Added a typeclass for rounding functions.

01858c76 · Guillaume Melquiond · de2f0164 · 01858c76 · 01858c76 · 01858c76
Commit 01858c76 authored Jul 22, 2011 by Guillaume Melquiond
12 changed files
--- a/src/Appli/Fappli_Axpy.v
+++ b/src/Appli/Fappli_Axpy.v
@@ -63,8 +63,8 @@ Qed.


 Definition MinOrMax x f := 
-   ((f = round beta (FLX_exp prec) rndDN x) 
-     \/ (f = round beta (FLX_exp prec) rndUP x)).
+   ((f = round beta (FLX_exp prec) Zfloor x) 
+     \/ (f = round beta (FLX_exp prec) Zceil x)).

 Theorem MinOrMax_opp: forall x f,
   MinOrMax x f <->  MinOrMax (-x) (-f).
@@ -85,7 +85,7 @@ Theorem implies_DN_lt_ulp:
  forall x f, format f ->
  (0 < f <= x)%R ->
  (Rabs (f-x) < ulp f)%R -> 
-  (f = round beta (FLX_exp prec) rndDN x)%R.
+  (f = round beta (FLX_exp prec) Zfloor x)%R.
 intros x f Hf Hxf1 Hxf2.
 apply sym_eq.
 replace x with (f+-(f-x))%R by ring.
@@ -160,8 +160,8 @@ Hypothesis Ha: format a.
 Hypothesis Hx: format x.
 Hypothesis Hy: format y.

-Notation t := (round beta (FLX_exp prec) (rndN choice) (a*x)).
-Notation u := (round beta (FLX_exp prec) (rndN choice) (t+y)).
+Notation t := (round beta (FLX_exp prec) (Znearest choice) (a*x)).
+Notation u := (round beta (FLX_exp prec) (Znearest choice) (t+y)).

 (*
 Axpy_aux1 : lemma Closest?(b)(a*x,t) => Closest?(b)(t+y,u) => 0 < u

--- a/src/Appli/Fappli_IEEE.v
+++ b/src/Appli/Fappli_IEEE.v
@@ -52,6 +52,7 @@ Hypothesis Hmax : (prec < emax)%Z.
 Let emin := (3 - emax - prec)%Z.
 Let fexp := FLT_exp emin prec.
 Instance fexp_correct : Valid_exp fexp := FLT_exp_valid emin prec.
+Instance fexp_monotone : Monotone_exp fexp := FLT_exp_monotone emin prec.

 Definition bounded_prec m e :=
  Zeq_bool (fexp (Z_of_nat (S (digits2_Pnat m)) + e)) e.
@@ -526,11 +527,11 @@ Inductive mode := mode_NE | mode_ZR | mode_DN | mode_UP | mode_NA.

 Definition round_mode m :=
  match m with
-  | mode_NE => rndNE
-  | mode_ZR => rndZR
-  | mode_DN => rndDN
-  | mode_UP => rndUP
-  | mode_NA => rndNA
+  | mode_NE => ZnearestE
+  | mode_ZR => Ztrunc
+  | mode_DN => Zfloor
+  | mode_UP => Zceil
+  | mode_NA => ZnearestA
  end.

 Definition choice_mode m sx mx lx :=
@@ -542,6 +543,11 @@ Definition choice_mode m sx mx lx :=
  | mode_NA => cond_incr (round_N true lx) mx
  end.

