux:(pred u < x <= u)%R(- u <= - x < succ (- u))%R

split;[now apply Ropp_le_contravar|idtac].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, u:RFu:F uHux:(pred u < x <= u)%R(- x < succ (- u))%R

rewrite succ_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx, u:RFu:F uHux:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpExp_not_FTZ fexp -> ulp (ulp 0) = ulp 0

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpExp_not_FTZ fexp -> ulp (ulp 0) = ulp 0

intros H; case (negligible_exp_spec').beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpnegligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) -> ulp (ulp 0) = ulp 01 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexp(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp (ulp 0) = ulp 0

intros (K1,K2).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpK1:negligible_exp = NoneK2:forall n : Z, (fexp n < n)%Zulp (ulp 0) = ulp 01 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpK1:negligible_exp = NoneK2:forall n : Z, (fexp n < n)%Z0%R = ulp 01 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpK1:negligible_exp = NoneK2:forall n : Z, (fexp n < n)%Zmatch negligible_exp with
| Some n => bpow (fexp n)
| None => 0%R
end = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexp(exists n : Z,
   negligible_exp = Some n /\ (n <= fexp n)%Z) ->
ulp (ulp 0) = ulp 0

now rewrite K1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zulp (ulp 0) = ulp 0

apply Rle_antisym.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(ulp (ulp 0) <= ulp 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(ulp 0 <= ulp (ulp 0))%R

replace (ulp 0) with (bpow (fexp n)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(ulp (bpow (fexp n)) <= bpow (fexp n))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) = ulp 0beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(ulp 0 <= ulp (ulp 0))%R

rewrite ulp_bpow.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(bpow (fexp (fexp n + 1)) <= bpow (fexp n))%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) = ulp 0beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(ulp 0 <= ulp (ulp 0))%R

apply bpow_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(fexp (fexp n + 1) <= fexp n)%Z2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) = ulp 0beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(ulp 0 <= ulp (ulp 0))%R

now apply valid_exp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) = ulp 01 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(ulp 0 <= ulp (ulp 0))%R

unfold ulp; rewrite Req_bool_true; try easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) =
match negligible_exp with
| Some n0 => bpow (fexp n0)
| None => 0%R
end1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Z(ulp 0 <= ulp (ulp 0))%R

rewrite Hn1; easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpH:Exp_not_FTZ fexpn:ZHn1:negligible_exp = Some nHn2:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall x : R,
F x ->
(0 < x)%R ->
ulp (succ x) = ulp x \/ succ x = bpow (mag beta x)

intros x Fx Hx.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%Rulp (succ x) = ulp x \/ succ x = bpow (mag beta x)

generalize (Rlt_le _ _ Hx); intros Hx'.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Rulp (succ x) = ulp x \/ succ x = bpow (mag beta x)

rewrite succ_eq_pos;[idtac|now left].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Rulp (x + ulp x) = ulp x \/
(x + ulp x)%R = bpow (mag beta x)

destruct (mag beta x) as (e,He); simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%Rulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e

rewrite Rabs_pos_eq in He; try easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:x <> 0%R -> (bpow (e - 1) <= x < bpow e)%Rulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e

specialize (He (Rgt_not_eq _ _ Hx)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%Rulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e

assert (H:(x+ulp x <= bpow e)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%R(x + ulp x <= bpow e)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x <= bpow e)%Rulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e

apply id_p_ulp_le_bpow; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%R(x < bpow e)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x <= bpow e)%Rulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e

apply He.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x <= bpow e)%Rulp (x + ulp x) = ulp x \/ (x + ulp x)%R = bpow e

destruct H;[left|now right].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Rulp (x + ulp x) = ulp x

rewrite ulp_neq_0 at 1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Rbpow (cexp beta fexp (x + ulp x)) = ulp x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Rbpow (cexp beta fexp (x + ulp x)) = ulp x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Rbpow (cexp beta fexp (x + ulp x)) = ulp x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%R(0 <= ulp x)%R

2: apply ulp_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Rbpow (cexp beta fexp (x + ulp x)) = ulp x

rewrite ulp_neq_0 at 2.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Rbpow (cexp beta fexp (x + ulp x)) =
bpow (cexp beta fexp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Rx <> 0%R

