SB and GM: add realization for FP arithmetic

Makefile.in

@@ -845,7 +845,10 @@ COQLIBS_INT = $(addprefix lib/coq/int/, $(COQLIBS_INT_FILES))
 COQLIBS_REAL_FILES = Abs FromInt MinMax Real Square
 COQLIBS_REAL = $(addprefix lib/coq/real/, $(COQLIBS_REAL_FILES))
 
-COQLIBS_FILES = $(COQLIBS_INT) $(COQLIBS_REAL)
+COQLIBS_FP_FILES = Rounding GenFloat Single Double
+COQLIBS_FP = $(addprefix lib/coq/floating_point/, $(COQLIBS_FP_FILES))
+
+COQLIBS_FILES = $(COQLIBS_INT) $(COQLIBS_REAL) $(COQLIBS_FP)
 
 COQV  = $(addsuffix .v,  $(COQLIBS_FILES))
 COQVO = $(addsuffix .vo, $(COQLIBS_FILES))
@@ -859,8 +862,10 @@ install_no_local::
 	mkdir -p $(LIBDIR)/why3/coq
 	mkdir -p $(LIBDIR)/why3/coq/int
 	mkdir -p $(LIBDIR)/why3/coq/real
+	mkdir -p $(LIBDIR)/why3/coq/floating_point
 	cp $(addsuffix .vo, $(COQLIBS_INT)) $(LIBDIR)/why3/coq/int/
 	cp $(addsuffix .vo, $(COQLIBS_REAL)) $(LIBDIR)/why3/coq/real/
+	cp $(addsuffix .vo, $(COQLIBS_FP)) $(LIBDIR)/why3/coq/floating_point/
 
 install_local: $(COQVO)
 

lib/coq/floating_point/Double.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require Import Rbasic_fun.
+Require int.Int.
+Require real.Real.
+Require real.Abs.
+Require real.FromInt.
+Require floating_point.Rounding.
+Require Import floating_point.GenFloat.
+
+Definition double : Type := t 53 1024.
+
+Definition round: floating_point.Rounding.mode -> R -> R := round 53 1024.
+
+Definition round_logic: floating_point.Rounding.mode -> R -> double := round_logic 53 1024 (refl_equal true) (refl_equal true).
+
+Definition value: double -> R := value 53 1024.
+
+Definition exact: double -> R := exact 53 1024.
+
+Definition model: double -> R := model 53 1024.
+
+Definition round_error(x:double): R := (Rabs ((value x) - (exact x))%R).
+
+Definition total_error(x:double): R := (Rabs ((value x) - (model x))%R).
+
+Definition no_overflow(m:floating_point.Rounding.mode) (x:R): Prop :=
+  ((Rabs (round m
+  x)) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Bounded_real_no_overflow : forall (m:floating_point.Rounding.mode)
+  (x:R),
+  ((Rabs x) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R ->
+  (no_overflow m x).
+(* YOU MAY EDIT THE PROOF BELOW *)
+exact (Bounded_real_no_overflow 53 1024 (refl_equal true) (refl_equal true)).
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_monotonic : forall (m:floating_point.Rounding.mode) (x:R) (y:R),
+  (x <= y)%R -> ((round m x) <= (round m y))%R.
+(* YOU MAY EDIT THE PROOF BELOW *)
+now apply Round_monotonic.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_idempotent : forall (m1:floating_point.Rounding.mode)
+  (m2:floating_point.Rounding.mode) (x:R), ((round m1 (round m2
+  x)) = (round m2 x)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+now apply Round_idempotent.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_value : forall (m:floating_point.Rounding.mode) (x:double),
+  ((round m (value x)) = (value x)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+now apply Round_value.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Bounded_value : forall (x:double),
+  ((Rabs (value x)) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R.
+(* YOU MAY EDIT THE PROOF BELOW *)
+now apply Bounded_value.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Exact_rounding_for_integers : forall (m:floating_point.Rounding.mode)
+  (i:Z), (((-9007199254740992%Z)%Z <= i)%Z /\ (i <= 9007199254740992%Z)%Z) ->
+  ((round m (IZR i)) = (IZR i)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros m i Hi.
