Merge of Flocq version 2.2.0.

More precise modeling of NaNs.


git-svn-id: https://yquem.inria.fr/compcert/svn/compcert/trunk@2303 fca1b0fc-160b-0410-b1d3-a4f43f01ea2e

3 files changed, 1301 insertions, 158 deletions

diff --git a/flocq/Appli/Fappli_IEEE.v b/flocq/Appli/Fappli_IEEE.v
index 63b150f..5e9897f 100644
--- a/flocq/Appli/Fappli_IEEE.v
+++ b/flocq/Appli/Fappli_IEEE.v
@@ -2,9 +2,9 @@
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
-Copyright (C) 2010-2011 Sylvie Boldo
+Copyright (C) 2010-2013 Sylvie Boldo
 #<br />#
-Copyright (C) 2010-2011 Guillaume Melquiond
+Copyright (C) 2010-2013 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
 modify it under the terms of the GNU Lesser General Public
@@ -33,7 +33,7 @@ Section AnyRadix.
 Inductive full_float :=
   | F754_zero : bool -> full_float
   | F754_infinity : bool -> full_float
-  | F754_nan : full_float
+  | F754_nan : bool -> positive -> full_float
   | F754_finite : bool -> positive -> Z -> full_float.
 
 Definition FF2R beta x :=
@@ -67,16 +67,20 @@ Definition bounded m e :=
 Definition valid_binary x :=
   match x with
   | F754_finite _ m e => bounded m e
+  | F754_nan _ pl => (Z_of_nat' (S (digits2_Pnat pl)) <? prec)%Z
   | _ => true
   end.
 
 (** Basic type used for representing binary FP numbers.
     Note that there is exactly one such object per FP datum.
     NaNs do not have any payload. They cannot be distinguished. *)
+
+Definition nan_pl := {pl | (Z_of_nat' (S (digits2_Pnat pl)) <? prec)%Z  = true}.
+
 Inductive binary_float :=
   | B754_zero : bool -> binary_float
   | B754_infinity : bool -> binary_float
-  | B754_nan : binary_float
+  | B754_nan : bool -> nan_pl -> binary_float
   | B754_finite : bool ->
     forall (m : positive) (e : Z), bounded m e = true -> binary_float.
 
@@ -85,7 +89,7 @@ Definition FF2B x :=
   | F754_finite s m e => B754_finite s m e
   | F754_infinity s => fun _ => B754_infinity s
   | F754_zero s => fun _ => B754_zero s
-  | F754_nan => fun _ => B754_nan
+  | F754_nan b pl => fun H => B754_nan b (exist _ pl H)
   end.
 
 Definition B2FF x :=
@@ -93,7 +97,7 @@ Definition B2FF x :=
   | B754_finite s m e _ => F754_finite s m e
   | B754_infinity s => F754_infinity s
   | B754_zero s => F754_zero s
-  | B754_nan => F754_nan
+  | B754_nan b (exist pl _) => F754_nan b pl
   end.
 
 Definition radix2 := Build_radix 2 (refl_equal true).
@@ -108,30 +112,30 @@ Theorem FF2R_B2FF :
   forall x,
   FF2R radix2 (B2FF x) = B2R x.
 Proof.
-now intros [sx|sx| |sx mx ex Hx].
+now intros [sx|sx|sx [plx Hplx]|sx mx ex Hx].
 Qed.
 
 Theorem B2FF_FF2B :
   forall x Hx,
   B2FF (FF2B x Hx) = x.
 Proof.
-now intros [sx|sx| |sx mx ex] Hx.
+now intros [sx|sx|sx plx|sx mx ex] Hx.
 Qed.
 
 Theorem valid_binary_B2FF :
   forall x,
   valid_binary (B2FF x) = true.
 Proof.
-now intros [sx|sx| |sx mx ex Hx].
+now intros [sx|sx|sx [plx Hplx]|sx mx ex Hx].
 Qed.
 
 Theorem FF2B_B2FF :
   forall x H,
   FF2B (B2FF x) H = x.
 Proof.
-intros [sx|sx| |sx mx ex Hx] H ; try easy.
-apply f_equal.
-apply eqbool_irrelevance.
+intros [sx|sx|sx [plx Hplx]|sx mx ex Hx] H ; try easy.
+simpl. apply f_equal, f_equal, eqbool_irrelevance.
+apply f_equal, eqbool_irrelevance.
 Qed.
 
 Theorem FF2B_B2FF_valid :
@@ -146,7 +150,7 @@ Theorem B2R_FF2B :
   forall x Hx,
   B2R (FF2B x Hx) = FF2R radix2 x.
 Proof.
-now intros [sx|sx| |sx mx ex] Hx.
+now intros [sx|sx|sx plx|sx mx ex] Hx.
 Qed.
 
 Theorem match_FF2B :
@@ -154,17 +158,17 @@ Theorem match_FF2B :
   match FF2B x Hx return T with
   | B754_zero sx => fz sx
   | B754_infinity sx => fi sx
-  | B754_nan => fn
+  | B754_nan b (exist p _) => fn b p
   | B754_finite sx mx ex _ => ff sx mx ex
   end =
   match x with
   | F754_zero sx => fz sx
   | F754_infinity sx => fi sx
-  | F754_nan => fn
+  | F754_nan b p => fn b p
   | F754_finite sx mx ex => ff sx mx ex
   end.
 Proof.
-now intros T fz fi fn ff [sx|sx| |sx mx ex] Hx.
+now intros T fz fi fn ff [sx|sx|sx plx|sx mx ex] Hx.
 Qed.
 
 Theorem canonic_canonic_mantissa :
@@ -189,19 +193,28 @@ Theorem generic_format_B2R :
   forall x,
   generic_format radix2 fexp (B2R x).
 Proof.
-intros [sx|sx| |sx mx ex Hx] ; try apply generic_format_0.
+intros [sx|sx|sx plx|sx mx ex Hx] ; try apply generic_format_0.
 simpl.
 apply generic_format_canonic.
 apply canonic_canonic_mantissa.
 now destruct (andb_prop _ _ Hx) as (H, _).
 Qed.
 
+Theorem FLT_format_B2R :
+  forall x,
+  FLT_format radix2 emin prec (B2R x).
+Proof with auto with typeclass_instances.
+intros x.
+apply FLT_format_generic...
+apply generic_format_B2R.
+Qed.
+
 Theorem B2FF_inj :
   forall x y : binary_float,
   B2FF x = B2FF y ->
   x = y.
 Proof.
-intros [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ; try easy.
+intros [sx|sx|sx [plx Hplx]|sx mx ex Hx] [sy|sy|sy [ply Hply]|sy my ey Hy] ; try easy.
 (* *)
 intros H.
 now inversion H.
@@ -212,11 +225,18 @@ now inversion H.
 intros H.
 inversion H.
 clear H.
+revert Hplx.
+rewrite H2.
+intros Hx.
+apply f_equal, f_equal, eqbool_irrelevance.
+(* *)
+intros H.
+inversion H.
+clear H.
 revert Hx.
 rewrite H2, H3.
 intros Hx.
-apply f_equal.
-apply eqbool_irrelevance.
+apply f_equal, eqbool_irrelevance.
 Qed.
 
 Definition is_finite_strict f :=
@@ -265,6 +285,29 @@ apply f_equal.
 apply eqbool_irrelevance.
 Qed.
 
+Definition Bsign x :=
+  match x with
+  | B754_nan s _ => s
+  | B754_zero s => s
+  | B754_infinity s => s
+  | B754_finite s _ _ _ => s
+  end.
+
+Definition sign_FF x :=
+  match x with
+  | F754_nan s _ => s
+  | F754_zero s => s
+  | F754_infinity s => s
+  | F754_finite s _ _ => s
+  end.
+
+Theorem Bsign_FF2B :
+  forall x H,
+  Bsign (FF2B x H) = sign_FF x.
+Proof.
+now intros [sx|sx|sx plx|sx mx ex] H.
+Qed.
+
 Definition is_finite f :=
   match f with
   | B754_finite _ _ _ _ => true
@@ -290,42 +333,93 @@ Theorem is_finite_FF_B2FF :
   forall x,
   is_finite_FF (B2FF x) = is_finite x.
 Proof.
+now intros [| |? []|].
+Qed.
+
+Theorem B2R_Bsign_inj:
+  forall x y : binary_float,
+    is_finite x = true ->
+    is_finite y = true ->
+    B2R x = B2R y ->
+    Bsign x = Bsign y ->
+    x = y.
+Proof.
+intros. destruct x, y; try (apply B2R_inj; now eauto).
+- simpl in H2. congruence.
+- symmetry in H1. apply Rmult_integral in H1.
+  destruct H1. apply eq_Z2R with (n:=0%Z) in H1. destruct b0; discriminate H1.
+  simpl in H1. pose proof (bpow_gt_0 radix2 e).
+  rewrite H1 in H3. apply Rlt_irrefl in H3. destruct H3.
+- apply Rmult_integral in H1.
+  destruct H1. apply eq_Z2R with (n:=0%Z) in H1. destruct b; discriminate H1.
+  simpl in H1. pose proof (bpow_gt_0 radix2 e).
+  rewrite H1 in H3. apply Rlt_irrefl in H3. destruct H3.
+Qed.
+
+Definition is_nan f :=
+  match f with
+  | B754_nan _ _ => true
+  | _ => false
+  end.
+
+Definition is_nan_FF f :=
+  match f with
+  | F754_nan _ _ => true
+  | _ => false
+  end.
+
+Theorem is_nan_FF2B :
+  forall x Hx,
+  is_nan (FF2B x Hx) = is_nan_FF x.
+Proof.
 now intros [| | |].
 Qed.
 
-Definition Bopp x :=
+Theorem is_nan_FF_B2FF :
+  forall x,
+  is_nan_FF (B2FF x) = is_nan x.
+Proof.
+now intros [| |? []|].
+Qed.
+
+Definition Bopp opp_nan x :=
   match x with
-  | B754_nan => x
+  | B754_nan sx plx =>
+    let '(sres, plres) := opp_nan sx plx in B754_nan sres plres
   | B754_infinity sx => B754_infinity (negb sx)
   | B754_finite sx mx ex Hx => B754_finite (negb sx) mx ex Hx
   | B754_zero sx => B754_zero (negb sx)
   end.
 
 Theorem Bopp_involutive :
-  forall x, Bopp (Bopp x) = x.
+  forall opp_nan x,
+  is_nan x = false ->
+  Bopp opp_nan (Bopp opp_nan x) = x.
 Proof.
-now intros [sx|sx| |sx mx ex Hx] ; simpl ; try rewrite Bool.negb_involutive.
+now intros opp_nan [sx|sx|sx plx|sx mx ex Hx] ; simpl ; try rewrite Bool.negb_involutive.
 Qed.
 