+Global Instance valid_rnd_round_mode : forall m, Valid_rnd (round_mode m).
+Proof.
+destruct m ; unfold round_mode ; auto with typeclass_instances.
+Qed.
+
 Definition overflow_to_inf m s :=
  match m with
  | mode_NE => true
@@ -573,7 +579,7 @@ Theorem binary_round_sign_correct :
    FF2R radix2 (binary_round_sign mode (Rlt_bool x 0) mx ex lx) = round radix2 fexp (round_mode mode) x
  else
    binary_round_sign mode (Rlt_bool x 0) mx ex lx = binary_overflow mode (Rlt_bool x 0).
-Proof.
+Proof with auto with typeclass_instances.
 intros m x mx ex lx Bx Ex.
 unfold binary_round_sign.
 rewrite shr_truncate. 2: easy.
@@ -627,9 +633,7 @@ rewrite <- ln_beta_F2R_digits, <- Hr, ln_beta_abs.
 rewrite H1b.
 rewrite canonic_exponent_abs.
 fold (canonic_exponent radix2 fexp (round radix2 fexp (round_mode m) x)).
-apply canonic_exponent_round.
-apply fexp_correct.
-apply FLT_exp_monotone.
+apply canonic_exponent_round...
 rewrite H1c.
 case (Rlt_bool x 0).
 apply Rlt_not_eq.
@@ -718,9 +722,8 @@ apply Rlt_trans with R0.
 now apply F2R_lt_0_compat.
 now apply F2R_gt_0_compat.
 rewrite <- Hr.
-apply generic_format_abs.
-apply generic_format_round.
-apply fexp_correct.
+apply generic_format_abs...
+apply generic_format_round...
 (* . not m1' < 0 *)
 elim Rgt_not_eq with (2 := Hr).
 apply Rlt_le_trans with R0.
@@ -822,7 +825,7 @@ Theorem Bmult_correct :
    B2FF (Bmult 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 ; [ apply refl_equal | apply bpow_gt_0 ] ).
+  try ( rewrite ?Rmult_0_r, ?Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ apply refl_equal | apply bpow_gt_0 | auto with typeclass_instances ] ).
 simpl.
 case Bmult_correct_aux.
 intros H1 H2.
@@ -978,22 +981,20 @@ Theorem Bplus_correct :
    B2R (Bplus m x y) = round radix2 fexp (round_mode m) (B2R x + B2R y)
  else
    (B2FF (Bplus m x y) = binary_overflow m (Bsign x) /\ Bsign x = Bsign y).
-Proof.
+Proof with auto with typeclass_instances.
 intros 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.
+rewrite Rplus_0_r, round_0, Rabs_R0, Rlt_bool_true...
 simpl.
 case (Bool.eqb sx sy) ; try easy.
 now case m.
 apply bpow_gt_0.
 (* *)
-rewrite Rplus_0_l, round_generic, Rlt_bool_true.
-apply refl_equal.
+rewrite Rplus_0_l, round_generic, Rlt_bool_true...
 apply B2R_lt_emax.
 apply generic_format_B2R.
 (* *)
-rewrite Rplus_0_r, round_generic, Rlt_bool_true.
-apply refl_equal.
+rewrite Rplus_0_r, round_generic, Rlt_bool_true...
 apply B2R_lt_emax.
 apply generic_format_B2R.
 (* *)
@@ -1058,15 +1059,13 @@ split.
 apply Rlt_le_trans with (F2R (Float radix2 (cond_Zopp true (Zpos mx)) ex)).
 rewrite <- opp_F2R.
 now apply Ropp_lt_contravar.
-apply round_monotone_l.
-apply fexp_correct.
+apply round_monotone_l...
 now apply generic_format_canonic.
 pattern (F2R (Float radix2 (cond_Zopp true (Zpos mx)) ex)) at 1 ; rewrite <- Rplus_0_r.
 apply Rplus_le_compat_l.
 now apply F2R_ge_0_compat.
 apply Rle_lt_trans with (2 := By).
-apply round_monotone_r.
-apply fexp_correct.
+apply round_monotone_r...
 now apply generic_format_canonic.
 rewrite <- (Rplus_0_l (F2R (Float radix2 (Zpos my) ey))).
 apply Rplus_le_compat_r.
@@ -1080,22 +1079,20 @@ split.
 apply Rlt_le_trans with (F2R (Float radix2 (cond_Zopp true (Zpos my)) ey)).
 rewrite <- opp_F2R.
 now apply Ropp_lt_contravar.
-apply round_monotone_l.
-apply fexp_correct.
+apply round_monotone_l...
 now apply generic_format_canonic.
 pattern (F2R (Float radix2 (cond_Zopp true (Zpos my)) ey)) at 1 ; rewrite <- Rplus_0_l.
 apply Rplus_le_compat_r.
 now apply F2R_ge_0_compat.
 apply Rle_lt_trans with (2 := Bx).
-apply round_monotone_r.
-apply fexp_correct.
+apply round_monotone_r...
 now apply generic_format_canonic.
 rewrite <- (Rplus_0_r (F2R (Float radix2 (Zpos mx) ex))).
 apply Rplus_le_compat_l.
 now apply F2R_le_0_compat.
 destruct mz as [|mz|mz].
 (* . mz = 0 *)
-rewrite F2R_0, round_0, Rabs_R0, Rlt_bool_true.
+rewrite F2R_0, round_0, Rabs_R0, Rlt_bool_true...
 now case m.
 apply bpow_gt_0.
 (* . mz > 0 *)
@@ -1253,7 +1250,7 @@ intros m x [sy|sy| |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 ; [ apply refl_equal | apply bpow_gt_0 ] ).
+  try ( rewrite Rmult_0_l, round_0, Rabs_R0, Rlt_bool_true ; [ apply refl_equal | apply bpow_gt_0 | auto with typeclass_instances ] ).
 simpl.
 case Bdiv_correct_aux.
 intros H1 H2.
@@ -1275,7 +1272,7 @@ Lemma Bsqrt_correct_aux :
    end in
  valid_binary z = true /\
  FF2R radix2 z = round radix2 fexp (round_mode m) (sqrt x).
-Proof.
+Proof with auto with typeclass_instances.
 intros m mx ex Hx.
 simpl.
 refine (_ (Fsqrt_core_correct radix2 prec (Zpos mx) ex _)) ; try easy.
@@ -1353,8 +1350,7 @@ apply generic_format_canonic.
 apply (canonic_bounded_prec false).
 apply (andb_prop _ _ Hx).
 (* .. *)
-apply round_monotone_l.
-apply fexp_correct.
+apply round_monotone_l...
 apply generic_format_0.
 apply sqrt_ge_0.
 rewrite Rabs_pos_eq.
@@ -1389,7 +1385,7 @@ Theorem Bsqrt_correct :
  forall m x,
  B2R (Bsqrt m x) = round radix2 fexp (round_mode m) (sqrt (B2R x)).
 Proof.
-intros m [sx|[|]| |sx mx ex Hx] ; try ( now simpl ; rewrite sqrt_0, round_0 ).
+intros m [sx|[|]| |sx mx ex Hx] ; try ( now simpl ; rewrite sqrt_0, round_0 ; auto with typeclass_instances ).
 simpl.
 case Bsqrt_correct_aux.
 intros H1 H2.
@@ -1399,6 +1395,7 @@ unfold sqrt.
 case Rcase_abs.
 intros _.
 apply round_0.
+auto with typeclass_instances.
 intros H.
 elim Rge_not_lt with (1 := H).
 now apply F2R_lt_0_compat.

