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src/Core/Fcore.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,6 +17,7 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * To ease the import *)
 Require Export Fcore_Raux.
 Require Export Fcore_defs.
 Require Export Fcore_float_prop.

src/Core/Fcore_FIX.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,6 +17,7 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * Fixed-point format *)
 Require Import Fcore_Raux.
 Require Import Fcore_defs.
 Require Import Fcore_rnd.
@@ -30,7 +32,7 @@ Notation bpow := (bpow beta).
 
 Variable emin : Z.
 
-(* fixed-point format *)
+(* fixed-point format with exponent emin *)
 Definition FIX_format (x : R) :=
   exists f : float beta,
   x = F2R f /\ (Fexp f = emin)%Z.
@@ -40,6 +42,7 @@ Definition FIX_RoundingModeP (rnd : R -> R):=
 
 Definition FIX_exp (e : Z) := emin.
 
+(** Properties of the FIX format *)
 Theorem FIX_exp_correct : valid_exp FIX_exp.
 Proof.
 intros k.

src/Core/Fcore_FLT.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,6 +17,7 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * Floating-point format with gradual underflow *)
 Require Import Fcore_Raux.
 Require Import Fcore_defs.
 Require Import Fcore_rnd.
@@ -44,6 +46,7 @@ Definition FLT_RoundingModeP (rnd : R -> R):=
 
 Definition FLT_exp e := Zmax (e - prec) emin.
 
+(** Properties of the FLT format *)
 Theorem FLT_exp_correct : valid_exp FLT_exp.
 Proof.
 intros k.
@@ -148,6 +151,7 @@ apply Rle_lt_trans with (1 := Hx).
 apply He.
 Qed.
 
+(** Links between FLT and FLX *)
 Theorem FLT_generic_format_FLX :
   forall x : R,
   (bpow (emin + prec - 1) <= Rabs x)%R ->
@@ -200,7 +204,7 @@ apply bpow_gt_0.
 Qed.
 
 
-
+(** Links between FLT and FIX (underflow) *)
 Theorem FLT_canonic_FIX :
   forall x, x <> R0 ->
   (Rabs x < bpow (emin + prec))%R ->
@@ -261,6 +265,7 @@ apply iff_sym.
 now apply FIX_format_generic.
 Qed.
 
+(** FLT is a nice format which is not FTZ and that allows a rounding to nearest, ties to even *)
 Theorem FLT_not_FTZ :
   not_FTZ_prop FLT_exp.
 Proof.

src/Core/Fcore_FLX.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,6 +17,7 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * Floating-point format without underflow *)
 Require Import Fcore_Raux.
 Require Import Fcore_defs.
 Require Import Fcore_rnd.
@@ -43,6 +45,7 @@ Definition FLX_RoundingModeP (rnd : R -> R):=
 
 Definition FLX_exp (e : Z) := (e - prec)%Z.
 
+(** Properties of the FLX format *)
 Theorem FLX_exp_correct : valid_exp FLX_exp.
 Proof.
 intros k.
@@ -142,7 +145,7 @@ apply FLX_format_generic.
 exact FLX_exp_correct.
 Qed.
 
-(* unbounded floating-point format with normal mantissas *)
+(** unbounded floating-point format with normal mantissas *)
 Definition FLXN_format (x : R) :=
   exists f : float beta,
   x = F2R f /\ (x <> R0 ->
@@ -238,6 +241,7 @@ now apply <- FLX_format_FLXN.
 exact FLX_exp_correct.
 Qed.
 
+(** FLX is a nice format which is not FTZ and that allows a rounding to nearest, ties to even *)
 Theorem FLX_not_FTZ :
   not_FTZ_prop FLX_exp.
 Proof.

src/Core/Fcore_FTZ.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,6 +17,7 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * Floating-point format with abrupt underflow *)
 Require Import Fcore_Raux.
 Require Import Fcore_defs.
 Require Import Fcore_rnd.
@@ -43,6 +45,7 @@ Definition FTZ_RoundingModeP (rnd : R -> R):=
 
 Definition FTZ_exp e := if Zlt_bool (e - prec) emin then (emin + prec - 1)%Z else (e - prec)%Z.
 