 Theorem B2R_Bopp :
-  forall x,
-  B2R (Bopp x) = (- B2R x)%R.
+  forall opp_nan x,
+  B2R (Bopp opp_nan x) = (- B2R x)%R.
 Proof.
-intros [sx|sx| |sx mx ex Hx] ; apply sym_eq ; try apply Ropp_0.
+intros opp_nan [sx|sx|sx plx|sx mx ex Hx]; apply sym_eq ; try apply Ropp_0.
+simpl. destruct opp_nan. apply Ropp_0.
 simpl.
 rewrite <- F2R_opp.
 now case sx.
 Qed.
 
-
-Theorem is_finite_Bopp: forall x,
-  is_finite (Bopp x) = is_finite x.
+Theorem is_finite_Bopp :
+  forall opp_nan x,
+  is_finite (Bopp opp_nan x) = is_finite x.
 Proof.
-now intros [| | |].
+intros opp_nan [| | |] ; try easy.
+intros s pl.
+simpl.
+now case opp_nan.
 Qed.
 
-
-
 Theorem bounded_lt_emax :
   forall mx ex,
   bounded mx ex = true ->
@@ -361,7 +455,7 @@ Theorem abs_B2R_lt_emax :
   forall x,
   (Rabs (B2R x) < bpow radix2 emax)%R.
 Proof.
-intros [sx|sx| |sx mx ex Hx] ; simpl ; try ( rewrite Rabs_R0 ; apply bpow_gt_0 ).
+intros [sx|sx|sx plx|sx mx ex Hx] ; simpl ; try ( rewrite Rabs_R0 ; apply bpow_gt_0 ).
 rewrite <- F2R_Zabs, abs_cond_Zopp.
 now apply bounded_lt_emax.
 Qed.
@@ -638,7 +732,7 @@ Definition binary_round_aux mode sx mx ex lx :=
   match shr_m mrs'' with
   | Z0 => F754_zero sx
   | Zpos m => if Zle_bool e'' (emax - prec) then F754_finite sx m e'' else binary_overflow mode sx
-  | _ => F754_nan (* dummy *)
+  | _ => F754_nan false xH (* dummy *)
   end.
 
 Theorem binary_round_aux_correct :
@@ -649,7 +743,7 @@ Theorem binary_round_aux_correct :
   valid_binary z = true /\
   if Rlt_bool (Rabs (round radix2 fexp (round_mode mode) x)) (bpow radix2 emax) then
     FF2R radix2 z = round radix2 fexp (round_mode mode) x /\
-    is_finite_FF z = true
+    is_finite_FF z = true /\ sign_FF z = Rlt_bool x 0
   else
     z = binary_overflow mode (Rlt_bool x 0).
 Proof with auto with typeclass_instances.
@@ -818,29 +912,6 @@ exact inbetween_int_UP_sign.
 exact inbetween_int_NA_sign.
 Qed.
 
-Definition Bsign x :=
-  match x with
-  | B754_nan => false
-  | B754_zero s => s
-  | B754_infinity s => s
-  | B754_finite s _ _ _ => s
-  end.
-
-Definition sign_FF x :=
-  match x with
-  | F754_nan => false
-  | F754_zero s => s
-  | F754_infinity s => s
-  | F754_finite s _ _ => s
-  end.
-
-Theorem Bsign_FF2B :
-  forall x H,
-  Bsign (FF2B x H) = sign_FF x.
-Proof.
-now intros [sx|sx| |sx mx ex] H.
-Qed.
-
 (** Multiplication *)
 
 Lemma Bmult_correct_aux :
@@ -851,7 +922,7 @@ Lemma Bmult_correct_aux :
   valid_binary z = true /\
   if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (x * y))) (bpow radix2 emax) then
     FF2R radix2 z = round radix2 fexp (round_mode m) (x * y) /\
-    is_finite_FF z = true
+    is_finite_FF z = true /\ sign_FF z = xorb sx sy
   else
     z = binary_overflow m (xorb sx sy).
 Proof.
@@ -896,15 +967,15 @@ apply Rlt_bool_false.
 now apply F2R_ge_0_compat.
 Qed.
 
-Definition Bmult m x y :=
+Definition Bmult mult_nan m x y :=
+  let f pl := B754_nan (fst pl) (snd pl) in
   match x, y with
-  | B754_nan, _ => x
-  | _, B754_nan => y
+  | B754_nan _ _, _ | _, B754_nan _ _ => f (mult_nan x y)
   | B754_infinity sx, B754_infinity sy => B754_infinity (xorb sx sy)
   | B754_infinity sx, B754_finite sy _ _ _ => B754_infinity (xorb sx sy)
   | B754_finite sx _ _ _, B754_infinity sy => B754_infinity (xorb sx sy)
-  | B754_infinity _, B754_zero _ => B754_nan
-  | B754_zero _, B754_infinity _ => B754_nan
+  | B754_infinity _, B754_zero _ => f (mult_nan x y)
+  | B754_zero _, B754_infinity _ => f (mult_nan x y)
   | B754_finite sx _ _ _, B754_zero sy => B754_zero (xorb sx sy)
   | B754_zero sx, B754_finite sy _ _ _ => B754_zero (xorb sx sy)
   | B754_zero sx, B754_zero sy => B754_zero (xorb sx sy)
@@ -913,36 +984,40 @@ Definition Bmult m x y :=
   end.
 
 Theorem Bmult_correct :
-  forall m x y,
+  forall mult_nan m x y,
   if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x * B2R y))) (bpow radix2 emax) then
-    B2R (Bmult m x y) = round radix2 fexp (round_mode m) (B2R x * B2R y) /\
-    is_finite (Bmult m x y) = andb (is_finite x) (is_finite y)
+    B2R (Bmult mult_nan m x y) = round radix2 fexp (round_mode m) (B2R x * B2R y) /\
+    is_finite (Bmult mult_nan m x y) = andb (is_finite x) (is_finite y) /\
+    (is_nan (Bmult mult_nan m x y) = false ->
+      Bsign (Bmult mult_nan m x y) = xorb (Bsign x) (Bsign y))
   else
-    B2FF (Bmult m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).
+    B2FF (Bmult mult_nan m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).
 Proof.
-intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ;
-  try ( rewrite ?Rmult_0_r, ?Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ split ; apply refl_equal | apply bpow_gt_0 | auto with typeclass_instances ] ).
+intros mult_nan m [sx|sx|sx plx|sx mx ex Hx] [sy|sy|sy ply|sy my ey Hy] ;
+  try ( rewrite ?Rmult_0_r, ?Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ now repeat constructor | apply bpow_gt_0 | now auto with typeclass_instances ] ).
 simpl.
 case Bmult_correct_aux.
 intros H1.
 case Rlt_bool.
-intros (H2, H3).
+intros (H2, (H3, H4)).
 split.
 now rewrite B2R_FF2B.
+split.
 now rewrite is_finite_FF2B.
+rewrite Bsign_FF2B. auto.
 intros H2.
 now rewrite B2FF_FF2B.
 Qed.
 
-Definition Bmult_FF m x y :=
+Definition Bmult_FF mult_nan m x y :=
+  let f pl := F754_nan (fst pl) (snd pl) in
   match x, y with
-  | F754_nan, _ => x
-  | _, F754_nan => y
+  | F754_nan _ _, _ | _, F754_nan _ _ => f (mult_nan x y)
   | F754_infinity sx, F754_infinity sy => F754_infinity (xorb sx sy)
   | F754_infinity sx, F754_finite sy _ _ => F754_infinity (xorb sx sy)
   | F754_finite sx _ _, F754_infinity sy => F754_infinity (xorb sx sy)
-  | F754_infinity _, F754_zero _ => F754_nan
-  | F754_zero _, F754_infinity _ => F754_nan
+  | F754_infinity _, F754_zero _ => f (mult_nan x y)
+  | F754_zero _, F754_infinity _ => f (mult_nan x y)
   | F754_finite sx _ _, F754_zero sy => F754_zero (xorb sx sy)
   | F754_zero sx, F754_finite sy _ _ => F754_zero (xorb sx sy)
   | F754_zero sx, F754_zero sy => F754_zero (xorb sx sy)
@@ -951,14 +1026,20 @@ Definition Bmult_FF m x y :=
   end.
 
 Theorem B2FF_Bmult :
+  forall mult_nan mult_nan_ff,
   forall m x y,
-  B2FF (Bmult m x y) = Bmult_FF m (B2FF x) (B2FF y).
+  mult_nan_ff (B2FF x) (B2FF y) = (let '(sr, exist plr _) := mult_nan x y in (sr, plr)) ->
+  B2FF (Bmult mult_nan m x y) = Bmult_FF mult_nan_ff m (B2FF x) (B2FF y).
 Proof.
-intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] ; try easy.
+intros mult_nan mult_nan_ff m x y Hmult_nan.
+unfold Bmult_FF. rewrite Hmult_nan.
+destruct x as [sx|sx|sx [plx Hplx]|sx mx ex Hx], y as [sy|sy|sy [ply Hply]|sy my ey Hy] ;
+  simpl; try match goal with |- context [mult_nan ?x ?y] =>
+               destruct (mult_nan x y) as [? []] end;
+  try easy.
 apply B2FF_FF2B.
 Qed.
 