--- a/src/Appli/Fappli_sqrt_FLT_ne.v
+++ b/src/Appli/Fappli_sqrt_FLT_ne.v
@@ -44,7 +44,7 @@ Theorem Fsqrt_FLT_ne_correct :
  Rnd_NE_pt beta (FLT_exp emin prec) (sqrt (F2R x)) (F2R (Fsqrt_FLT_ne x)).
 Proof with auto with typeclass_instances.
 intros x.
-replace (F2R (Fsqrt_FLT_ne x)) with (round beta (FLT_exp emin prec) rndNE (sqrt (F2R x))).
+replace (F2R (Fsqrt_FLT_ne x)) with (round beta (FLT_exp emin prec) ZnearestE (sqrt (F2R x))).
 apply round_NE_pt...
 unfold Fsqrt_FLT_ne.
 destruct x as (mx, ex).
@@ -53,7 +53,7 @@ case (Zle_bool mx 0) ; intros Hm.
 (* mx = 0 *)
 rewrite F2R_0.
 replace (sqrt (F2R (Float beta mx ex))) with R0.
-apply round_0.
+apply round_0...
 destruct (Zle_lt_or_eq _ _ Hm) as [Hm'|Hm'].
 unfold sqrt.
 case Rcase_abs ; intros Hs.

--- a/src/Calc/Fcalc_round.v
+++ b/src/Calc/Fcalc_round.v
@@ -38,7 +38,7 @@ Notation format := (generic_format beta fexp).