+(** Properties of the FTZ format *)
 Theorem FTZ_exp_correct : valid_exp FTZ_exp.
 Proof.
 intros k.
@@ -235,6 +238,7 @@ Qed.
 
 Section FTZ_round.
 
+(** Rounding with FTZ *)
 Hypothesis rnd : Zround.
 
 Definition Zrnd_FTZ x :=

src/Core/Fcore_Raux.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,11 +17,13 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * Missing definitions/lemmas *)
 Require Export Reals.
 Require Export ZArith.
 
 Section Rmissing.
 
+(** About R *)
 Theorem Rle_0_minus :
   forall x y, (x <= y)%R -> (0 <= y - x)%R.
 Proof.
@@ -296,6 +299,7 @@ End Rmissing.
 
 Section Zmissing.
 
+(** About Z *)
 Theorem Zopp_le_cancel :
   forall x y : Z,
   (-y <= -x)%Z -> Zle x y.
@@ -327,6 +331,7 @@ End Zmissing.
 
 Section Z2R.
 
+(** Z2R function (Z -> R) *)
 Fixpoint P2R (p : positive) :=
   match p with
   | xH => 1%R
@@ -505,6 +510,7 @@ End Z2R.
 
 Section Zcompare.
 
+(** Useful comparisons *)
 Inductive Zeq_bool_prop (x y : Z) : bool -> Prop :=
   | Zeq_bool_true : x = y -> Zeq_bool_prop x y true
   | Zeq_bool_false : x <> y -> Zeq_bool_prop x y false.
@@ -910,6 +916,7 @@ Qed.
 End Req_bool.
 Section Floor_Ceil.
 
+(** Zfloor and Zceil *)
 Definition Zfloor (x : R) := (up x - 1)%Z.
 
 Theorem Zfloor_lb :
@@ -1193,6 +1200,7 @@ End Floor_Ceil.
 
 Section Even_Odd.
 
+(** Zeven, used for rounding to nearest, ties to even *)
 Definition Zeven (n : Z) :=
   match n with
   | Zpos (xO _) => true
@@ -1287,6 +1295,7 @@ End Proof_Irrelevance.
 
 Section pow.
 
+(** The radix must be greater than 1 *)
 Record radix := { radix_val :> Z ; radix_prop : Zle_bool 2 radix_val = true }.
 
 Theorem radix_val_inj :
@@ -1320,6 +1329,7 @@ easy.
 now apply Zle_bool_imp_le.
 Qed.
 
+(** Well-used function called bpow for computing the power function #&beta;#^e *)
 Definition bpow e :=
   match e with
   | Zpos p => Z2R (Zpower_pos r p)
@@ -1559,6 +1569,7 @@ rewrite <- Ropp_mult_distr_l_reverse.
 now rewrite <- Z2R_opp.
 Qed.
 
+(** Another well-used function for having the logarithm of a real number x to the base #&beta;# *) 
 Record ln_beta_prop x := {
   ln_beta_val :> Z ;
    _ : (x <> 0)%R -> (bpow (ln_beta_val - 1)%Z <= Rabs x < bpow ln_beta_val)%R

src/Core/Fcore_defs.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,10 +17,12 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * Basic definitions: float and rounding property *)
 Require Import Fcore_Raux.
 
 Section Def.
 
+(** Definition of a floating-point number *)
 Record float (beta : radix) := Float { Fnum : Z ; Fexp : Z }.
 
 Implicit Arguments Fnum [[beta]].
@@ -30,6 +33,7 @@ Variable beta : radix.
 Definition F2R (f : float beta) :=
   (Z2R (Fnum f) * bpow beta (Fexp f))%R.
 
+(** Requirements on a rounding mode property *)
 Definition MonotoneP (rnd : R -> R) :=
   forall x y : R,
   (x <= y)%R -> (rnd x <= rnd y)%R.
@@ -59,7 +63,7 @@ Implicit Arguments F2R [[beta]].
 