-
 Definition shl_align mx ex ex' :=
   match (ex' - ex)%Z with
   | Zneg d => (shift_pos d mx, ex')
@@ -1049,7 +1130,8 @@ Theorem binary_round_correct :
   let x := F2R (Float radix2 (cond_Zopp sx (Zpos mx)) ex) in
   if Rlt_bool (Rabs (round radix2 fexp (round_mode m) x)) (bpow radix2 emax) then
     FF2R radix2 z = round radix2 fexp (round_mode m) x /\
-    is_finite_FF z = true
+    is_finite_FF z = true /\
+    sign_FF z = sx
   else
     z = binary_overflow m sx.
 Proof.
@@ -1084,22 +1166,35 @@ Theorem binary_normalize_correct :
   forall m mx ex szero,
   if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (F2R (Float radix2 mx ex)))) (bpow radix2 emax) then
     B2R (binary_normalize m mx ex szero) = round radix2 fexp (round_mode m) (F2R (Float radix2 mx ex)) /\
-    is_finite (binary_normalize m mx ex szero) = true
+    is_finite (binary_normalize m mx ex szero) = true /\
+    Bsign (binary_normalize m mx ex szero) =
+      match Rcompare (F2R (Float radix2 mx ex)) 0 with
+        | Eq => szero
+        | Lt => true
+        | Gt => false
+      end
   else
     B2FF (binary_normalize m mx ex szero) = binary_overflow m (Rlt_bool (F2R (Float radix2 mx ex)) 0).
 Proof with auto with typeclass_instances.
 intros m mx ez szero.
 destruct mx as [|mz|mz] ; simpl.
 rewrite F2R_0, round_0, Rabs_R0, Rlt_bool_true...
+split... split...
+rewrite Rcompare_Eq...
 apply bpow_gt_0.
 (* . mz > 0 *)
 generalize (binary_round_correct m false mz ez).
 simpl.
 case Rlt_bool_spec.
-intros _ (Vz, (Rz, Rz')).
+intros _ (Vz, (Rz, (Rz', Rz''))).
 split.
 now rewrite B2R_FF2B.
+split.
 now rewrite is_finite_FF2B.
+rewrite Bsign_FF2B, Rz''.
+rewrite Rcompare_Gt...
+apply F2R_gt_0_compat.
+simpl. zify; omega.
 intros Hz' (Vz, Rz).
 rewrite B2FF_FF2B, Rz.
 apply f_equal.
@@ -1110,10 +1205,15 @@ now apply F2R_ge_0_compat.
 generalize (binary_round_correct m true mz ez).
 simpl.
 case Rlt_bool_spec.
-intros _ (Vz, (Rz, Rz')).
+intros _ (Vz, (Rz, (Rz', Rz''))).
 split.
 now rewrite B2R_FF2B.
+split.
 now rewrite is_finite_FF2B.
+rewrite Bsign_FF2B, Rz''.
+rewrite Rcompare_Lt...
+apply F2R_lt_0_compat.
+simpl. zify; omega.
 intros Hz' (Vz, Rz).
 rewrite B2FF_FF2B, Rz.
 apply f_equal.
@@ -1123,12 +1223,12 @@ now apply F2R_lt_0_compat.
 Qed.
 
 (** Addition *)
-Definition Bplus m x y :=
+Definition Bplus plus_nan m x y :=
+  let f pl := B754_nan (fst pl) (snd pl) in
   match x, y with
-  | B754_nan, _ => x
-  | _, B754_nan => y
+  | B754_nan _ _, _ | _, B754_nan _ _ => f (plus_nan x y)
   | B754_infinity sx, B754_infinity sy =>
-    if Bool.eqb sx sy then x else B754_nan
+    if Bool.eqb sx sy then x else f (plus_nan x y)
   | B754_infinity _, _ => x
   | _, B754_infinity _ => y
   | B754_zero sx, B754_zero sy =>
@@ -1143,28 +1243,47 @@ Definition Bplus m x y :=
   end.
 
 Theorem Bplus_correct :
-  forall m x y,
+  forall plus_nan m x y,
   is_finite x = true ->
   is_finite y = true ->
   if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x + B2R y))) (bpow radix2 emax) then
-    B2R (Bplus m x y) = round radix2 fexp (round_mode m) (B2R x + B2R y) /\
-    is_finite (Bplus m x y) = true
+    B2R (Bplus plus_nan m x y) = round radix2 fexp (round_mode m) (B2R x + B2R y) /\
+    is_finite (Bplus plus_nan m x y) = true /\
+    Bsign (Bplus plus_nan m x y) =
+      match Rcompare (B2R x + B2R y) 0 with
+        | Eq => match m with mode_DN => orb (Bsign x) (Bsign y)
+                                 | _ => andb (Bsign x) (Bsign y) end
+        | Lt => true
+        | Gt => false
+      end
   else
-    (B2FF (Bplus m x y) = binary_overflow m (Bsign x) /\ Bsign x = Bsign y).
+    (B2FF (Bplus plus_nan m x y) = binary_overflow m (Bsign x) /\ Bsign x = Bsign y).
 Proof with auto with typeclass_instances.
-intros m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] Fx Fy ; try easy.
+intros plus_nan m [sx|sx| |sx mx ex Hx] [sy|sy| |sy my ey Hy] Fx Fy ; try easy.
 (* *)
 rewrite Rplus_0_r, round_0, Rabs_R0, Rlt_bool_true...
 simpl.
-case (Bool.eqb sx sy) ; try easy.
-now case m.
+rewrite Rcompare_Eq by auto.
+destruct sx, sy; try easy; now case m.
 apply bpow_gt_0.
 (* *)
 rewrite Rplus_0_l, round_generic, Rlt_bool_true...
+split... split...
+simpl. unfold F2R.
+erewrite <- Rmult_0_l, Rcompare_mult_r.
+rewrite Rcompare_Z2R with (y:=0%Z).
+destruct sy...
+apply bpow_gt_0.
 apply abs_B2R_lt_emax.
 apply generic_format_B2R.
 (* *)
 rewrite Rplus_0_r, round_generic, Rlt_bool_true...
+split... split...
+simpl. unfold F2R.
+erewrite <- Rmult_0_l, Rcompare_mult_r.
+rewrite Rcompare_Z2R with (y:=0%Z).
+destruct sx...
+apply bpow_gt_0.
 apply abs_B2R_lt_emax.
 apply generic_format_B2R.
 (* *)
@@ -1264,7 +1383,19 @@ now apply F2R_le_0_compat.
 (* . *)
 generalize (binary_normalize_correct m mz ez szero).
 case Rlt_bool_spec.
-easy.
+split; try easy. split; try easy.
+destruct (Rcompare_spec (F2R (beta:=radix2) {| Fnum := mz; Fexp := ez |}) 0); try easy.
+rewrite H1 in Hp.
+apply Rplus_opp_r_uniq in Hp.
+rewrite <- F2R_Zopp in Hp.
+eapply canonic_unicity in Hp.
+inversion Hp. destruct sy, sx, m; try discriminate H3; easy.
+apply canonic_canonic_mantissa.
+apply Bool.andb_true_iff in Hy. easy.
+replace (-cond_Zopp sx (Z.pos mx))%Z with  (cond_Zopp (negb sx) (Z.pos mx))
+  by (destruct sx; auto).
+apply canonic_canonic_mantissa.
+apply Bool.andb_true_iff in Hx. easy.
 intros Hz' Vz.
 specialize (Sz Hz').
 split.
@@ -1273,26 +1404,32 @@ now apply f_equal.
 apply Sz.
 Qed.
 
-Definition Bminus m x y := Bplus m x (Bopp y).
+Definition Bminus minus_nan m x y := Bplus minus_nan m x (Bopp pair y).
 
 Theorem Bminus_correct :
-  forall m x y,
+  forall minus_nan m x y,
   is_finite x = true ->
   is_finite y = true ->
   if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x - B2R y))) (bpow radix2 emax) then
-    B2R (Bminus m x y) = round radix2 fexp (round_mode m) (B2R x - B2R y) /\
-    is_finite (Bminus m x y) = true
+    B2R (Bminus minus_nan m x y) = round radix2 fexp (round_mode m) (B2R x - B2R y) /\
+    is_finite (Bminus minus_nan m x y) = true /\
+    Bsign (Bminus minus_nan m x y) =
+      match Rcompare (B2R x - B2R y) 0 with
+        | Eq => match m with mode_DN => orb (Bsign x) (negb (Bsign y))
+                                 | _ => andb (Bsign x) (negb (Bsign y)) end
+        | Lt => true
+        | Gt => false
+      end
   else
-    (B2FF (Bminus m x y) = binary_overflow m (Bsign x) /\ Bsign x = negb (Bsign y)).
+    (B2FF (Bminus minus_nan m x y) = binary_overflow m (Bsign x) /\ Bsign x = negb (Bsign y)).
 Proof with auto with typeclass_instances.
-intros m x y Fx Fy.
-replace (negb (Bsign y)) with (Bsign (Bopp y)).
+intros m minus_nan x y Fx Fy.
+replace (negb (Bsign y)) with (Bsign (Bopp pair y)).
 unfold Rminus.
-rewrite <- B2R_Bopp.
+erewrite <- B2R_Bopp.
 apply Bplus_correct.
 exact Fx.
-now rewrite is_finite_Bopp.
-now destruct y as [ | | | ].
+rewrite is_finite_Bopp. auto. now destruct y as [ | | | ].
 Qed.
 
 (** Division *)
@@ -1316,12 +1453,12 @@ Lemma Bdiv_correct_aux :
     let '(mz, ez, lz) := Fdiv_core_binary (Zpos mx) ex (Zpos my) ey in
     match mz with
     | Zpos mz => binary_round_aux m (xorb sx sy) mz ez lz
-    | _ => F754_nan (* dummy *)
+    | _ => F754_nan false xH (* dummy *)
     end in
   valid_binary z = true /\
   if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (x / y))) (bpow radix2 emax) then
     FF2R radix2 z = round radix2 fexp (round_mode m) (x / y) /\
-    is_finite_FF z = true
+    is_finite_FF z = true /\ sign_FF z = xorb sx sy
   else
     z = binary_overflow m (xorb sx sy).
 Proof.
@@ -1406,45 +1543,49 @@ apply Rinv_0_lt_compat.
 now apply F2R_gt_0_compat.
 Qed.
 
-Definition Bdiv m x y :=
+Definition Bdiv div_nan m x y :=
+  let f pl := B754_nan (fst pl) (snd pl) in
   match x, y with
-  | B754_nan, _ => x
-  | _, B754_nan => y
-  | B754_infinity sx, B754_infinity sy => B754_nan
+  | B754_nan _ _, _ | _, B754_nan _ _ => f (div_nan x y)
+  | B754_infinity sx, B754_infinity sy => f (div_nan x y)
   | B754_infinity sx, B754_finite sy _ _ _ => B754_infinity (xorb sx sy)
   | B754_finite sx _ _ _, B754_infinity sy => B754_zero (xorb sx sy)
   | B754_infinity sx, B754_zero sy => B754_infinity (xorb sx sy)
   | B754_zero sx, B754_infinity sy => B754_zero (xorb sx sy)
   | B754_finite sx _ _ _, B754_zero sy => B754_infinity (xorb sx sy)
   | B754_zero sx, B754_finite sy _ _ _ => B754_zero (xorb sx sy)
-  | B754_zero sx, B754_zero sy => B754_nan
+  | B754_zero sx, B754_zero sy => f (div_nan x y)
   | B754_finite sx mx ex _, B754_finite sy my ey _ =>
     FF2B _ (proj1 (Bdiv_correct_aux m sx mx ex sy my ey))
   end.
 