 Theorem inbetween_float_round :
  forall rnd choice,
-  ( forall x m l, inbetween_int m x l -> Zrnd rnd x = choice m l ) ->
+  ( forall x m l, inbetween_int m x l -> rnd x = choice m l ) ->
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e x l ->
@@ -58,7 +58,7 @@ Definition cond_incr (b : bool) m := if b then (m + 1)%Z else m.
 Theorem inbetween_float_round_sign :
  forall rnd choice,
  ( forall x m l, inbetween_int m (Rabs x) l ->
-    Zrnd rnd x = cond_Zopp (Rlt_bool x 0) (choice (Rlt_bool x 0) m l) ) ->
+    rnd x = cond_Zopp (Rlt_bool x 0) (choice (Rlt_bool x 0) m l) ) ->
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e (Rabs x) l ->
@@ -93,7 +93,7 @@ Qed.
 Theorem inbetween_int_DN :
  forall x m l,
  inbetween_int m x l ->
-  Zrnd rndDN x = m.
+  Zfloor x = m.
 Proof.
 intros x m l Hl.
 refine (Zfloor_imp m _ _).
@@ -106,7 +106,7 @@ Theorem inbetween_float_DN :
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e x l ->
-  round beta fexp rndDN x = F2R (Float beta m e).
+  round beta fexp Zfloor x = F2R (Float beta m e).
 Proof.
 apply inbetween_float_round with (choice := fun m l => m).
 exact inbetween_int_DN.
@@ -121,7 +121,7 @@ Definition round_sign_DN s l :=
 Theorem inbetween_int_DN_sign :
  forall x m l,
  inbetween_int m (Rabs x) l ->
-  Zrnd rndDN x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_sign_DN (Rlt_bool x 0) l) m).
+  Zfloor x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_sign_DN (Rlt_bool x 0) l) m).
 Proof.
 intros x m l Hl.
 unfold Rabs in Hl.
@@ -158,7 +158,7 @@ Theorem inbetween_float_DN_sign :
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e (Rabs x) l ->
-  round beta fexp rndDN x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) (cond_incr (round_sign_DN (Rlt_bool x 0) l) m)) e).
+  round beta fexp Zfloor x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) (cond_incr (round_sign_DN (Rlt_bool x 0) l) m)) e).
 Proof.
 apply inbetween_float_round_sign with (choice := fun s m l => cond_incr (round_sign_DN s l) m).
 exact inbetween_int_DN_sign.
@@ -175,7 +175,7 @@ Definition round_UP l :=
 Theorem inbetween_int_UP :
  forall x m l,
  inbetween_int m x l ->
-  Zrnd rndUP x = cond_incr (round_UP l) m.
+  Zceil x = cond_incr (round_UP l) m.
 Proof.
 intros x m l Hl.
 assert (Hl': l = loc_Exact \/ (l <> loc_Exact /\ round_UP l = true)).
@@ -199,7 +199,7 @@ Theorem inbetween_float_UP :
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e x l ->
-  round beta fexp rndUP x = F2R (Float beta (cond_incr (round_UP l) m) e).
+  round beta fexp Zceil x = F2R (Float beta (cond_incr (round_UP l) m) e).
 Proof.
 apply inbetween_float_round with (choice := fun m l => cond_incr (round_UP l) m).
 exact inbetween_int_UP.
@@ -214,7 +214,7 @@ Definition round_sign_UP s l :=
 Theorem inbetween_int_UP_sign :
  forall x m l,
  inbetween_int m (Rabs x) l ->
-  Zrnd rndUP x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_sign_UP (Rlt_bool x 0) l) m).
+  Zceil x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_sign_UP (Rlt_bool x 0) l) m).
 Proof.
 intros x m l Hl.
 unfold Rabs in Hl.
@@ -249,7 +249,7 @@ Theorem inbetween_float_UP_sign :
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e (Rabs x) l ->