+now apply Exact_rounding_for_integers.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_down_le : forall (x:R), ((round floating_point.Rounding.Down
+  x) <= x)%R.
+(* YOU MAY EDIT THE PROOF BELOW *)
+now apply Round_down_le.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_up_ge : forall (x:R), (x <= (round floating_point.Rounding.Up
+  x))%R.
+(* YOU MAY EDIT THE PROOF BELOW *)
+now apply Round_up_ge.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_down_neg : forall (x:R), ((round floating_point.Rounding.Down
+  (-x)%R) = (-(round floating_point.Rounding.Up x))%R).
+(* YOU MAY EDIT THE PROOF BELOW *)
+now apply Round_down_neg.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_up_neg : forall (x:R), ((round floating_point.Rounding.Up
+  (-x)%R) = (-(round floating_point.Rounding.Down x))%R).
+(* YOU MAY EDIT THE PROOF BELOW *)
+now apply Round_up_neg.
+Qed.
+(* DO NOT EDIT BELOW *)

lib/coq/floating_point/GenFloat.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require Import Rbasic_fun.
+Require int.Int.
+Require real.Real.
+Require real.Abs.
+Require real.FromInt.
+Require floating_point.Rounding.
+
+Require Import Fcore.
+Require Import Fappli_IEEE.
+Require Import int.Abs.
+
+Section GenFloat.
+Global Coercion B2R_coercion prec emax := @B2R prec emax.
+
+Variable prec emax : Z.
+Hypothesis Hprec : Zlt_bool 0 prec = true.
+Hypothesis Hemax : Zlt_bool prec emax = true.
+Let emin := (3 - emax - prec)%Z.
+Let fexp := FLT_exp emin prec.
+Lemma Hprec': (0 < prec)%Z. revert Hprec. now case Zlt_bool_spec. Qed.
+Lemma Hemax': (prec < emax)%Z. revert Hemax. now case Zlt_bool_spec. Qed.
+Let binary_round_correct := binary_round_sign_shl_correct prec emax Hprec' Hemax'.
+
+
+Record t : Set := mk_fp {
+  binary : binary_float prec emax;
+  value := (binary : R);
+  exact : R;
+  model : R
+}.
+
+
+Record t_strict: Set := mk_fp_strict {
+  datum :> t;
+  finite : is_finite prec emax (binary datum) = true
+}.
+
+Import Rounding.
+Definition rnd_of_mode (m:mode) :=
+  match m with
+  |  NearestTiesToEven => mode_NE
+  |  ToZero            => mode_ZR
+  |  Up                => mode_UP
+  |  Down              => mode_DN
+  |  NearestTiesToAway => mode_NA
+  end.
+
+
+
+Definition r_to_fp rnd x : binary_float prec emax :=
+  let r := round radix2 fexp (round_mode rnd) x in
+  let m := Ztrunc (scaled_mantissa radix2 fexp r) in
+  let e := canonic_exponent radix2 fexp r in
+  match m with
+  | Z0 => B754_zero prec emax false
+  | Zpos m => FF2B _ _ _ (proj1 (binary_round_correct rnd false m e))
+  | Zneg m => FF2B _ _ _ (proj1 (binary_round_correct rnd true m e))
+  end.
+
+Lemma is_finite_FF2B :
+  forall f H,
+  is_finite prec emax (FF2B prec emax f H) =
+    match f with
+    | F754_finite _ _ _ => true
+    | F754_zero _ => true
+    | _ => false
+    end.
+Proof.
+now intros [| | |].
+Qed.
+
+Theorem r_to_fp_correct :
+  forall rnd x,
+  let r := round radix2 fexp (round_mode rnd) x in
+  (Rabs r < bpow radix2 emax)%R ->
+  is_finite prec emax (r_to_fp rnd x) = true /\
+  r_to_fp rnd x = r :>R.