 Theorem Bdiv_correct :
-  forall m x y,
+  forall div_nan m x y,
   B2R y <> R0 ->
   if Rlt_bool (Rabs (round radix2 fexp (round_mode m) (B2R x / B2R y))) (bpow radix2 emax) then
-    B2R (Bdiv m x y) = round radix2 fexp (round_mode m) (B2R x / B2R y) /\
-    is_finite (Bdiv m x y) = is_finite x
+    B2R (Bdiv div_nan m x y) = round radix2 fexp (round_mode m) (B2R x / B2R y) /\
+    is_finite (Bdiv div_nan m x y) = is_finite x /\
+    (is_nan (Bdiv div_nan m x y) = false ->
+      Bsign (Bdiv div_nan m x y) = xorb (Bsign x) (Bsign y))
   else
-    B2FF (Bdiv m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).
+    B2FF (Bdiv div_nan m x y) = binary_overflow m (xorb (Bsign x) (Bsign y)).
 Proof.
-intros m x [sy|sy| |sy my ey Hy] Zy ; try now elim Zy.
+intros div_nan m x [sy|sy|sy ply|sy my ey Hy] Zy ; try now elim Zy.
 revert x.
 unfold Rdiv.
-intros [sx|sx| |sx mx ex Hx] ;
-  try ( rewrite Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ split ; apply refl_equal | apply bpow_gt_0 | auto with typeclass_instances ] ).
+intros [sx|sx|sx plx|sx mx ex Hx] ;
+  try ( rewrite Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ now repeat constructor | apply bpow_gt_0 | auto with typeclass_instances ] ).
 simpl.
 case Bdiv_correct_aux.
 intros H1.
 unfold Rdiv.
 case Rlt_bool.
-intros (H2, H3).
+intros (H2, (H3, H4)).
 split.
 now rewrite B2R_FF2B.
+split.
 now rewrite is_finite_FF2B.
+rewrite Bsign_FF2B. congruence.
 intros H2.
 now rewrite B2FF_FF2B.
 Qed.
@@ -1473,11 +1614,11 @@ Lemma Bsqrt_correct_aux :
     let '(mz, ez, lz) := Fsqrt_core_binary (Zpos mx) ex in
     match mz with
     | Zpos mz => binary_round_aux m false mz ez lz
-    | _ => F754_nan (* dummy *)
+    | _ => F754_nan false xH (* dummy *)
     end in
   valid_binary z = true /\
   FF2R radix2 z = round radix2 fexp (round_mode m) (sqrt x) /\
-  is_finite_FF z = true.
+  is_finite_FF z = true /\ sign_FF z = false.
 Proof with auto with typeclass_instances.
 intros m mx ex Hx.
 replace (Fsqrt_core_binary (Zpos mx) ex) with (Fsqrt_core radix2 prec (Zpos mx) ex).
@@ -1578,28 +1719,30 @@ now case mz.
 apply sqrt_ge_0.
 Qed.
 
-Definition Bsqrt m x :=
+Definition Bsqrt sqrt_nan m x :=
+  let f pl := B754_nan (fst pl) (snd pl) in
   match x with
-  | B754_nan => x
+  | B754_nan sx plx => f (sqrt_nan x)
   | B754_infinity false => x
-  | B754_infinity true => B754_nan
-  | B754_finite true _ _ _ => B754_nan
+  | B754_infinity true => f (sqrt_nan x)
+  | B754_finite true _ _ _ => f (sqrt_nan x)
   | B754_zero _ => x
   | B754_finite sx mx ex Hx =>
     FF2B _ (proj1 (Bsqrt_correct_aux m mx ex Hx))
   end.
 
 Theorem Bsqrt_correct :
-  forall m x,
-  B2R (Bsqrt m x) = round radix2 fexp (round_mode m) (sqrt (B2R x)) /\
-  is_finite (Bsqrt m x) = match x with B754_zero _ => true | B754_finite false _ _ _ => true | _ => false end.
+  forall sqrt_nan m x,
+  B2R (Bsqrt sqrt_nan m x) = round radix2 fexp (round_mode m) (sqrt (B2R x)) /\
+  is_finite (Bsqrt sqrt_nan m x) = match x with B754_zero _ => true | B754_finite false _ _ _ => true | _ => false end /\
+  (is_nan (Bsqrt sqrt_nan m x) = false -> Bsign (Bsqrt sqrt_nan m x) = Bsign x).
 Proof.
-intros m [sx|[|]| |sx mx ex Hx] ; try ( now simpl ; rewrite sqrt_0, round_0 ; auto with typeclass_instances ).
+intros sqrt_nan m [sx|[|]| |sx mx ex Hx] ; try ( now simpl ; rewrite sqrt_0, round_0 ; intuition auto with typeclass_instances ).
 simpl.
 case Bsqrt_correct_aux.
-intros H1 (H2, H3).
+intros H1 (H2, (H3, H4)).
 case sx.
-refine (conj _ (refl_equal false)).
+refine (conj _ (conj (refl_equal false) _)).
 apply sym_eq.
 unfold sqrt.
 case Rcase_abs.
@@ -1609,9 +1752,12 @@ auto with typeclass_instances.
 intros H.
 elim Rge_not_lt with (1 := H).
 now apply F2R_lt_0_compat.
+easy.
 split.
 now rewrite B2R_FF2B.
+split.
 now rewrite is_finite_FF2B.
+intro. rewrite Bsign_FF2B. auto.
 Qed.
 
 End Binary.
diff --git a/flocq/Appli/Fappli_IEEE_bits.v b/flocq/Appli/Fappli_IEEE_bits.v
index 06ed21e..a41fba9 100644
--- a/flocq/Appli/Fappli_IEEE_bits.v
+++ b/flocq/Appli/Fappli_IEEE_bits.v
@@ -2,9 +2,9 @@
 This file is part of the Flocq formalization of floating-point
 arithmetic in Coq: http://flocq.gforge.inria.fr/
 
-Copyright (C) 2011 Sylvie Boldo
+Copyright (C) 2011-2013 Sylvie Boldo
 #<br />#
-Copyright (C) 2011 Guillaume Melquiond
+Copyright (C) 2011-2013 Guillaume Melquiond
 
 This library is free software; you can redistribute it and/or
 modify it under the terms of the GNU Lesser General Public
@@ -172,7 +172,7 @@ Definition bits_of_binary_float (x : binary_float) :=
   match x with
   | B754_zero sx => join_bits sx 0 0
   | B754_infinity sx => join_bits sx 0 (Zpower 2 ew - 1)
-  | B754_nan => join_bits false (Zpower 2 mw - 1) (Zpower 2 ew - 1)
+  | B754_nan sx (exist plx _) => join_bits sx (Zpos plx) (Zpower 2 ew - 1)
   | B754_finite sx mx ex _ =>
     if Zle_bool (Zpower 2 mw) (Zpos mx) then
       join_bits sx (Zpos mx - Zpower 2 mw) (ex - emin + 1)
@@ -184,7 +184,7 @@ Definition split_bits_of_binary_float (x : binary_float) :=
   match x with
   | B754_zero sx => (sx, 0, 0)%Z
   | B754_infinity sx => (sx, 0, Zpower 2 ew - 1)%Z
-  | B754_nan => (false, Zpower 2 mw - 1, Zpower 2 ew - 1)%Z
+  | B754_nan sx (exist plx _) => (sx, Zpos plx, Zpower 2 ew - 1)%Z
   | B754_finite sx mx ex _ =>
     if Zle_bool (Zpower 2 mw) (Zpos mx) then
       (sx, Zpos mx - Zpower 2 mw, ex - emin + 1)%Z
@@ -196,8 +196,16 @@ Theorem split_bits_of_binary_float_correct :
   forall x,
   split_bits (bits_of_binary_float x) = split_bits_of_binary_float x.
 Proof.
-intros [sx|sx| |sx mx ex Hx] ;
+intros [sx|sx|sx [plx Hplx]|sx mx ex Hx] ;
   try ( simpl ; apply split_join_bits ; split ; try apply Zle_refl ; try apply Zlt_pred ; trivial ; omega ).
+simpl. apply split_join_bits; split; try (zify; omega).
+destruct (digits2_Pnat_correct plx).
+rewrite Zpower_nat_Z in H0.
+eapply Zlt_le_trans. apply H0.
+change 2%Z with (radix_val radix2). apply Zpower_le.
+rewrite Z.ltb_lt in Hplx.
+unfold prec in *. zify; omega.
+(* *)
 unfold bits_of_binary_float, split_bits_of_binary_float.
 assert (Hf: (emin <= ex /\ Zdigits radix2 (Zpos mx) <= prec)%Z).
 destruct (andb_prop _ _ Hx) as (Hx', _).
@@ -246,14 +254,18 @@ Definition binary_float_of_bits_aux x :=
     match mx with
     | Z0 => F754_zero sx
     | Zpos px => F754_finite sx px emin
-    | Zneg _ => F754_nan (* dummy *)
+    | Zneg _ => F754_nan false xH (* dummy *)
     end
   else if Zeq_bool ex (Zpower 2 ew - 1) then
-    if Zeq_bool mx 0 then F754_infinity sx else F754_nan
+    match mx with
+      | Z0 => F754_infinity sx
+      | Zpos plx => F754_nan sx plx
+      | Zneg _ => F754_nan false xH (* dummy *)
+    end
   else
     match (mx + Zpower 2 mw)%Z with
     | Zpos px => F754_finite sx px (ex + emin - 1)
-    | _ => F754_nan (* dummy *)
+    | _ => F754_nan false xH (* dummy *)
     end.
 
 Lemma binary_float_of_bits_aux_correct :
@@ -292,9 +304,20 @@ cut (0 < emax)%Z. clear -H Hew ; omega.
 apply (Zpower_gt_0 radix2).
 clear -Hew ; omega.
 apply bpow_gt_0.
+simpl. intros. rewrite Z.ltb_lt. unfold prec. zify; omega.
 case Zeq_bool_spec ; intros He2.
-now case Zeq_bool.
+case_eq (x mod 2 ^ mw)%Z; try easy.
+(* nan *)
+intros plx Eqplx. apply Z.ltb_lt.
+rewrite Z_of_nat_S_digits2_Pnat.
+assert (forall a b, a <= b -> a < b+1)%Z by (intros; omega). apply H. clear H.
+apply Zdigits_le_Zpower. simpl.
+rewrite <- Eqplx. edestruct Z_mod_lt; eauto.
+change 2%Z with (radix_val radix2).
+apply Z.lt_gt, Zpower_gt_0. omega.
+simpl. intros. rewrite Z.ltb_lt. unfold prec. zify; omega.
 case_eq (x mod 2^mw + 2^mw)%Z ; try easy.
+simpl. intros. rewrite Z.ltb_lt. unfold prec. zify; omega.
 (* normal *)
 intros px Hm.
 assert (prec = Zdigits radix2 (Zpos px)).
@@ -365,6 +388,7 @@ apply Zlt_gt.
 apply (Zpower_gt_0 radix2).
 now apply Zlt_le_weak.
 apply bpow_gt_0.
+simpl. intros. rewrite Z.ltb_lt. unfold prec. zify; omega.
 Qed.
 