-  round beta fexp rndUP x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) (cond_incr (round_sign_UP (Rlt_bool x 0) l) m)) e).
+  round beta fexp Zceil x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) (cond_incr (round_sign_UP (Rlt_bool x 0) l) m)) e).
 Proof.
 apply inbetween_float_round_sign with (choice := fun s m l => cond_incr (round_sign_UP s l) m).
 exact inbetween_int_UP_sign.
@@ -266,15 +266,15 @@ Definition round_ZR (s : bool) l :=
 Theorem inbetween_int_ZR :
  forall x m l,
  inbetween_int m x l ->
-  Zrnd rndZR x = cond_incr (round_ZR (Zlt_bool m 0) l) m.
-Proof.
+  Ztrunc x = cond_incr (round_ZR (Zlt_bool m 0) l) m.
+Proof with auto with typeclass_instances.
 intros x m l Hl.
 inversion_clear Hl as [Hx|l' Hx Hl'].
 (* Exact *)
 rewrite Hx.
-now rewrite Zrnd_Z2R.
+rewrite Zrnd_Z2R...
 (* not Exact *)
-unfold Zrnd, rndZR, Ztrunc.
+unfold Ztrunc.
 assert (Hm: Zfloor x = m).
 apply Zfloor_imp.
 exact (conj (Rlt_le _ _ (proj1 Hx)) (proj2 Hx)).
@@ -300,7 +300,7 @@ Theorem inbetween_float_ZR :
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e x l ->
-  round beta fexp rndZR x = F2R (Float beta (cond_incr (round_ZR (Zlt_bool m 0) l) m) e).
+  round beta fexp Ztrunc x = F2R (Float beta (cond_incr (round_ZR (Zlt_bool m 0) l) m) e).
 Proof.
 apply inbetween_float_round with (choice := fun m l => cond_incr (round_ZR (Zlt_bool m 0) l) m).
 exact inbetween_int_ZR.
@@ -309,7 +309,7 @@ Qed.
 Theorem inbetween_int_ZR_sign :
  forall x m l,
  inbetween_int m (Rabs x) l ->
-  Zrnd rndZR x = cond_Zopp (Rlt_bool x 0) m.
+  Ztrunc x = cond_Zopp (Rlt_bool x 0) m.
 Proof.
 intros x m l Hl.
 simpl.
@@ -339,7 +339,7 @@ Theorem inbetween_float_ZR_sign :
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e (Rabs x) l ->
-  round beta fexp rndZR x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) m) e).
+  round beta fexp Ztrunc x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) m) e).
 Proof.
 apply inbetween_float_round_sign with (choice := fun s m l => m).
 exact inbetween_int_ZR_sign.
@@ -358,15 +358,15 @@ Definition round_N (p : bool) l :=
 Theorem inbetween_int_N :
  forall choice x m l,
  inbetween_int m x l ->
-  Zrnd (rndN choice) x = cond_incr (round_N (choice m) l) m.
-Proof.
+  Znearest choice x = cond_incr (round_N (choice m) l) m.
+Proof with auto with typeclass_instances.
 intros choice x m l Hl.
 inversion_clear Hl as [Hx|l' Hx Hl'].
 (* Exact *)
 rewrite Hx.
-now rewrite Zrnd_Z2R.
+rewrite Zrnd_Z2R...
 (* not Exact *)
-unfold Zrnd, rndNE, rndN, Znearest.
+unfold Znearest.
 assert (Hm: Zfloor x = m).
 apply Zfloor_imp.
 exact (conj (Rlt_le _ _ (proj1 Hx)) (proj2 Hx)).
@@ -387,8 +387,8 @@ Qed.
 Theorem inbetween_int_N_sign :
  forall choice x m l,
  inbetween_int m (Rabs x) l ->
-  Zrnd (rndN choice) x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_N (if Rlt_bool x 0 then negb (choice (-(m + 1))%Z) else choice m) l) m).
-Proof.
+  Znearest choice x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_N (if Rlt_bool x 0 then negb (choice (-(m + 1))%Z) else choice m) l) m).
+Proof with auto with typeclass_instances.
 intros choice x m l Hl.
 simpl.
 unfold Rabs in Hl.
@@ -401,7 +401,7 @@ rewrite Znearest_opp.
 apply f_equal.
 inversion_clear Hl as [Hx|l' Hx Hl'].