2: now apply Rgt_not_eq.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Rbpow (cexp beta fexp (x + ulp x)) =
bpow (cexp beta fexp x)

f_equal; unfold cexp; f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Rmag beta (x + ulp x) = mag beta x

apply trans_eq with e.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Rmag beta (x + ulp x) = e1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Re = mag beta x

apply mag_unique_pos; split; try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%R(bpow (e - 1) <= x + ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Re = mag beta x

apply Rle_trans with (1:=proj1 He).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%R(x <= x + ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Re = mag beta x

rewrite <- (Rplus_0_r x) at 1; apply Rplus_le_compat_l.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%R(0 <= ulp x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Re = mag beta x

apply ulp_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpx:RFx:F xHx:(0 < x)%RHx':(0 <= x)%Re:ZHe:(bpow (e - 1) <= x < bpow e)%RH:(x + ulp x < bpow e)%Re = mag beta x

now apply sym_eq, mag_unique_pos.
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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpExp_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpExp_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%Rulp (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:F xulp (round beta fexp rnd x) = ulp x \/
round beta fexp rnd x = bpow (mag beta x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xulp (round beta fexp rnd x) = ulp x \/
round beta fexp rnd x = bpow (mag beta x)

rewrite round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xulp (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zfloor xulp (round beta fexp Zfloor x) = ulp x \/
round beta fexp Zfloor x = bpow (mag beta x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

left.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zfloor xulp (round beta fexp Zfloor x) = ulp x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

apply ulp_DN; now left...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xulp (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil x(0 <= round beta fexp Zfloor x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 <= round beta fexp Zfloor x)%Rulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

apply round_ge_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xF 02 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil x(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 <= round beta fexp Zfloor x)%Rulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

apply generic_format_0...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil x(0 <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 <= round beta fexp Zfloor x)%Rulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

apply Rlt_le...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 <= round beta fexp Zfloor x)%Rulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

destruct M as [M|M].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%Rulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

rewrite <- (succ_DN_eq_UP x)...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%Rulp (succ (round beta fexp Zfloor x)) = ulp x \/
succ (round beta fexp Zfloor x) = bpow (mag beta x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RF (round beta fexp Zfloor x)4 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%R(0 < round beta fexp Zfloor x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RY: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RY: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

apply generic_format_round...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%R(0 < round beta fexp Zfloor x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RY: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RY: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RY: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)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RY: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

rewrite ulp_DN in Y...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RY:ulp (succ (round beta fexp Zfloor x)) =
ulp (round beta fexp Zfloor x)(0 <= x)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RY: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

now apply Rlt_le.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RY: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)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

right; rewrite Y.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:(0 < round beta fexp Zfloor x)%RY: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)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

apply f_equal, mag_DN...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (round beta fexp Zceil x) = ulp x \/
round beta fexp Zceil x = bpow (mag beta x)

left; rewrite <- (succ_DN_eq_UP x)...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (succ (round beta fexp Zfloor x)) = ulp x

rewrite <- M, succ_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp (ulp 0) = ulp x

rewrite ulp_ulp_0...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xulp 0 = ulp x

case (negligible_exp_spec').beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xnegligible_exp = None /\
(forall n : Z, (fexp n < n)%Z) -> ulp 0 = ulp x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xK1:negligible_exp = NoneK2:forall n : Z, (fexp n < n)%Zulp 0 = ulp x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xK1:negligible_exp = NoneK2:forall n : Z, (fexp n < n)%Zx <> 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xK1:negligible_exp = NoneK2:forall n : Z, (fexp n < n)%Zx = 0%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xK1:negligible_exp = NoneK2:forall n : Z, (fexp n < n)%Zx = 0%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zulp 0 = ulp x

replace (ulp 0) with (bpow (fexp n)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) = ulp x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) = ulp 0

2: unfold ulp; rewrite Req_bool_true; try easy.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) = ulp x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) =
match negligible_exp with
| Some n0 => bpow (fexp n0)
| None => 0%R
end

2: now rewrite Hn1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) = ulp x

rewrite ulp_neq_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) = bpow (cexp beta fexp x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zx <> 0%R