 Definition binary_float_of_bits x :=
@@ -380,7 +404,7 @@ unfold binary_float_of_bits.
 rewrite B2FF_FF2B.
 unfold binary_float_of_bits_aux.
 rewrite split_bits_of_binary_float_correct.
-destruct x as [sx|sx| |sx mx ex Bx].
+destruct x as [sx|sx|sx [plx Hplx]|sx mx ex Bx].
 apply refl_equal.
 (* *)
 simpl.
@@ -391,12 +415,7 @@ now apply (Zpower_gt_1 radix2).
 (* *)
 simpl.
 rewrite Zeq_bool_false.
-rewrite Zeq_bool_true.
-rewrite Zeq_bool_false.
-apply refl_equal.
-cut (1 < 2^mw)%Z. clear ; omega.
-now apply (Zpower_gt_1 radix2).
-apply refl_equal.
+rewrite Zeq_bool_true; auto.
 cut (1 < 2^ew)%Z. clear ; omega.
 now apply (Zpower_gt_1 radix2).
 (* *)
@@ -442,7 +461,6 @@ Qed.
 Theorem bits_of_binary_float_of_bits :
   forall x,
   (0 <= x < 2^(mw+ew+1))%Z ->
-  binary_float_of_bits x <> B754_nan prec emax ->
   bits_of_binary_float (binary_float_of_bits x) = x.
 Proof.
 intros x Hx.
@@ -462,28 +480,28 @@ now apply Zlt_gt.
 case Zeq_bool_spec ; intros He1.
 (* subnormal *)
 case_eq mx.
-intros Hm Jx _ _.
+intros Hm Jx _.
 now rewrite He1 in Jx.
-intros px Hm Jx _ _.
+intros px Hm Jx _.
 rewrite Zle_bool_false.
 now rewrite <- He1.
 now rewrite <- Hm.
-intros px Hm _ _ _.
+intros px Hm _ _.
 apply False_ind.
 apply Zle_not_lt with (1 := proj1 Bm).
 now rewrite Hm.
 case Zeq_bool_spec ; intros He2.
 (* infinity/nan *)
-case Zeq_bool_spec ; intros Hm.
-now rewrite Hm, He2.
-intros _ Cx Nx.
-now elim Nx.
+case_eq mx; intros Hm.
+now rewrite He2.
+now rewrite He2.
+intros. zify; omega.
 (* normal *)
 case_eq (mx + 2 ^ mw)%Z.
 intros Hm.
 apply False_ind.
 clear -Bm Hm ; omega.
-intros p Hm Jx Cx _.
+intros p Hm Jx Cx.
 rewrite <- Hm.
 rewrite Zle_bool_true.
 now ring_simplify (mx + 2^mw - 2^mw)%Z (ex + emin - 1 - emin + 1)%Z.
diff --git a/flocq/Appli/Fappli_rnd_odd.v b/flocq/Appli/Fappli_rnd_odd.v
new file mode 100644
index 0000000..b4a2c52
--- /dev/null
+++ b/flocq/Appli/Fappli_rnd_odd.v
@@ -0,0 +1,979 @@
+(**
+This file is part of the Flocq formalization of floating-point
+arithmetic in Coq: http://flocq.gforge.inria.fr/
+
+Copyright (C) 2010-2013 Sylvie Boldo
+#<br />#
+Copyright (C) 2010-2013 Guillaume Melquiond
+
+This library is free software; you can redistribute it and/or
+modify it under the terms of the GNU Lesser General Public
+License as published by the Free Software Foundation; either
+version 3 of the License, or (at your option) any later version.
+
+This library is distributed in the hope that it will be useful,
+but WITHOUT ANY WARRANTY; without even the implied warranty of
+MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+COPYING file for more details.
+*)
+
+(** * Rounding to odd and its properties, including the equivalence
+      between rnd_NE and double rounding with rnd_odd and then rnd_NE *)
+
+
+Require Import Fcore.
+Require Import Fcalc_ops.
+
+Definition Zrnd_odd x :=  match Req_EM_T x (Z2R (Zfloor x))  with
+  | left _   => Zfloor x
+  | right _  => match (Zeven (Zfloor x)) with
+      | true => Zceil x
+      | false => Zfloor x
+     end
+  end.
+
+
+
+Global Instance valid_rnd_odd : Valid_rnd Zrnd_odd.
+Proof.
+split.
+(* . *)
+intros x y Hxy.
+assert (Zfloor x <= Zrnd_odd y)%Z.
+(* .. *)
+apply Zle_trans with (Zfloor y).
+now apply Zfloor_le.
+unfold Zrnd_odd; destruct (Req_EM_T  y (Z2R (Zfloor y))).
+now apply Zle_refl.
+case (Zeven (Zfloor y)).
+apply le_Z2R.
+apply Rle_trans with y.
+apply Zfloor_lb.
+apply Zceil_ub.
+now apply Zle_refl.
+unfold Zrnd_odd at 1.
+(* . *)
+destruct (Req_EM_T  x (Z2R (Zfloor x))) as [Hx|Hx].
+(* .. *)
+apply H.
+(* .. *)
+case_eq (Zeven (Zfloor x)); intros Hx2.
+2: apply H.
+unfold Zrnd_odd; destruct (Req_EM_T  y (Z2R (Zfloor y))) as [Hy|Hy].
+apply Zceil_glb.
+now rewrite <- Hy.
+case_eq (Zeven (Zfloor y)); intros Hy2.
+now apply Zceil_le.
+apply Zceil_glb.
+assert (H0:(Zfloor x <= Zfloor y)%Z) by now apply Zfloor_le.
+case (Zle_lt_or_eq _ _  H0); intros H1.
+apply Rle_trans with (1:=Zceil_ub _).
+rewrite Zceil_floor_neq.
+apply Z2R_le; omega.
+now apply sym_not_eq.
+contradict Hy2.
+rewrite <- H1, Hx2; discriminate.
+(* . *)
+intros n; unfold Zrnd_odd.
+rewrite Zfloor_Z2R, Zceil_Z2R.
+destruct (Req_EM_T  (Z2R n) (Z2R n)); trivial.
+case (Zeven n); trivial.
+Qed.
+
+
+
+Lemma Zrnd_odd_Zodd: forall x, x <> (Z2R (Zfloor x)) ->
+  (Zeven (Zrnd_odd x)) = false.
+Proof.
+intros x Hx; unfold Zrnd_odd.
+destruct (Req_EM_T  x (Z2R (Zfloor x))) as [H|H].
+now contradict H.
+case_eq (Zeven (Zfloor x)).
+(* difficult case *)
+intros H'.
+rewrite Zceil_floor_neq.
+rewrite Zeven_plus, H'.
+reflexivity.
+now apply sym_not_eq.
+trivial.
+Qed.
+
+
+
+
+Section Fcore_rnd_odd.
+
+Variable beta : radix.
+
+Notation bpow e := (bpow beta e).
+
+Variable fexp : Z -> Z.
+
+Context { valid_exp : Valid_exp fexp }.
+Context { exists_NE_ : Exists_NE beta fexp }.
+
+Notation format := (generic_format beta fexp).
+Notation canonic := (canonic beta fexp).
+Notation cexp := (canonic_exp beta fexp).
+
+
+Definition Rnd_odd_pt (x f : R) :=
+  format f /\ ((f = x)%R \/
+    ((Rnd_DN_pt format x f \/ Rnd_UP_pt format x f) /\
+    exists g : float beta, f = F2R g /\ canonic g /\ Zeven (Fnum g) = false)).
+
+Definition Rnd_odd (rnd : R -> R) :=
+  forall x : R, Rnd_odd_pt x (rnd x).
+
+
+Theorem Rnd_odd_pt_sym :   forall x f : R,
+  Rnd_odd_pt (-x) (-f) -> Rnd_odd_pt x f.
+Proof with auto with typeclass_instances.
+intros x f (H1,H2).
+split.
+replace f with (-(-f))%R by ring.
+now apply generic_format_opp.
+destruct H2.
+left.
+replace f with (-(-f))%R by ring.
+rewrite H; ring.
+right.
+destruct H as (H2,(g,(Hg1,(Hg2,Hg3)))).
+split.
+destruct H2.
+right.
+replace f with (-(-f))%R by ring.
+replace x with (-(-x))%R by ring.
+apply Rnd_DN_UP_pt_sym...
+apply generic_format_opp.
+left.
+replace f with (-(-f))%R by ring.
+replace x with (-(-x))%R by ring.
+apply Rnd_UP_DN_pt_sym...
+apply generic_format_opp.
+exists (Float beta (-Fnum g) (Fexp g)).
+split.
+rewrite F2R_Zopp.
+replace f with (-(-f))%R by ring.
+rewrite Hg1; reflexivity.
+split.
+now apply canonic_opp.
+simpl.
+now rewrite Zeven_opp.
+Qed.
+
+
+Theorem round_odd_opp :
+  forall x,
+  (round beta fexp Zrnd_odd  (-x) = (- round beta fexp Zrnd_odd x))%R.
+Proof.
+intros x; unfold round.
+rewrite <- F2R_Zopp.
+unfold F2R; simpl.
+apply f_equal2; apply f_equal.
+rewrite scaled_mantissa_opp.
+generalize (scaled_mantissa beta fexp x); intros r.
+unfold Zrnd_odd.
+case (Req_EM_T (- r) (Z2R (Zfloor (- r)))).
+case (Req_EM_T r (Z2R (Zfloor r))).
+intros Y1 Y2.
+apply eq_Z2R.
+now rewrite Z2R_opp, <- Y1, <-Y2.
+intros Y1 Y2.
+absurd (r=Z2R (Zfloor r)); trivial.
+pattern r at 2; replace r with (-(-r))%R by ring.
+rewrite Y2, <- Z2R_opp.
+rewrite Zfloor_Z2R.
+rewrite Z2R_opp, <- Y2.
+ring.
+case (Req_EM_T r (Z2R (Zfloor r))).
+intros Y1 Y2.
+absurd (-r=Z2R (Zfloor (-r)))%R; trivial.
+pattern r at 2; rewrite Y1.
+rewrite <- Z2R_opp, Zfloor_Z2R.
+now rewrite Z2R_opp, <- Y1.
+intros Y1 Y2.
+unfold Zceil; rewrite Ropp_involutive.
+replace  (Zeven (Zfloor (- r))) with (negb (Zeven (Zfloor r))).
+case (Zeven (Zfloor r));  simpl; ring.
+apply trans_eq with (Zeven (Zceil r)).
+rewrite Zceil_floor_neq.
+rewrite Zeven_plus.
+simpl; reflexivity.
+now apply sym_not_eq.
+rewrite <- (Zeven_opp (Zfloor (- r))).