 Section RND.
 
-(* property of being a round toward -inf *)
+(** property of being a round toward -inf *)
 Definition Rnd_DN_pt (F : R -> Prop) (x f : R) :=
   F f /\ (f <= x)%R /\
   forall g : R, F g -> (g <= x)%R -> (g <= f)%R.
@@ -67,7 +71,7 @@ Definition Rnd_DN_pt (F : R -> Prop) (x f : R) :=
 Definition Rnd_DN (F : R -> Prop) (rnd : R -> R) :=
   forall x : R, Rnd_DN_pt F x (rnd x).
 
-(* property of being a round toward +inf *)
+(** property of being a round toward +inf *)
 Definition Rnd_UP_pt (F : R -> Prop) (x f : R) :=
   F f /\ (x <= f)%R /\
   forall g : R, F g -> (x <= g)%R -> (f <= g)%R.
@@ -75,7 +79,7 @@ Definition Rnd_UP_pt (F : R -> Prop) (x f : R) :=
 Definition Rnd_UP (F : R -> Prop) (rnd : R -> R) :=
   forall x : R, Rnd_UP_pt F x (rnd x).
 
-(* property of being a round toward zero *)
+(** property of being a round toward zero *)
 Definition Rnd_ZR_pt (F : R -> Prop) (x f : R) :=
   ( (0 <= x)%R -> Rnd_DN_pt F x f ) /\
   ( (x <= 0)%R -> Rnd_UP_pt F x f ).
@@ -83,7 +87,7 @@ Definition Rnd_ZR_pt (F : R -> Prop) (x f : R) :=
 Definition Rnd_ZR (F : R -> Prop) (rnd : R -> R) :=
   forall x : R, Rnd_ZR_pt F x (rnd x).
 
-(* property of being a round to nearest *)
+(** property of being a round to nearest *)
 Definition Rnd_N_pt (F : R -> Prop) (x f : R) :=
   F f /\
   forall g : R, F g -> (Rabs (f - x) <= Rabs (g - x))%R.

src/Core/Fcore_float_prop.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,6 +17,7 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * Basic properties of floating-point formats: lemmas about mantissa, exponent... *)
 Require Import Fcore_Raux.
 Require Import Fcore_defs.
 
@@ -25,6 +27,7 @@ Variable beta : radix.
 
 Notation bpow e := (bpow beta e).
 
+(** Basic facts *)
 Theorem F2R_le_reg :
   forall e m1 m2 : Z,
   (F2R (Float beta m1 e) <= F2R (Float beta m2 e))%R ->
@@ -85,6 +88,32 @@ apply Zle_antisym ;
   apply Rle_refl.
 Qed.
 
+Theorem abs_F2R :
+  forall m e : Z,
+  Rabs (F2R (Float beta m e)) = F2R (Float beta (Zabs m) e).
+Proof.
+intros m e.
+unfold F2R.
+rewrite Rabs_mult.
+rewrite <- Z2R_abs.
+simpl.
+apply f_equal.
+apply Rabs_right.
+apply Rle_ge.
+apply bpow_ge_0.
+Qed.
+
+Theorem opp_F2R :
+  forall m e : Z,
+  Ropp (F2R (Float beta m e)) = F2R (Float beta (Zopp m) e).
+Proof.
+intros m e.
+unfold F2R. simpl.
+rewrite <- Ropp_mult_distr_l_reverse.
+now rewrite Z2R_opp.
+Qed.
+
+(** Sign facts *)
 Theorem F2R_0 :
   forall e : Z,
   F2R (Float beta 0 e) = R0.
@@ -184,6 +213,7 @@ rewrite <- F2R_0 with (Fexp f).
 now apply F2R_lt_compat.
 Qed.
 
+(** Floats and bpow *)
 Theorem F2R_bpow :
   forall e : Z,
   F2R (Float beta 1 e) = bpow e.
@@ -276,54 +306,6 @@ apply Z2R_le.
 omega.
 Qed.
 