+Proof.
+intros rnd x r Bx.
+unfold r_to_fp. fold r.
+assert (Gx: generic_format radix2 fexp r).
+apply generic_format_round.
+apply FLT_exp_correct.
+exact Hprec'.
+assert (Hr: Z2R (Ztrunc (scaled_mantissa radix2 fexp r)) = scaled_mantissa radix2 fexp r).
+apply sym_eq.
+now apply scaled_mantissa_generic.
+revert Hr.
+case_eq (Ztrunc (scaled_mantissa radix2 fexp r)).
+(* *)
+intros _ Hx.
+repeat split.
+apply Rmult_eq_reg_r with (bpow radix2 (- canonic_exponent radix2 fexp r)).
+now rewrite Rmult_0_l.
+apply Rgt_not_eq.
+apply bpow_gt_0.
+(* *)
+intros p Hp Hx.
+case binary_round_correct ; intros Hv.
+unfold F2R, Fnum, Fexp, cond_Zopp.
+rewrite Hx, scaled_mantissa_bpow.
+rewrite round_generic with (1 := Gx).
+rewrite Rlt_bool_true with (1 := Bx).
+intros H.
+split.
+rewrite is_finite_FF2B.
+revert H.
+assert (0 <> r)%R.
+intros H.
+rewrite <- H, scaled_mantissa_0 in Hx.
+now apply (Z2R_neq 0 (Zpos p)).
+now case binary_round_sign_shl.
+now rewrite B2R_FF2B.
+(* *)
+intros p Hp Hx.
+case binary_round_correct ; intros Hv.
+unfold F2R, Fnum, Fexp, cond_Zopp, Zopp.
+rewrite Hx, scaled_mantissa_bpow.
+rewrite round_generic with (1 := Gx).
+rewrite Rlt_bool_true with (1 := Bx).
+intros H.
+split.
+rewrite is_finite_FF2B.
+revert H.
+assert (0 <> r)%R.
+intros H.
+rewrite <- H, scaled_mantissa_0 in Hx.
+now apply (Z2R_neq 0 (Zneg p)).
+now case binary_round_sign_shl.
+now rewrite B2R_FF2B.
+Qed.
+
+Theorem r_to_fp_format :
+  forall rnd x,
+  FLT_format radix2 emin prec x ->
+  (Rabs x < bpow radix2 emax)%R ->
+  r_to_fp rnd x = x :>R.
+Proof.
+intros rnd x Fx Bx.
+assert (Gx: generic_format radix2 fexp x).
+apply -> FLT_format_generic.
+apply Fx.
+exact Hprec'.
+pattern x at 2 ; rewrite <- round_generic with (rnd := round_mode rnd) (1 := Gx).
+refine (proj2 (r_to_fp_correct _ _ _)).
+now rewrite round_generic with (1 := Gx).
+Qed.
+
+
+
+Definition r_to_fp_aux (m:mode) (r r1 r2:R) :=
+  mk_fp (r_to_fp (rnd_of_mode m) r) r1 r2.
+
+
+Definition round: floating_point.Rounding.mode -> R -> R :=
+   (fun m => round radix2 fexp (round_mode (rnd_of_mode m))).
+
+Definition round_logic: floating_point.Rounding.mode -> R -> t :=
+   fun m r => r_to_fp_aux m r r r.
+
+
+Definition round_error(x:t): R := (Rabs ((value x) - (exact x))%R).
+
+Definition total_error(x:t): R := (Rabs ((value x) - (model x))%R).
+
+Definition max: R := F2R (Float radix2 (Zpower 2 prec -1) (emax-prec)).
+
+
+Definition no_overflow(m:floating_point.Rounding.mode) (x:R): Prop :=
+  ((Rabs (round m x)) <= max)%R.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Bounded_real_no_overflow : forall (m:floating_point.Rounding.mode)
+  (x:R), ((Rabs x) <= max)%R -> (no_overflow m x).