+reflexivity.
+apply canonic_exp_opp.
+Qed.
+
+
+
+Theorem round_odd_pt :
+  forall x,
+  Rnd_odd_pt x (round beta fexp Zrnd_odd x).
+Proof with auto with typeclass_instances.
+cut (forall x : R, (0 < x)%R -> Rnd_odd_pt x (round beta fexp Zrnd_odd x)).
+intros H x; case (Rle_or_lt 0 x).
+intros H1; destruct H1.
+now apply H.
+rewrite <- H0.
+rewrite round_0...
+split.
+apply generic_format_0.
+now left.
+intros Hx; apply Rnd_odd_pt_sym.
+rewrite <- round_odd_opp.
+apply H.
+auto with real.
+(* *)
+intros x Hxp.
+generalize (generic_format_round beta fexp Zrnd_odd x).
+set (o:=round beta fexp Zrnd_odd x).
+intros Ho.
+split.
+assumption.
+(* *)
+case (Req_dec o x); intros Hx.
+now left.
+right.
+assert (o=round beta fexp Zfloor x \/ o=round beta fexp Zceil x).
+unfold o, round, F2R;simpl.
+case (Zrnd_DN_or_UP Zrnd_odd  (scaled_mantissa beta fexp x))...
+intros H; rewrite H; now left.
+intros H; rewrite H; now right.
+split.
+destruct H; rewrite H.
+left; apply round_DN_pt...
+right; apply round_UP_pt...
+(* *)
+unfold o, Zrnd_odd, round.
+case (Req_EM_T (scaled_mantissa beta fexp x)
+     (Z2R (Zfloor (scaled_mantissa beta fexp x)))).
+intros T.
+absurd (o=x); trivial.
+apply round_generic...
+unfold generic_format, F2R; simpl.
+rewrite <- (scaled_mantissa_mult_bpow beta fexp) at 1.
+apply f_equal2; trivial; rewrite T at 1.
+apply f_equal, sym_eq, Ztrunc_floor.
+apply Rmult_le_pos.
+now left.
+apply bpow_ge_0.
+intros L.
+case_eq (Zeven (Zfloor (scaled_mantissa beta fexp x))).
+(* . *)
+generalize (generic_format_round beta fexp Zceil x).
+unfold generic_format.
+set (f:=round beta fexp Zceil x).
+set (ef := canonic_exp beta fexp f).
+set (mf := Ztrunc (scaled_mantissa beta fexp f)).
+exists (Float beta mf ef).
+unfold Fcore_generic_fmt.canonic.
+rewrite <- H0.
+repeat split; try assumption.
+apply trans_eq with (negb (Zeven (Zfloor (scaled_mantissa beta fexp x)))).
+2: rewrite H1; reflexivity.
+apply trans_eq with (negb (Zeven (Fnum
+  (Float beta  (Zfloor (scaled_mantissa beta fexp x)) (cexp x))))).
+2: reflexivity.
+case (Rle_lt_or_eq_dec 0 (round beta fexp Zfloor x)).
+rewrite <- round_0 with beta fexp Zfloor...
+apply round_le...
+now left.
+intros Y.
+generalize (DN_UP_parity_generic beta fexp)...
+unfold DN_UP_parity_prop.
+intros T; apply T with x; clear T.
+unfold generic_format.
+rewrite <- (scaled_mantissa_mult_bpow beta fexp x) at 1.
+unfold F2R; simpl.
+apply Rmult_neq_compat_r.
+apply Rgt_not_eq, bpow_gt_0.
+rewrite Ztrunc_floor.
+assumption.
+apply Rmult_le_pos.
+now left.
+apply bpow_ge_0.
+unfold Fcore_generic_fmt.canonic.
+simpl.
+apply sym_eq, canonic_exp_DN...
+unfold Fcore_generic_fmt.canonic.
+rewrite <- H0; reflexivity.
+reflexivity.
+apply trans_eq with (round beta fexp Ztrunc (round beta fexp Zceil x)).
+reflexivity.
+apply round_generic...
+intros Y.
+replace  (Fnum {| Fnum := Zfloor (scaled_mantissa beta fexp x); Fexp := cexp x |})
+   with (Fnum (Float beta 0 (fexp (ln_beta beta 0)))).
+generalize (DN_UP_parity_generic beta fexp)...
+unfold DN_UP_parity_prop.
+intros T; apply T with x; clear T.
+unfold generic_format.
+rewrite <- (scaled_mantissa_mult_bpow beta fexp x) at 1.
+unfold F2R; simpl.
+apply Rmult_neq_compat_r.
+apply Rgt_not_eq, bpow_gt_0.
+rewrite Ztrunc_floor.
+assumption.
+apply Rmult_le_pos.
+now left.
+apply bpow_ge_0.
+apply canonic_0.
+unfold Fcore_generic_fmt.canonic.
+rewrite <- H0; reflexivity.
+rewrite <- Y; unfold F2R; simpl; ring.
+apply trans_eq with (round beta fexp Ztrunc (round beta fexp Zceil x)).
+reflexivity.
+apply round_generic...
+simpl.
+apply eq_Z2R, Rmult_eq_reg_r with (bpow (cexp x)).
+unfold round, F2R in Y; simpl in Y; rewrite <- Y.
+simpl; ring.
+apply Rgt_not_eq, bpow_gt_0.
+(* . *)
+intros Y.
+case (Rle_lt_or_eq_dec 0 (round beta fexp Zfloor x)).
+rewrite <- round_0 with beta fexp Zfloor...
+apply round_le...
+now left.
+intros Hrx.
+set (ef := canonic_exp beta fexp x).
+set (mf := Zfloor (scaled_mantissa beta fexp x)).
+exists (Float beta mf ef).
+unfold Fcore_generic_fmt.canonic.
+repeat split; try assumption.
+simpl.
+apply trans_eq with (cexp (round beta fexp Zfloor x )).
+apply sym_eq, canonic_exp_DN...
+reflexivity.
+intros Hrx; contradict Y.
+replace (Zfloor (scaled_mantissa beta fexp x)) with 0%Z.
+simpl; discriminate.
+apply eq_Z2R, Rmult_eq_reg_r with (bpow (cexp x)).
+unfold round, F2R in Hrx; simpl in Hrx; rewrite <- Hrx.
+simpl; ring.
+apply Rgt_not_eq, bpow_gt_0.
+Qed.
+
+End Fcore_rnd_odd.
+
+Section Odd_prop_aux.
+
+Variable beta : radix.
+Hypothesis Even_beta: Zeven (radix_val beta)=true.
+
+Notation bpow e := (bpow beta e).
+
+Variable fexp : Z -> Z.
+Variable fexpe : Z -> Z.
+
+Context { valid_exp : Valid_exp fexp }.
+Context { exists_NE_ : Exists_NE beta fexp }. (* for underflow reason *)
+Context { valid_expe : Valid_exp fexpe }.
+Context { exists_NE_e : Exists_NE beta fexpe }. (* for defining rounding to odd *)
+
+Hypothesis fexpe_fexp: forall e, (fexpe e <= fexp e -2)%Z.
+
+
+Lemma generic_format_fexpe_fexp: forall x,
+ generic_format beta fexp x ->  generic_format beta fexpe x.
+Proof.
+intros x Hx.
+apply generic_inclusion_ln_beta with fexp; trivial; intros Hx2.
+generalize (fexpe_fexp (ln_beta beta x)).
+omega.
+Qed.
+
+
+
+Lemma exists_even_fexp_lt: forall (c:Z->Z), forall (x:R),
+      (exists f:float beta, F2R f = x /\ (c (ln_beta beta x) < Fexp f)%Z) ->
+      exists f:float beta, F2R f =x /\ canonic beta c f /\ Zeven (Fnum f) = true.
+Proof with auto with typeclass_instances.
+intros c x (g,(Hg1,Hg2)).
+exists (Float beta
+     (Fnum g*Z.pow (radix_val beta) (Fexp g - c (ln_beta beta x)))
+     (c (ln_beta beta x))).
+assert (F2R (Float beta
+     (Fnum g*Z.pow (radix_val beta) (Fexp g - c (ln_beta beta x)))
+     (c (ln_beta beta x))) = x).
+unfold F2R; simpl.
+rewrite Z2R_mult, Z2R_Zpower.
+rewrite Rmult_assoc, <- bpow_plus.
+rewrite <- Hg1; unfold F2R.
+apply f_equal, f_equal.
+ring.
+omega.
+split; trivial.
+split.
+unfold canonic, canonic_exp.
+now rewrite H.
+simpl.
+rewrite Zeven_mult.
+rewrite Zeven_Zpower.
+rewrite Even_beta.
+apply Bool.orb_true_intro.
+now right.
+omega.
+Qed.
+
+
+Variable choice:Z->bool.
+Variable x:R.
+
+
+Variable d u: float beta.
+Hypothesis Hd: Rnd_DN_pt (generic_format beta fexp) x (F2R d).
+Hypothesis Cd: canonic beta fexp d.
+Hypothesis Hu: Rnd_UP_pt (generic_format beta fexp) x (F2R u).
+Hypothesis Cu: canonic beta fexp u.
+
+Hypothesis xPos: (0 < x)%R.
+
+
+Let m:= ((F2R d+F2R u)/2)%R.
+
+
+Lemma d_eq: F2R d= round beta fexp Zfloor x.
+Proof with auto with typeclass_instances.
+apply Rnd_DN_pt_unicity with (generic_format beta fexp) x...
+apply round_DN_pt...
+Qed.
+
+
+Lemma u_eq: F2R u= round beta fexp Zceil x.
+Proof with auto with typeclass_instances.
+apply Rnd_UP_pt_unicity with (generic_format beta fexp) x...
+apply round_UP_pt...
+Qed.
+
+
+Lemma d_ge_0: (0 <= F2R d)%R.
+Proof with auto with typeclass_instances.
+rewrite d_eq; apply round_ge_generic...
+apply generic_format_0.
+now left.
+Qed.
+
+
+
+Lemma ln_beta_d:  (0< F2R d)%R ->
+    (ln_beta beta (F2R d) = ln_beta beta x :>Z).
+Proof with auto with typeclass_instances.
+intros Y.
+rewrite d_eq; apply ln_beta_round_DN...
+now rewrite <- d_eq.
+Qed.
+
+