2: apply Rgt_not_eq...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zbpow (fexp n) = bpow (cexp beta fexp x)

unfold cexp; f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Zfexp n = fexp (mag beta x)

destruct (mag beta x) as (e,He); simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%Rfexp n = fexp e

apply sym_eq, valid_exp...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(e <= fexp n)%Z

assert (e <= fexp e)%Z.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(e <= fexp e)%Z1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%Z(e <= fexp n)%Z

apply exp_small_round_0_pos with beta Zfloor x...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(bpow (e - 1) <= x < bpow e)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%Z(e <= fexp n)%Z

rewrite <- (Rabs_pos_eq x).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(bpow (e - 1) <= Rabs x < bpow e)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(0 <= x)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%Z(e <= fexp n)%Z

apply He, Rgt_not_eq...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%R(0 <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%Z(e <= fexp n)%Z

apply Rlt_le...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%Z(e <= fexp n)%Z

replace (fexp n) with (fexp e); try assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RHx:(0 < x)%RFx:~ F xHr:round beta fexp rnd x = round beta fexp Zceil xM:0%R = round beta fexp Zfloor xn:ZHn1:negligible_exp = Some nHn2:(n <= fexp n)%Ze:ZHe:x <> 0%R -> (bpow (e - 1) <= Rabs x < bpow e)%RH:(e <= fexp e)%Zfexp 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpExp_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpExp_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:Rulp (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%R(0 < - x)%R3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rulp (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rround 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rulp (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)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rround 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

rewrite ulp_opp, <- ulp_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rulp (- 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)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rround 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

rewrite <- round_opp, Ropp_involutive.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rulp (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)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rround 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

intros Y;now left.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rround 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)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

rewrite mag_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%Rround 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)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

intros Y; right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%RY:round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta x)Rabs (round beta fexp rnd x) = bpow (mag beta x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

rewrite <- (Ropp_involutive x) at 1.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%RY:round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta x)Rabs (round beta fexp rnd (- - x)) = bpow (mag beta x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

rewrite round_opp, Y.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%RY:round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta x)Rabs (- bpow (mag beta x)) = bpow (mag beta x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

rewrite Rabs_Ropp, Rabs_right...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x < 0)%RY:round beta fexp (Zrnd_opp rnd) (- x) =
bpow (mag beta x)(bpow (mag beta x) >= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

apply Rle_ge, bpow_ge_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%R \/ (x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

destruct Zx as [Zx|Zx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:x = 0%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

left; rewrite Zx; rewrite round_0...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
Rabs (round beta fexp rnd x) = bpow (mag beta x)

rewrite Rabs_right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x > 0)%Rulp (round beta fexp rnd x) = ulp x \/
round beta fexp rnd x = bpow (mag beta x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x > 0)%R(round beta fexp rnd x >= 0)%R

apply ulp_round_pos...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x > 0)%R(round beta fexp rnd x >= 0)%R

apply Rle_ge, round_ge_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x > 0)%RF 01 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(x > 0)%R(0 <= x)%R

apply generic_format_0...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpNot_FTZ_:Exp_not_FTZ fexprnd:R -> ZZrnd:Valid_rnd rndx:RZx:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, (x <= succ (round beta fexp rnd x))%R

Proof.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall rnd : R -> Z,
Valid_rnd rnd ->
forall x : R, (x <= succ (round beta fexp rnd x))%R

intros rnd Vrnd x.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:R(x <= succ (round beta fexp rnd x))%R

apply (Rle_trans _ (round beta fexp Raux.Zceil x)).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:R(x <= round beta fexp Zceil x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:R(round beta fexp Zceil x <=
 succ (round beta fexp rnd x))%R

{beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:R(x <= round beta fexp Zceil x)%R
 now apply round_UP_pt. }beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZVrnd:Valid_rnd rndx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:RHr:round beta fexp rnd x = round beta fexp Zfloor x(round beta fexp Zceil x <=
 succ (round beta fexp Zfloor x))%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:RHr:round beta fexp rnd x = round beta fexp Zceil x(round beta fexp Zceil x <=
 succ (round beta fexp Zceil x))%R

{beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:RHr: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexprnd:R -> ZVrnd:Valid_rnd rndx:RHr: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.