+(* YOU MAY EDIT THE PROOF BELOW *)
+Proof.
+intros m x Hx.
+apply Rabs_le.
+assert (generic_format radix2 fexp max).
+apply generic_format_canonic_exponent.
+unfold canonic_exponent.
+rewrite ln_beta_F2R.
+rewrite (ln_beta_unique _ _ prec).
+ring_simplify (prec + (emax - prec))%Z.
+unfold fexp, FLT_exp.
+rewrite Zmax_l.
+apply Zle_refl.
+unfold emin.
+generalize Hprec' Hemax' ; clear; omega.
+rewrite <- Z2R_abs, Zabs_eq, <- 2!Z2R_Zpower.
+split.
+apply Z2R_le.
+apply Zlt_succ_le.
+change (2 ^ prec - 1)%Z with (Zpred (2^prec))%Z.
+rewrite <- Zsucc_pred.
+apply lt_Z2R.
+change 2%Z with (radix_val radix2).
+rewrite 2!Z2R_Zpower.
+apply bpow_lt.
+apply Zlt_pred.
+apply Zlt_le_weak.
+exact Hprec'.
+generalize Hprec' ; clear ; omega.
+apply Z2R_lt.
+apply Zlt_pred.
+apply Zlt_le_weak.
+exact Hprec'.
+generalize Hprec' ; clear ; omega.
+apply Zlt_succ_le.
+change (2 ^ prec - 1)%Z with (Zpred (2^prec))%Z.
+rewrite <- Zsucc_pred.
+change 2%Z with (radix_val radix2).
+apply Zpower_gt_0.
+apply Zlt_le_weak.
+exact Hprec'.
+apply Zgt_not_eq.
+cut (2 <= 2^prec)%Z.
+clear ; omega.
+change (radix2 ^ 1 <= radix2 ^ prec)%Z.
+apply le_Z2R.
+rewrite 2!Z2R_Zpower.
+apply bpow_le.
+generalize Hprec' ; clear ; omega.
+apply Zlt_le_weak.
+exact Hprec'.
+easy.
+generalize (Rabs_le_inv _ _ Hx).
+split.
+erewrite <- round_generic with (x := Ropp max).
+apply round_monotone with (2 := proj1 H0).
+apply FLT_exp_correct; exact Hprec'.
+now apply generic_format_opp.
+rewrite <- round_generic with (rnd := round_mode (rnd_of_mode m)) (1 := H).
+apply round_monotone with (2 := proj2 H0).
+apply FLT_exp_correct; exact Hprec'.
+Qed.
+
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_monotonic : forall (m:floating_point.Rounding.mode) (x:R) (y:R),
+  (x <= y)%R -> ((round m x) <= (round m y))%R.
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros m x y Hxy.
+apply round_monotone with (2 := Hxy).
+apply FLT_exp_correct; exact Hprec'.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_idempotent : forall (m1:floating_point.Rounding.mode)
+  (m2:floating_point.Rounding.mode) (x:R), ((round m1 (round m2
+  x)) = (round m2 x)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros m1 m2 x.
+apply round_generic.
+apply generic_format_round.
+apply FLT_exp_correct; exact Hprec'.
+Qed.
+
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_value : forall (m:floating_point.Rounding.mode) (x:t), ((round m
+  (value x)) = (value x)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+Proof.
+intros m x.
+apply round_generic.
+apply generic_format_B2R.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+(* This is not an axiom: this is proved in flocq 2.0 *)
+Axiom le_pred_lt_bis: forall (beta : radix) (fexp : Z -> Z),
+       valid_exp fexp ->
+       forall x y : R,
+       F beta fexp x ->
+       F beta fexp y -> (0 < y)%R -> (x < y)%R -> (x <= pred beta fexp y)%R.
+
+
+Lemma Bounded_value : forall (x:t), ((Rabs (value x)) <= max)%R.
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x.
+replace max with (pred radix2 fexp (bpow radix2 emax)).