 rewrite Hx.
-apply Znearest_Z2R.
+apply Zrnd_Z2R...
 assert (Hm: Zfloor (-x) = m).
 apply Zfloor_imp.
 exact (conj (Rlt_le _ _ (proj1 Hx)) (proj2 Hx)).
@@ -425,7 +425,7 @@ rewrite Rlt_bool_false with (1 := Zx).
 simpl.
 inversion_clear Hl as [Hx|l' Hx Hl'].
 rewrite Hx.
-apply Znearest_Z2R.
+apply Zrnd_Z2R...
 assert (Hm: Zfloor x = m).
 apply Zfloor_imp.
 exact (conj (Rlt_le _ _ (proj1 Hx)) (proj2 Hx)).
@@ -449,7 +449,7 @@ Qed.
 Theorem inbetween_int_NE :
  forall x m l,
  inbetween_int m x l ->
-  Zrnd rndNE x = cond_incr (round_N (negb (Zeven m)) l) m.
+  ZnearestE x = cond_incr (round_N (negb (Zeven m)) l) m.
 Proof.
 intros x m l Hl.
 now apply inbetween_int_N with (choice := fun x => negb (Zeven x)).
@@ -459,7 +459,7 @@ Theorem inbetween_float_NE :
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e x l ->
-  round beta fexp rndNE x = F2R (Float beta (cond_incr (round_N (negb (Zeven m)) l) m) e).
+  round beta fexp ZnearestE x = F2R (Float beta (cond_incr (round_N (negb (Zeven m)) l) m) e).
 Proof.
 apply inbetween_float_round with (choice := fun m l => cond_incr (round_N (negb (Zeven m)) l) m).
 exact inbetween_int_NE.
@@ -468,7 +468,7 @@ Qed.
 Theorem inbetween_int_NE_sign :
  forall x m l,
  inbetween_int m (Rabs x) l ->
-  Zrnd rndNE x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_N (negb (Zeven m)) l) m).
+  ZnearestE x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_N (negb (Zeven m)) l) m).
 Proof.
 intros x m l Hl.
 erewrite inbetween_int_N_sign with (choice := fun x => negb (Zeven x)).
@@ -484,7 +484,7 @@ Theorem inbetween_float_NE_sign :
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e (Rabs x) l ->
-  round beta fexp rndNE x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) (cond_incr (round_N (negb (Zeven m)) l) m)) e).
+  round beta fexp ZnearestE x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) (cond_incr (round_N (negb (Zeven m)) l) m)) e).
 Proof.
 apply inbetween_float_round_sign with (choice := fun s m l => cond_incr (round_N (negb (Zeven m)) l) m).
 exact inbetween_int_NE_sign.
@@ -495,7 +495,7 @@ Qed.
 Theorem inbetween_int_NA :
  forall x m l,
  inbetween_int m x l ->
-  Zrnd rndNA x = cond_incr (round_N (Zle_bool 0 m) l) m.
+  ZnearestA x = cond_incr (round_N (Zle_bool 0 m) l) m.
 Proof.
 intros x m l Hl.
 now apply inbetween_int_N with (choice := fun x => Zle_bool 0 x).
@@ -505,7 +505,7 @@ Theorem inbetween_float_NA :
  forall x m l,
  let e := canonic_exponent beta fexp x in
  inbetween_float beta m e x l ->
-  round beta fexp rndNA x = F2R (Float beta (cond_incr (round_N (Zle_bool 0 m) l) m) e).
+  round beta fexp ZnearestA x = F2R (Float beta (cond_incr (round_N (Zle_bool 0 m) l) m) e).
 Proof.
 apply inbetween_float_round with (choice := fun m l => cond_incr (round_N (Zle_bool 0 m) l) m).
 exact inbetween_int_NA.
@@ -514,7 +514,7 @@ Qed.
 Theorem inbetween_int_NA_sign :
  forall x m l,
  inbetween_int m (Rabs x) l ->
-  Zrnd rndNA x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_N true l) m).
+  ZnearestA x = cond_Zopp (Rlt_bool x 0) (cond_incr (round_N true l) m).
 Proof.
 intros x m l Hl.
 erewrite inbetween_int_N_sign with (choice := Zle_bool 0).
@@ -829,19 +829,21 @@ Qed.