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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%Rsucc u = 0%R /\ u = 0%R \/ (u < succ u)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RP:(u < succ u)%Rsucc u = 0%R /\ u = 0%R \/ (u < succ u)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RP:u = succ usucc u = 0%R /\ u = 0%R \/ (u < succ u)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:succ u = 0%R /\ u = 0%R \/ (u < succ u)%R(round beta fexp (Znearest choice) v <= u)%R

now right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RP:u = succ usucc u = 0%R /\ u = 0%R \/ (u < succ u)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RP:u = succ uZu:u = 0%Rsucc u = 0%R /\ u = 0%R \/ (u < succ u)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RP:u = succ uZu:u <> 0%Rsucc u = 0%R /\ u = 0%R \/ (u < succ u)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:succ u = 0%R /\ u = 0%R \/ (u < succ u)%R(round beta fexp (Znearest choice) v <= u)%R

left; split; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RP:u = succ uZu:u = 0%Rsucc u = 0%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RP:u = succ uZu:u <> 0%Rsucc u = 0%R /\ u = 0%R \/ (u < succ u)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:succ u = 0%R /\ u = 0%R \/ (u < succ u)%R(round beta fexp (Znearest choice) v <= u)%R

now rewrite <- P.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RP:u = succ uZu:u <> 0%Rsucc u = 0%R /\ u = 0%R \/ (u < succ u)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV1:succ u = 0%RV2:u = 0%R(round beta fexp (Znearest choice) v <= u)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%R(round beta fexp (Znearest choice) v <= u)%R

rewrite V2; apply round_le_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV1:succ u = 0%RV2:u = 0%RF 02 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV1:succ u = 0%RV2:u = 0%R(v <= 0)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%R(round beta fexp (Znearest choice) v <= u)%R

apply generic_format_0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV1:succ u = 0%RV2:u = 0%R(v <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%R(round beta fexp (Znearest choice) v <= u)%R

left; apply Rlt_le_trans with (1:=H).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV1:succ u = 0%RV2:u = 0%R((u + succ u) / 2 <= 0)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%R(round beta fexp (Znearest choice) v <= u)%R

rewrite V1,V2; right; field.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT:(u < (u + succ u) / 2 < succ u)%R(round beta fexp (Znearest choice) v <= u)%R

destruct T as (T1,T2).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_N_pt F v (round beta fexp (Znearest choice) v)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_N_pt F ((u + succ u) / 2) u

apply round_N_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_N_pt F ((u + succ u) / 2) u

apply Rnd_N_pt_DN with (succ u)%R.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_DN_pt F ((u + succ u) / 2) u2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_UP_pt F ((u + succ u) / 2) (succ u)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_DN_pt F ((u + succ u) / 2)
  (round beta fexp Zfloor ((u + succ u) / 2))3 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%Rround beta fexp Zfloor ((u + succ u) / 2) = ubeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_UP_pt F ((u + succ u) / 2) (succ u)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%R((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R

apply round_DN_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%Rround beta fexp Zfloor ((u + succ u) / 2) = u2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_UP_pt F ((u + succ u) / 2) (succ u)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%R(u <= (u + succ u) / 2 < succ u)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_UP_pt F ((u + succ u) / 2) (succ u)beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%R((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R

split; try left; assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_UP_pt F ((u + succ u) / 2) (succ u)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RRnd_UP_pt F ((u + succ u) / 2)
  (round beta fexp Zceil ((u + succ u) / 2))2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%Rround beta fexp Zceil ((u + succ u) / 2) = succ ubeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%R((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R

apply round_UP_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%Rround beta fexp Zceil ((u + succ u) / 2) = succ u1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%RF (succ u)2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%R(pred (succ u) < (u + succ u) / 2 <= succ u)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%R((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R

apply generic_format_succ...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%R(pred (succ u) < (u + succ u) / 2 <= succ u)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%R((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R

rewrite pred_succ; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%R(u < (u + succ u) / 2 <= succ u)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((u + succ u) / 2 < succ u)%R((u + succ u) / 2 - u <= succ u - (u + succ u) / 2)%R