+apply le_pred_lt_bis.
+apply FLT_exp_correct; exact Hprec'.
+apply generic_format_abs.
+apply generic_format_B2R.
+apply generic_format_bpow.
+unfold fexp, FLT_exp, emin.
+clear ; zify ; generalize Hprec' Hemax' ; omega.
+apply bpow_gt_0.
+apply B2R_lt_emax.
+unfold pred.
+rewrite ln_beta_bpow.
+ring_simplify (emax+1-1)%Z.
+rewrite Req_bool_true.
+2: easy.
+unfold fexp, FLT_exp, emin.
+rewrite Zmax_l.
+unfold max, F2R; simpl.
+pattern emax at 1; replace emax with (prec+(emax-prec))%Z by ring.
+rewrite bpow_plus.
+change 2%Z with (radix_val radix2).
+rewrite Z2R_minus, Z2R_Zpower.
+simpl; ring.
+apply Zlt_le_weak.
+exact Hprec'.
+clear; generalize Hprec' Hemax' ; omega.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+Definition max_representable_integer: Z := Zpower 2 prec.
+
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Exact_rounding_for_integers : forall (m:floating_point.Rounding.mode)
+  (i:Z), (((-max_representable_integer)%Z <= i)%Z /\
+  (i <= max_representable_integer)%Z) -> ((round m (IZR i)) = (IZR i)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+Proof.
+intros m z Hz.
+apply round_generic.
+assert (Zabs z <= max_representable_integer)%Z.
+apply Abs_le with (1:=Hz).
+destruct (Zle_lt_or_eq _ _ H) as [Bz|Bz] ; clear H Hz.
+apply FLT_format_generic.
+exact Hprec'.
+exists (Float radix2 z 0).
+unfold F2R. simpl.
+split.
+rewrite Z2R_IZR.
+now rewrite Rmult_1_r.
+split. easy.
+clear;unfold emin; generalize Hprec' Hemax'; omega.
+unfold max_representable_integer in Bz.
+change 2%Z with (radix_val radix2) in Bz.
+destruct z as [|z|z] ; unfold Zabs in Bz.
+apply generic_format_0.
+rewrite Bz.
+rewrite <- Z2R_IZR, Z2R_Zpower.
+apply generic_format_bpow.
+unfold fexp, FLT_exp, emin.
+clear; generalize Hprec' Hemax'; zify.
+omega.
+apply Zlt_le_weak.
+apply Hprec'.
+change (Zneg z) with (Zopp (Zpos z)).
+rewrite Bz, <- Z2R_IZR, Z2R_opp.
+rewrite Z2R_Zpower.
+apply generic_format_opp.
+apply generic_format_bpow.
+unfold fexp, FLT_exp, emin.
+clear; generalize Hprec' Hemax'; zify.
+omega.
+apply Zlt_le_weak.
+apply Hprec'.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_down_le : forall (x:R), ((round floating_point.Rounding.Down
+  x) <= x)%R.
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x.
+eapply round_DN_pt.
+apply FLT_exp_correct; exact Hprec'.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_up_ge : forall (x:R), (x <= (round floating_point.Rounding.Up
+  x))%R.
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x.
+eapply round_UP_pt.
+apply FLT_exp_correct; exact Hprec'.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_down_neg : forall (x:R), ((round floating_point.Rounding.Down
+  (-x)%R) = (-(round floating_point.Rounding.Up x))%R).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x.
+apply round_opp.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Lemma Round_up_neg : forall (x:R), ((round floating_point.Rounding.Up
+  (-x)%R) = (-(round floating_point.Rounding.Down x))%R).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros x.
+pattern x at 2 ; rewrite <- Ropp_involutive.
+rewrite Round_down_neg.
+now rewrite Ropp_involutive.
+Qed.
+
+End GenFloat.
+(* DO NOT EDIT BELOW *)

lib/coq/floating_point/Rounding.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Inductive mode  :=
+  | NearestTiesToEven : mode
+  | ToZero : mode