 Section round_dir.

-Variable rnd: Zround.
+Variable rnd : R -> Z.
+Context { valid_rnd : Valid_rnd rnd }.
+
 Variable choice : Z -> location -> Z.
 Hypothesis inbetween_int_valid :
  forall x m l,
  inbetween_int m x l ->
-  Zrnd rnd x = choice m l.
+  rnd x = choice m l.

 Theorem round_any_correct :
  forall x m e l,
  inbetween_float beta m e x l ->
  (e = canonic_exponent beta fexp x \/ (l = loc_Exact /\ format x)) ->
  round beta fexp rnd x = F2R (Float beta (choice m l) e).
-Proof.
+Proof with auto with typeclass_instances.
 intros x m e l Hin [He|(Hl,Hf)].
 rewrite He in Hin |- *.
 apply inbetween_float_round with (2 := Hin).
@@ -851,7 +853,7 @@ inversion_clear Hin.
 rewrite Hl.
 replace (choice m loc_Exact) with m.
 rewrite <- H.
-now apply round_generic.
+apply round_generic...
 rewrite <- (Zrnd_Z2R rnd m) at 1.
 apply inbetween_int_valid.
 now constructor.
@@ -877,19 +879,21 @@ End round_dir.

 Section round_dir_sign.

-Variable rnd: Zround.
+Variable rnd : R -> Z.
+Context { valid_rnd : Valid_rnd rnd }.
+
 Variable choice : bool -> Z -> location -> Z.
 Hypothesis inbetween_int_valid :
  forall x m l,
  inbetween_int m (Rabs x) l ->
-  Zrnd rnd x = cond_Zopp (Rlt_bool x 0) (choice (Rlt_bool x 0) m l).
+  rnd x = cond_Zopp (Rlt_bool x 0) (choice (Rlt_bool x 0) m l).

 Theorem round_sign_any_correct :
  forall x m e l,
  inbetween_float beta m e (Rabs x) l ->
  (e = canonic_exponent beta fexp x \/ (l = loc_Exact /\ format x)) ->
  round beta fexp rnd x = F2R (Float beta (cond_Zopp (Rlt_bool x 0) (choice (Rlt_bool x 0) m l)) e).
-Proof.
+Proof with auto with typeclass_instances.
 intros x m e l Hin [He|(Hl,Hf)].
 rewrite He in Hin |- *.
 apply inbetween_float_round_sign with (2 := Hin).
@@ -905,7 +909,7 @@ rewrite Rlt_bool_true with (1 := Zx).
 simpl.
 rewrite <- opp_F2R.
 rewrite <- H, Ropp_involutive.
-now apply round_generic.
+apply round_generic...
 rewrite Rlt_bool_false.
 simpl.
 rewrite <- H.

--- a/src/Core/Fcore_FTZ.v
+++ b/src/Core/Fcore_FTZ.v
@@ -236,30 +236,16 @@ Qed.
 Section FTZ_round.

 (** Rounding with FTZ *)
-Hypothesis rnd : Zround.
+Variable rnd : R -> Z.
+Context { valid_rnd : Valid_rnd rnd }.

 Definition Zrnd_FTZ x :=
-  if Rle_bool R1 (Rabs x) then Zrnd rnd x else Z0.
+  if Rle_bool R1 (Rabs x) then rnd x else Z0.

-Theorem Z_FTZ_Z2R :
-  forall n, Zrnd_FTZ (Z2R n) = n.
-Proof.
-intros n.
-unfold Zrnd_FTZ.
-rewrite Zrnd_Z2R.
-case Rle_bool_spec.
-easy.
-rewrite <- Z2R_abs.
-intros H.
-generalize (lt_Z2R _ 1 H).
-clear.
-now case n ; trivial ; simpl ; intros [p|p|].
-Qed.
-
-Theorem Z_FTZ_monotone :
-  forall x y, (x <= y)%R ->
-  (Zrnd_FTZ x <= Zrnd_FTZ y)%Z.
-Proof.
+Global Instance valid_rnd_FTZ : Valid_rnd Zrnd_FTZ.
+Proof with auto with typeclass_instances.
+split.
+(* *)
 intros x y Hxy.
 unfold Zrnd_FTZ.
 case Rle_bool_spec ; intros Hx ;
@@ -268,7 +254,7 @@ case Rle_bool_spec ; intros Hx ;
 (* 1 <= |x| *)
 now apply Zrnd_monotone.
 rewrite <- (Zrnd_Z2R rnd 0).
-apply Zrnd_monotone.
+apply Zrnd_monotone...
 apply Rle_trans with (Z2R (-1)). 2: now apply Z2R_le.
 destruct (Rabs_ge_inv _ _ Hx) as [Hx1|Hx1].
 exact Hx1.
@@ -278,7 +264,7 @@ apply Rle_trans with (1 := Hxy).
 apply RRle_abs.
 (* |x| < 1 *)
 rewrite <- (Zrnd_Z2R rnd 0).
-apply Zrnd_monotone.
+apply Zrnd_monotone...
 apply Rle_trans with (Z2R 1).
 now apply Z2R_le.
 destruct (Rabs_ge_inv _ _ Hy) as [Hy1|Hy1].
@@ -286,14 +272,23 @@ elim Rle_not_lt with (1 := Hy1).
 apply Rlt_le_trans with (2 := Hxy).
 apply (Rabs_def2 _ _ Hx).
 exact Hy1.
+(* *)