split; try left; assumption.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:(v < (u + succ u) / 2)%RV:(u < succ u)%RT1:(u < (u + succ u) / 2)%RT2:((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((u + pred u) / 2 < v)%R(u <= round beta fexp (Znearest choice) v)%R

rewrite <- (Ropp_involutive v).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((u + pred u) / 2 < v)%R(u <= round beta fexp (Znearest choice) (- - v))%R

rewrite round_N_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((u + pred u) / 2 < v)%RF (- u)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((u + pred u) / 2 < v)%R(- v < (- u + succ (- u)) / 2)%R

now apply generic_format_opp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((u + pred u) / 2 < v)%R(- v < (- u + succ (- u)) / 2)%R

apply Ropp_lt_cancel.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((u + pred u) / 2 < v)%R(- ((- u + succ (- u)) / 2) < - - v)%R

rewrite Ropp_involutive.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((u + pred u) / 2 < v)%R(- ((- u + succ (- u)) / 2) < v)%R

apply Rle_lt_trans with (2:=H).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((u + pred u) / 2 < v)%R(- ((- u + succ (- u)) / 2) <= (u + pred u) / 2)%R

unfold pred.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolu, v:RFu:F uH:((u + pred u) / 2 < v)%R(- ((- u + succ (- u)) / 2) <= (u + - succ (- u)) / 2)%R

right; field.
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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%Rround beta fexp (Znearest choice) x = d

apply Rle_antisym.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%R(round beta fexp (Znearest choice) x <= d)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%R(d <= round beta fexp (Znearest choice) x)%R

destruct (generic_format_EM beta fexp x) as [Fx|Fx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%RFx:F x(round beta fexp (Znearest choice) x <= d)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%RFx:~ F x(round beta fexp (Znearest choice) x <= d)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%R(d <= round beta fexp (Znearest choice) x)%R

rewrite round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%RFx:F x(x <= d)%R2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%RFx:~ F x(round beta fexp (Znearest choice) x <= d)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%R(d <= round beta fexp (Znearest choice) x)%R

apply round_DN_pt; trivial; now right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%RFx:~ F x(round beta fexp (Znearest choice) x <= d)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%R(d <= round beta fexp (Znearest choice) x)%R

apply round_N_le_midp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%RFx:~ F xF d2 more goalsbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%RFx:~ F x(x < (d + succ d) / 2)%Rbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%R(d <= round beta fexp (Znearest choice) x)%R

apply round_DN_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%RFx:~ F x(x < (d + succ d) / 2)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%R(d <= round beta fexp (Znearest choice) x)%R

apply Rlt_le_trans with (1:=H).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%RFx:~ F x((d + u) / 2 <= (d + succ d) / 2)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%R(d <= round beta fexp (Znearest choice) x)%R

right; apply f_equal2; trivial; apply f_equal.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%RFx:~ F xu = succ d1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(x < (d + u) / 2)%R(d <= round beta fexp (Znearest choice) x)%R

now apply sym_eq, succ_DN_eq_UP.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:(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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:(x < (d + u) / 2)%Rround beta fexp (Znearest choice) x = d

assert (H0:(d = round beta fexp Zfloor x)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:(x < (d + u) / 2)%Rd = round beta fexp Zfloor x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:(x < (d + u) / 2)%RH0:d = round beta fexp Zfloor xround beta fexp (Znearest choice) x = d

apply Rnd_DN_pt_unique with (1:=Hd).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:(x < (d + u) / 2)%RRnd_DN_pt F x (round beta fexp Zfloor x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:(x < (d + u) / 2)%RH0:d = round beta fexp Zfloor xround beta fexp (Znearest choice) x = d

apply round_DN_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:(x < (d + u) / 2)%RH0:d = round beta fexp Zfloor xround beta fexp (Znearest choice) x = d

rewrite H0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:(x < (d + u) / 2)%RH0:d = round beta fexp Zfloor xround beta fexp (Znearest choice) x =
round beta fexp Zfloor x

apply round_N_eq_DN.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:(x < (d + u) / 2)%RH0:d = round beta fexp Zfloor x(x <
 (round beta fexp Zfloor x + round beta fexp Zceil x) /
 2)%R