+Lemma Fexp_d: (0 < F2R d)%R -> Fexp d =fexp (ln_beta beta x).
+Proof with auto with typeclass_instances.
+intros Y.
+now rewrite Cd, <- ln_beta_d.
+Qed.
+
+
+
+Lemma format_bpow_x: (0 < F2R d)%R
+    -> generic_format beta fexp  (bpow (ln_beta beta x)).
+Proof with auto with typeclass_instances.
+intros Y.
+apply generic_format_bpow.
+apply valid_exp.
+rewrite <- Fexp_d; trivial.
+apply Zlt_le_trans with  (ln_beta beta (F2R d))%Z.
+rewrite Cd; apply ln_beta_generic_gt...
+now apply Rgt_not_eq.
+apply Hd.
+apply ln_beta_le; trivial.
+apply Hd.
+Qed.
+
+
+Lemma format_bpow_d: (0 < F2R d)%R ->
+  generic_format beta fexp (bpow (ln_beta beta (F2R d))).
+Proof with auto with typeclass_instances.
+intros Y; apply generic_format_bpow.
+apply valid_exp.
+apply ln_beta_generic_gt...
+now apply Rgt_not_eq.
+now apply generic_format_canonic.
+Qed.
+
+
+Lemma d_le_m: (F2R d <= m)%R.
+apply Rmult_le_reg_l with 2%R.
+auto with real.
+apply Rplus_le_reg_l with (-F2R d)%R.
+apply Rle_trans with (F2R d).
+right; ring.
+apply Rle_trans with (F2R u).
+apply Rle_trans with x.
+apply Hd.
+apply Hu.
+right; unfold m; field.
+Qed.
+
+Lemma m_le_u: (m <= F2R u)%R.
+apply Rmult_le_reg_l with 2%R.
+auto with real.
+apply Rplus_le_reg_l with (-F2R u)%R.
+apply Rle_trans with (F2R d).
+right; unfold m; field.
+apply Rle_trans with (F2R u).
+apply Rle_trans with x.
+apply Hd.
+apply Hu.
+right; ring.
+Qed.
+
+Lemma ln_beta_m: (0 < F2R d)%R -> (ln_beta beta m =ln_beta beta (F2R d) :>Z).
+Proof with auto with typeclass_instances.
+intros dPos; apply ln_beta_unique_pos.
+split.
+apply Rle_trans with (F2R d).
+destruct (ln_beta beta (F2R d)) as (e,He).
+simpl.
+rewrite Rabs_right in He.
+apply He.
+now apply Rgt_not_eq.
+apply Rle_ge; now left.
+apply d_le_m.
+case m_le_u; intros H.
+apply Rlt_le_trans with (1:=H).
+rewrite u_eq.
+apply round_le_generic...
+apply generic_format_bpow.
+apply valid_exp.
+apply ln_beta_generic_gt...
+now apply Rgt_not_eq.
+now apply generic_format_canonic.
+case (Rle_or_lt x (bpow (ln_beta beta (F2R d)))); trivial; intros Z.
+absurd ((bpow (ln_beta beta (F2R d)) <= (F2R d)))%R.
+apply Rlt_not_le.
+destruct  (ln_beta beta (F2R d)) as (e,He).
+simpl in *; rewrite Rabs_right in He.
+apply He.
+now apply Rgt_not_eq.
+apply Rle_ge; now left.
+apply Rle_trans with (round beta fexp Zfloor x).
+2: right; apply sym_eq, d_eq.
+apply round_ge_generic...
+apply generic_format_bpow.
+apply valid_exp.
+apply ln_beta_generic_gt...
+now apply Rgt_not_eq.
+now apply generic_format_canonic.
+now left.
+replace m with (F2R d).
+destruct  (ln_beta beta (F2R d)) as (e,He).
+simpl in *; rewrite Rabs_right in He.
+apply He.
+now apply Rgt_not_eq.
+apply Rle_ge; now left.
+assert (F2R d = F2R u).
+apply Rmult_eq_reg_l with (/2)%R.
+apply Rplus_eq_reg_l with (/2*F2R u)%R.
+apply trans_eq with m.
+unfold m, Rdiv; ring.
+rewrite H; field.
+auto with real.
+apply Rgt_not_eq, Rlt_gt; auto with real.
+unfold m; rewrite <- H0; field.
+Qed.
+
+
+Lemma ln_beta_m_0: (0 = F2R d)%R
+    -> (ln_beta beta m =ln_beta beta (F2R u)-1:>Z)%Z.
+Proof with auto with typeclass_instances.
+intros Y.
+apply ln_beta_unique_pos.
+unfold m; rewrite <- Y, Rplus_0_l.
+rewrite u_eq.
+destruct (ln_beta beta x) as (e,He).
+rewrite Rabs_right in He.
+rewrite round_UP_small_pos with (ex:=e).
+rewrite ln_beta_bpow.
+ring_simplify (fexp e + 1 - 1)%Z.
+split.
+unfold Zminus; rewrite bpow_plus.
+unfold Rdiv; apply Rmult_le_compat_l.
+apply bpow_ge_0.
+simpl; unfold Z.pow_pos; simpl.
+rewrite Zmult_1_r; apply Rinv_le.
+auto with real.
+apply (Z2R_le 2).
+specialize (radix_gt_1 beta).
+omega.
+apply Rlt_le_trans with (bpow (fexp e)*1)%R.
+2: right; ring.
+unfold Rdiv; apply Rmult_lt_compat_l.
+apply bpow_gt_0.
+rewrite <- Rinv_1 at 3.
+apply Rinv_lt; auto with real.
+now apply He, Rgt_not_eq.
+apply exp_small_round_0_pos with beta (Zfloor) x...
+now apply He, Rgt_not_eq.
+now rewrite <- d_eq, Y.
+now left.
+Qed.
+
+
+
+
+
+Lemma u'_eq:  (0 < F2R d)%R -> exists f:float beta, F2R f = F2R u /\ (Fexp f = Fexp d)%Z.
+Proof with auto with typeclass_instances.
+intros Y.
+rewrite u_eq; unfold round.
+eexists; repeat split.
+simpl; now rewrite Fexp_d.
+Qed.
+
+
+
+
+Lemma m_eq: (0 < F2R d)%R ->  exists f:float beta,
+   F2R f = m /\ (Fexp f = fexp (ln_beta beta x) -1)%Z.
+Proof with auto with typeclass_instances.
+intros Y.
+specialize (Zeven_ex (radix_val beta)); rewrite Even_beta.
+intros (b, Hb); rewrite Zplus_0_r in Hb.
+destruct u'_eq as (u', (Hu'1,Hu'2)); trivial.
+exists (Fmult beta (Float beta b (-1)) (Fplus beta d u'))%R.
+split.
+rewrite F2R_mult, F2R_plus, Hu'1.
+unfold m; rewrite Rmult_comm.
+unfold Rdiv; apply f_equal.
+unfold F2R; simpl; unfold Z.pow_pos; simpl.
+rewrite Zmult_1_r, Hb, Z2R_mult.
+simpl; field.
+apply Rgt_not_eq, Rmult_lt_reg_l with (Z2R 2).
+simpl; auto with real.
+rewrite Rmult_0_r, <-Z2R_mult, <-Hb.
+apply radix_pos.
+apply trans_eq with (-1+Fexp (Fplus beta d u'))%Z.
+unfold Fmult.
+destruct  (Fplus beta d u'); reflexivity.
+rewrite Zplus_comm; unfold Zminus; apply f_equal2.
+2: reflexivity.
+rewrite Fexp_Fplus.
+rewrite Z.min_l.
+now rewrite Fexp_d.
+rewrite Hu'2; omega.
+Qed.
+
+Lemma m_eq_0: (0 = F2R d)%R ->  exists f:float beta,
+   F2R f = m /\ (Fexp f = fexp (ln_beta beta (F2R u)) -1)%Z.
+Proof with auto with typeclass_instances.
+intros Y.
+specialize (Zeven_ex (radix_val beta)); rewrite Even_beta.
+intros (b, Hb); rewrite Zplus_0_r in Hb.
+exists (Fmult beta (Float beta b (-1)) u)%R.
+split.
+rewrite F2R_mult; unfold m; rewrite <- Y, Rplus_0_l.
+rewrite Rmult_comm.
+unfold Rdiv; apply f_equal.
+unfold F2R; simpl; unfold Z.pow_pos; simpl.
+rewrite Zmult_1_r, Hb, Z2R_mult.
+simpl; field.
+apply Rgt_not_eq, Rmult_lt_reg_l with (Z2R 2).
+simpl; auto with real.
+rewrite Rmult_0_r, <-Z2R_mult, <-Hb.
+apply radix_pos.
+apply trans_eq with (-1+Fexp u)%Z.
+unfold Fmult.
+destruct u; reflexivity.
+rewrite Zplus_comm, Cu; unfold Zminus; now apply f_equal2.
+Qed.
+
+Lemma fexp_m_eq_0:  (0 = F2R d)%R ->
+  (fexp (ln_beta beta (F2R u)-1) < fexp (ln_beta beta (F2R u))+1)%Z.
+Proof with auto with typeclass_instances.
+intros Y.
+assert ((fexp (ln_beta beta (F2R u) - 1) <= fexp (ln_beta beta (F2R u))))%Z.
+2: omega.
+destruct (ln_beta beta x) as (e,He).
+rewrite Rabs_right in He.
+2: now left.
+assert (e <= fexp e)%Z.
+apply exp_small_round_0_pos with beta (Zfloor) x...
+now apply He, Rgt_not_eq.
+now rewrite <- d_eq, Y.
+rewrite u_eq, round_UP_small_pos with (ex:=e); trivial.
+2: now apply He, Rgt_not_eq.
+rewrite ln_beta_bpow.
+ring_simplify (fexp e + 1 - 1)%Z.
+replace (fexp (fexp e)) with (fexp e).
+case exists_NE_; intros V.
+contradict V; rewrite Even_beta; discriminate.
+rewrite (proj2 (V e)); omega.
+apply sym_eq, valid_exp; omega.
+Qed.
+
+Lemma Fm:  generic_format beta fexpe m.
+case (d_ge_0); intros Y.
+(* *)
+destruct m_eq as (g,(Hg1,Hg2)); trivial.
+apply generic_format_F2R' with g.
+now apply sym_eq.
+intros H; unfold canonic_exp; rewrite Hg2.
+rewrite ln_beta_m; trivial.
+rewrite <- Fexp_d; trivial.
+rewrite Cd.
+unfold canonic_exp.
+generalize (fexpe_fexp (ln_beta beta (F2R d))).
+omega.
+(* *)
+destruct m_eq_0 as (g,(Hg1,Hg2)); trivial.
+apply generic_format_F2R' with g.
+assumption.
+intros H; unfold canonic_exp; rewrite Hg2.