-Theorem ln_beta_F2R_bounds :
-  forall x m e, (0 < m)%Z ->
-  (F2R (Float beta m e) <= x < F2R (Float beta (m + 1) e))%R ->
-  ln_beta beta x = ln_beta beta (F2R (Float beta m e)) :> Z.
-Proof.
-intros x m e Hp (Hx,Hx2).
-destruct (ln_beta beta (F2R (Float beta m e))) as (ex, He).
-simpl.
-apply ln_beta_unique.
-assert (Hp1: (0 < F2R (Float beta m e))%R).
-now apply F2R_gt_0_compat.
-specialize (He (Rgt_not_eq _ _ Hp1)).
-rewrite Rabs_pos_eq in He. 2: now apply Rlt_le.
-destruct He as (He1, He2).
-assert (Hx1: (0 < x)%R).
-now apply Rlt_le_trans with (2 := Hx).
-rewrite Rabs_pos_eq. 2: now apply Rlt_le.
-split.
-now apply Rle_trans with (1 := He1).
-apply Rlt_le_trans with (1 := Hx2).
-now apply F2R_p1_le_bpow.
-Qed.
-
-Theorem abs_F2R :
-  forall m e : Z,
-  Rabs (F2R (Float beta m e)) = F2R (Float beta (Zabs m) e).
-Proof.
-intros m e.
-unfold F2R.
-rewrite Rabs_mult.
-rewrite <- Z2R_abs.
-simpl.
-apply f_equal.
-apply Rabs_right.
-apply Rle_ge.
-apply bpow_ge_0.
-Qed.
-
-Theorem opp_F2R :
-  forall m e : Z,
-  Ropp (F2R (Float beta m e)) = F2R (Float beta (Zopp m) e).
-Proof.
-intros m e.
-unfold F2R. simpl.
-rewrite <- Ropp_mult_distr_l_reverse.
-now rewrite Z2R_opp.
-Qed.
-
 Theorem F2R_lt_bpow :
   forall f : float beta, forall e',
   (Zabs (Fnum f) < Zpower beta (e' - Fexp f))%Z ->
@@ -385,6 +367,30 @@ now apply Z2R_lt.
 exact Hp.
 Qed.
 
+(** Floats and ln_beta *)
+Theorem ln_beta_F2R_bounds :
+  forall x m e, (0 < m)%Z ->
+  (F2R (Float beta m e) <= x < F2R (Float beta (m + 1) e))%R ->
+  ln_beta beta x = ln_beta beta (F2R (Float beta m e)) :> Z.
+Proof.
+intros x m e Hp (Hx,Hx2).
+destruct (ln_beta beta (F2R (Float beta m e))) as (ex, He).
+simpl.
+apply ln_beta_unique.
+assert (Hp1: (0 < F2R (Float beta m e))%R).
+now apply F2R_gt_0_compat.
+specialize (He (Rgt_not_eq _ _ Hp1)).
+rewrite Rabs_pos_eq in He. 2: now apply Rlt_le.
+destruct He as (He1, He2).
+assert (Hx1: (0 < x)%R).
+now apply Rlt_le_trans with (2 := Hx).
+rewrite Rabs_pos_eq. 2: now apply Rlt_le.
+split.
+now apply Rle_trans with (1 := He1).
+apply Rlt_le_trans with (1 := Hx2).
+now apply F2R_p1_le_bpow.
+Qed.
+
 Theorem ln_beta_F2R :
   forall m e : Z,
   m <> Z0 ->

src/Core/Fcore_generic_fmt.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,6 +17,7 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * What is a real number belonging to a format, and many properties. *)
 Require Import Fcore_Raux.
 Require Import Fcore_defs.
 Require Import Fcore_rnd.
@@ -29,6 +31,7 @@ Notation bpow e := (bpow beta e).
 
 Variable fexp : Z -> Z.
 
+(** To be a good fexp *)
 Definition valid_exp :=
   forall k : Z,
   ( (fexp k < k)%Z -> (fexp (k + 1) <= k)%Z ) /\
@@ -50,6 +53,7 @@ Definition scaled_mantissa x :=
 Definition generic_format (x : R) :=
   x = F2R (Float beta (Ztrunc (scaled_mantissa x)) (canonic_exponent x)).
 