rewrite <- H0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:(x < (d + u) / 2)%RH0: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:(x < (d + u) / 2)%RH0:d = round beta fexp Zfloor xRnd_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%Rround beta fexp (Znearest choice) x = u

apply Rle_antisym.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%R(round beta fexp (Znearest choice) x <= u)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%R(u <= round beta fexp (Znearest choice) x)%R

apply round_le_generic; try apply round_UP_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%R(u <= round beta fexp (Znearest choice) x)%R

destruct (generic_format_EM beta fexp x) as [Fx|Fx].beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%RFx:F x(u <= round beta fexp (Znearest choice) x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%RFx:~ F x(u <= round beta fexp (Znearest choice) x)%R

rewrite round_generic...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%RFx:F x(u <= x)%R1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%RFx:~ F x(u <= round beta fexp (Znearest choice) x)%R

apply round_UP_pt; trivial; now right.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%RFx:~ F x(u <= round beta fexp (Znearest choice) x)%R

apply round_N_ge_midp.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%RFx:~ F xF u1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%RFx:~ F x((u + pred u) / 2 < x)%R

apply round_UP_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%RFx:~ F x((u + pred u) / 2 < x)%R

apply Rle_lt_trans with (2:=H).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%RFx:~ F x((u + pred u) / 2 <= (d + u) / 2)%R

right; apply f_equal2; trivial; rewrite Rplus_comm; apply f_equal2; trivial.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx:Rd:=round beta fexp Zfloor x:Ru:=round beta fexp Zceil x:RH:((d + u) / 2 < x)%RFx:~ F xpred 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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpforall (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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:((d + u) / 2 < x)%Rround beta fexp (Znearest choice) x = u

assert (H0:(u = round beta fexp Zceil x)%R).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:((d + u) / 2 < x)%Ru = round beta fexp Zceil x1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:((d + u) / 2 < x)%RH0:u = round beta fexp Zceil xround beta fexp (Znearest choice) x = u

apply Rnd_UP_pt_unique with (1:=Hu).beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:((d + u) / 2 < x)%RRnd_UP_pt F x (round beta fexp Zceil x)1 more goalbeta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:((d + u) / 2 < x)%RH0:u = round beta fexp Zceil xround beta fexp (Znearest choice) x = u

apply round_UP_pt...beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:((d + u) / 2 < x)%RH0:u = round beta fexp Zceil xround beta fexp (Znearest choice) x = u

rewrite H0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:((d + u) / 2 < x)%RH0:u = round beta fexp Zceil xround beta fexp (Znearest choice) x =
round beta fexp Zceil x

apply round_N_eq_UP.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:((d + u) / 2 < x)%RH0:u = round beta fexp Zceil x((round beta fexp Zfloor x + round beta fexp Zceil x) /
 2 < x)%R

rewrite <- H0.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:((d + u) / 2 < x)%RH0: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpchoice:Z -> boolx, d, u:RHd:Rnd_DN_pt F x dHu:Rnd_UP_pt F x uH:((d + u) / 2 < x)%RH0:u = round beta fexp Zceil xRnd_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpMonotone_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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice1, choice2:Z -> boolx:Rlet rx := round beta fexp (Znearest choice2) x in
(x <= round beta fexp (Znearest choice1) (rx + ulp rx))%R

simpl.beta:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice1, choice2:Z -> boolx: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice1, choice2:Z -> boolx:Rrx:=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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice1, choice2:Z -> boolx:Rrx:=round beta fexp (Znearest choice2) x:RVrnd1: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice1, choice2:Z -> boolx:Rrx:=round beta fexp (Znearest choice2) x:RVrnd1: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice1, choice2:Z -> boolx:Rrx:=round beta fexp (Znearest choice2) x:RVrnd1: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:radixfexp:Z -> Zvalid_exp:Valid_exp fexpHm:Monotone_exp fexpchoice1, choice2:Z -> boolx:Rrx:=round beta fexp (Znearest choice2) x:RVrnd1:Valid_rnd (Znearest choice1)Vrnd2:Valid_rnd (Znearest choice2)F (rx + ulp rx)

now apply generic_format_plus_ulp, generic_format_round.
Qed.

End Fcore_ulp.