+rewrite ln_beta_m_0; try assumption.
+apply Zle_trans with (1:=fexpe_fexp _).
+assert (fexp (ln_beta beta (F2R u)-1) < fexp (ln_beta beta (F2R u))+1)%Z;[idtac|omega].
+now apply fexp_m_eq_0.
+Qed.
+
+
+
+Lemma Zm:
+   exists g : float beta, F2R g = m /\ canonic beta fexpe g /\ Zeven (Fnum g) = true.
+Proof with auto with typeclass_instances.
+case (d_ge_0); intros Y.
+(* *)
+destruct m_eq as (g,(Hg1,Hg2)); trivial.
+apply exists_even_fexp_lt.
+exists g; split; trivial.
+rewrite Hg2.
+rewrite ln_beta_m; trivial.
+rewrite <- Fexp_d; trivial.
+rewrite Cd.
+unfold canonic_exp.
+generalize (fexpe_fexp  (ln_beta beta (F2R d))).
+omega.
+(* *)
+destruct m_eq_0 as (g,(Hg1,Hg2)); trivial.
+apply exists_even_fexp_lt.
+exists g; split; trivial.
+rewrite Hg2.
+rewrite ln_beta_m_0; trivial.
+apply Zle_lt_trans with (1:=fexpe_fexp _).
+assert (fexp (ln_beta beta (F2R u)-1) < fexp (ln_beta beta (F2R u))+1)%Z;[idtac|omega].
+now apply fexp_m_eq_0.
+Qed.
+
+
+Lemma DN_odd_d_aux: forall z, (F2R d<= z< F2R u)%R ->
+    Rnd_DN_pt (generic_format beta fexp) z (F2R d).
+Proof with auto with typeclass_instances.
+intros z Hz1.
+replace (F2R d) with (round beta fexp Zfloor z).
+apply round_DN_pt...
+case (Rnd_DN_UP_pt_split _ _ _ _ Hd Hu (round beta fexp Zfloor z)).
+apply generic_format_round...
+intros Y; apply Rle_antisym; trivial.
+apply round_DN_pt...
+apply Hd.
+apply Hz1.
+intros Y; absurd (z < z)%R.
+auto with real.
+apply Rlt_le_trans with (1:=proj2 Hz1), Rle_trans with (1:=Y).
+apply round_DN_pt...
+Qed.
+
+Lemma UP_odd_d_aux: forall z, (F2R d< z <= F2R u)%R ->
+    Rnd_UP_pt (generic_format beta fexp) z (F2R u).
+Proof with auto with typeclass_instances.
+intros z Hz1.
+replace (F2R u) with (round beta fexp Zceil z).
+apply round_UP_pt...
+case (Rnd_DN_UP_pt_split _ _ _ _ Hd Hu (round beta fexp Zceil z)).
+apply generic_format_round...
+intros Y; absurd (z < z)%R.
+auto with real.
+apply Rle_lt_trans with (2:=proj1 Hz1), Rle_trans with (2:=Y).
+apply round_UP_pt...
+intros Y; apply Rle_antisym; trivial.
+apply round_UP_pt...
+apply Hu.
+apply Hz1.
+Qed.
+
+
+Theorem round_odd_prop_pos: 
+  round beta fexp (Znearest choice) (round beta fexpe Zrnd_odd x) =
+               round beta fexp (Znearest choice) x.
+Proof with auto with typeclass_instances.
+set (o:=round beta fexpe Zrnd_odd x).
+case (generic_format_EM beta fexp x); intros Hx.
+replace o with x; trivial.
+unfold o; apply sym_eq, round_generic...
+now apply generic_format_fexpe_fexp.
+assert (K1:(F2R d <= o)%R).
+apply round_ge_generic...
+apply generic_format_fexpe_fexp, Hd.
+apply Hd.
+assert (K2:(o <= F2R u)%R).
+apply round_le_generic...
+apply generic_format_fexpe_fexp, Hu.
+apply Hu.
+assert (P:(x <> m -> o=m -> (forall P:Prop, P))).
+intros Y1 Y2.
+assert (H:(Rnd_odd_pt beta fexpe x o)).
+apply round_odd_pt...
+destruct H as (_,H); destruct H.
+absurd (x=m)%R; try trivial.
+now rewrite <- Y2, H.
+destruct H as (_,(k,(Hk1,(Hk2,Hk3)))).
+destruct Zm as (k',(Hk'1,(Hk'2,Hk'3))).
+absurd (true=false).
+discriminate.
+rewrite <- Hk3, <- Hk'3.
+apply f_equal, f_equal.
+apply canonic_unicity with fexpe...
+now rewrite Hk'1, <- Y2.
+assert (generic_format beta fexp o -> (forall P:Prop, P)).
+intros Y.
+assert (H:(Rnd_odd_pt beta fexpe x o)).
+apply round_odd_pt...
+destruct H as (_,H); destruct H.
+absurd (generic_format beta fexp x); trivial.
+now rewrite <- H.
+destruct H as (_,(k,(Hk1,(Hk2,Hk3)))).
+destruct (exists_even_fexp_lt fexpe o) as (k',(Hk'1,(Hk'2,Hk'3))).
+eexists; split.
+apply sym_eq, Y.
+simpl; unfold canonic_exp.
+apply Zle_lt_trans with (1:=fexpe_fexp _).
+omega.
+absurd (true=false).
+discriminate.
+rewrite <- Hk3, <- Hk'3.
+apply f_equal, f_equal.
+apply canonic_unicity with fexpe...
+now rewrite Hk'1, <- Hk1.
+case K1; clear K1; intros K1.
+2: apply H; rewrite <- K1; apply Hd.
+case K2; clear K2; intros K2.
+2: apply H; rewrite K2; apply Hu.
+case (Rle_or_lt  x m); intros Y;[destruct Y|idtac].
+(* . *)
+apply trans_eq with (F2R d).
+apply round_N_DN_betw with (F2R u)...
+apply DN_odd_d_aux; split; try left; assumption.
+apply UP_odd_d_aux; split; try left; assumption.
+split.
+apply round_ge_generic...
+apply generic_format_fexpe_fexp, Hd.
+apply Hd.
+assert (o <= (F2R d + F2R u) / 2)%R.
+apply round_le_generic...
+apply Fm.
+now left.
+destruct H1; trivial.
+apply P.
+now apply Rlt_not_eq.
+trivial.
+apply sym_eq, round_N_DN_betw with (F2R u)...
+split.
+apply Hd.
+exact H0.
+(* . *)
+replace o with x.
+reflexivity.
+apply sym_eq, round_generic...
+rewrite H0; apply Fm.
+(* . *)
+apply trans_eq with (F2R u).
+apply round_N_UP_betw with (F2R d)...
+apply DN_odd_d_aux; split; try left; assumption.
+apply UP_odd_d_aux; split; try left; assumption.
+split.
+assert ((F2R d + F2R u) / 2 <= o)%R.
+apply round_ge_generic...
+apply Fm.
+now left.
+destruct H0; trivial.
+apply P.
+now apply Rgt_not_eq.
+rewrite <- H0; trivial.
+apply round_le_generic...
+apply generic_format_fexpe_fexp, Hu.
+apply Hu.
+apply sym_eq, round_N_UP_betw with (F2R d)...
+split.
+exact Y.
+apply Hu.
+Qed.
+
+
+End Odd_prop_aux.
+
+Section Odd_prop.
+
+Variable beta : radix.
+Hypothesis Even_beta: Zeven (radix_val beta)=true.
+
+Variable fexp : Z -> Z.
+Variable fexpe : Z -> Z.
+Variable choice:Z->bool.
+
+Context { valid_exp : Valid_exp fexp }.
+Context { exists_NE_ : Exists_NE beta fexp }. (* for underflow reason *)
+Context { valid_expe : Valid_exp fexpe }.
+Context { exists_NE_e : Exists_NE beta fexpe }. (* for defining rounding to odd *)
+
+Hypothesis fexpe_fexp: forall e, (fexpe e <= fexp e -2)%Z.
+
+
+Theorem canonizer: forall f, generic_format beta fexp f
+   -> exists g : float beta, f = F2R g /\ canonic beta fexp g.
+Proof with auto with typeclass_instances.
+intros f Hf.
+exists (Float beta (Ztrunc (scaled_mantissa beta fexp f)) (canonic_exp beta fexp f)).
+assert (L:(f = F2R (Float beta (Ztrunc (scaled_mantissa beta fexp f)) (canonic_exp beta fexp f)))).
+apply trans_eq with (round beta fexp Ztrunc f).
+apply sym_eq, round_generic...
+reflexivity.
+split; trivial.
+unfold canonic; rewrite <- L.
+reflexivity.
+Qed.
+
+
+
+
+Theorem round_odd_prop: forall x, 
+  round beta fexp (Znearest choice) (round beta fexpe Zrnd_odd x) =
+               round beta fexp (Znearest choice) x.
+Proof with auto with typeclass_instances.
+intros x.
+case (total_order_T x 0); intros H; [case H; clear H; intros H | idtac].
+rewrite <- (Ropp_involutive x).
+rewrite round_odd_opp.
+rewrite 2!round_N_opp.
+apply f_equal.
+destruct (canonizer (round beta fexp Zfloor (-x))) as (d,(Hd1,Hd2)).
+apply generic_format_round...
+destruct (canonizer (round beta fexp Zceil (-x))) as (u,(Hu1,Hu2)).
+apply generic_format_round...
+apply round_odd_prop_pos with d u...
+rewrite <- Hd1; apply round_DN_pt...
+rewrite <- Hu1; apply round_UP_pt...
+auto with real.
+(* . *)
+rewrite H; repeat rewrite round_0...
+(* . *)
+destruct (canonizer (round beta fexp Zfloor x)) as (d,(Hd1,Hd2)).
+apply generic_format_round...
+destruct (canonizer (round beta fexp Zceil x)) as (u,(Hu1,Hu2)).
+apply generic_format_round...
+apply round_odd_prop_pos with d u...
+rewrite <- Hd1; apply round_DN_pt...
+rewrite <- Hu1; apply round_UP_pt...
+Qed.
+
+
+End Odd_prop.