+(** Basic facts *)
 Theorem generic_format_0 :
   generic_format 0.
 Proof.
@@ -228,6 +232,7 @@ apply Rle_trans with (2 := proj1 Hx).
 apply bpow_ge_0.
 Qed.
 
+(** Properties when the real number is "small" (kind of subnormal) *)
 Theorem mantissa_small_pos :
   forall x ex,
   (bpow (ex - 1) <= x < bpow ex)%R ->
@@ -297,6 +302,7 @@ assert (H := mantissa_small_pos x ex Hx He).
 split ; try apply Rlt_le ; apply H.
 Qed.
 
+(** Generic facts about any format *)
 Theorem generic_format_discrete :
   forall x m,
   let e := canonic_exponent x in
@@ -386,6 +392,7 @@ Qed.
 
 Section Fcore_generic_round_pos.
 
+(** * Rounding functions: R -> Z *)
 Record Zround := mkZround {
   Zrnd : R -> Z ;
   Zrnd_monotone : forall x y, (x <= y)%R -> (Zrnd x <= Zrnd y)%Z ;
@@ -420,6 +427,7 @@ rewrite Zfloor_Z2R, Zrnd_Z2R in Hx.
 apply Zlt_irrefl with (1 := Hx).
 Qed.
 
+(** * the most useful one: R -> F *)
 Definition round x :=
   F2R (Float beta (Zrnd (scaled_mantissa x)) (canonic_exponent x)).
 
@@ -698,6 +706,7 @@ Qed.
 
 End Zround_opp.
 
+(** IEEE-754 roundings: up, down and to zero *)
 Definition rndDN := mkZround Zfloor Zfloor_le Zfloor_Z2R.
 Definition rndUP := mkZround Zceil Zceil_le Zceil_Z2R.
 Definition rndZR := mkZround Ztrunc Ztrunc_le Ztrunc_Z2R.
@@ -1066,6 +1075,7 @@ End not_FTZ.
 
 Section Znearest.
 
+(** Roundings to nearest: when in the middle, use the choice function *)
 Variable choice : R -> bool.
 
 Definition Znearest x :=

src/Core/Fcore_rnd.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,6 +17,7 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * Roundings: properties and/or functions *)
 Require Import Fcore_Raux.
 Require Import Fcore_defs.
 

src/Core/Fcore_rnd_ne.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,6 +17,7 @@ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 COPYING file for more details.
 *)
 
+(** * Rounding to nearest, ties to even: existence, unicity... *)
 Require Import Fcore_Raux.
 Require Import Fcore_defs.
 Require Import Fcore_rnd.

src/Core/Fcore_ulp.v

-(*
+(**
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
 Copyright (C) 2010 Sylvie Boldo
+#<br />#
 Copyright (C) 2010 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
@@ -16,6 +17,7 @@ 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 *)
 Require Import Fcore_Raux.
 Require Import Fcore_defs.
 Require Import Fcore_rnd.
@@ -122,6 +124,7 @@ rewrite H.
 now rewrite scaled_mantissa_bpow.
 Qed.
 
+(** The successor of x is x + ulp x *)
 Theorem succ_le_bpow :
   forall x e, (0 < x)%R -> F x ->
   (x < bpow e)%R ->
@@ -325,7 +328,7 @@ split; trivial.
 apply Rlt_le; now apply Rlt_Rminus.
 Qed.
 
-
+(** Error of a rounding, expressed in number of ulps *)
 Theorem ulp_error :
   forall Zrnd x,
   (Rabs (round beta fexp Zrnd x - x) < ulp x)%R.
@@ -568,6 +571,7 @@ apply ulp_half_error.
 rewrite round_DN_opp; apply Ropp_0_gt_lt_contravar; apply Rlt_gt; assumption.
 Qed.
 
+(** Predecessor *)
 Definition pred x :=
   if Req_bool x (bpow (ln_beta beta x - 1)) then
     (x - bpow (fexp (ln_beta beta x - 1)))%R