Update Coq realizations.

lib/coq/bool/Bool.v

@@ -5,7 +5,7 @@ Require BuiltIn.
 
 (* Why3 goal *)
 Lemma andb_def : forall (x:bool) (y:bool),
-  ((andb x y) = match x with
+  ((Init.Datatypes.andb x y) = match x with
   | true => y
   | false => false
   end).
@@ -16,7 +16,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma orb_def : forall (x:bool) (y:bool),
-  ((orb x y) = match x with
+  ((Init.Datatypes.orb x y) = match x with
   | false => y
   | true => true
   end).
@@ -26,7 +26,8 @@ apply refl_equal.
 Qed.
 
 (* Why3 goal *)
-Lemma xorb_def : forall (x:bool) (y:bool), ((xorb x y) = match (x,
+Lemma xorb_def : forall (x:bool) (y:bool),
+  ((Init.Datatypes.xorb x y) = match (x,
   y) with
   | (true, false) => true
   | (false, true) => true
@@ -39,7 +40,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma notb_def : forall (x:bool),
-  ((negb x) = match x with
+  ((Init.Datatypes.negb x) = match x with
   | false => true
   | true => false
   end).
@@ -49,7 +50,8 @@ apply refl_equal.
 Qed.
 
 (* Why3 goal *)
-Lemma implb_def : forall (x:bool) (y:bool), ((implb x y) = match (x,
+Lemma implb_def : forall (x:bool) (y:bool),
+  ((Init.Datatypes.implb x y) = match (x,
   y) with
   | (true, false) => false
   | (_, _) => true

lib/coq/floating_point/Double.v

 (* This file is generated by Why3's Coq-realize driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import Rbasic_fun.
+Require Reals.Rbasic_fun.
 Require BuiltIn.
 Require int.Int.
 Require real.Real.
@@ -34,21 +34,21 @@ Defined.
 
 (* Why3 assumption *)
 Definition round_error (x:floating_point.DoubleFormat.double): R :=
-  (Rabs ((value x) - (exact x))%R).
+  (Reals.Rbasic_fun.Rabs ((value x) - (exact x))%R).
 
 (* Why3 assumption *)
 Definition total_error (x:floating_point.DoubleFormat.double): R :=
-  (Rabs ((value x) - (model x))%R).
+  (Reals.Rbasic_fun.Rabs ((value x) - (model x))%R).
 
 (* Why3 assumption *)
 Definition no_overflow (m:floating_point.Rounding.mode) (x:R): Prop :=
-  ((Rabs (round m
+  ((Reals.Rbasic_fun.Rabs (round m
   x)) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R.
 
 (* Why3 goal *)
 Lemma Bounded_real_no_overflow : forall (m:floating_point.Rounding.mode)
   (x:R),
-  ((Rabs x) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R ->
+  ((Reals.Rbasic_fun.Rabs x) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R ->
   (no_overflow m x).
 exact (Bounded_real_no_overflow 53 1024 (refl_equal true) (refl_equal true)).
 Qed.
@@ -74,14 +74,14 @@ Qed.
 
 (* Why3 goal *)
 Lemma Bounded_value : forall (x:floating_point.DoubleFormat.double),
-  ((Rabs (value x)) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R.
+  ((Reals.Rbasic_fun.Rabs (value x)) <= (9007199254740991 * 19958403095347198116563727130368385660674512604354575415025472424372118918689640657849579654926357010893424468441924952439724379883935936607391717982848314203200056729510856765175377214443629871826533567445439239933308104551208703888888552684480441575071209068757560416423584952303440099278848)%R)%R.
 now apply Bounded_value.
 Qed.
 
 (* Why3 goal *)
 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)).
+  ((round m (Reals.Raxioms.IZR i)) = (Reals.Raxioms.IZR i)).
 intros m i Hi.
 now apply Exact_rounding_for_integers.
 Qed.
@@ -160,9 +160,9 @@ Definition div_post (m:floating_point.Rounding.mode)
   (x:floating_point.DoubleFormat.double)
   (y:floating_point.DoubleFormat.double)
   (res:floating_point.DoubleFormat.double): Prop := ((value res) = (round m
-  (Rdiv (value x) (value y))%R)) /\
-  (((exact res) = (Rdiv (exact x) (exact y))%R) /\
-  ((model res) = (Rdiv (model x) (model y))%R)).
+  ((value x) / (value y))%R)) /\
+  (((exact res) = ((exact x) / (exact y))%R) /\
+  ((model res) = ((model x) / (model y))%R)).
 
 (* Why3 assumption *)
 Definition neg_post (x:floating_point.DoubleFormat.double)

lib/coq/floating_point/Single.v

 (* This file is generated by Why3's Coq-realize driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import Rbasic_fun.
+Require Reals.Rbasic_fun.
 Require BuiltIn.
 Require int.Int.
 Require real.Real.
@@ -34,15 +34,16 @@ Defined.
 
 (* Why3 assumption *)
 Definition round_error (x:floating_point.SingleFormat.single): R :=
-  (Rabs ((value x) - (exact x))%R).
+  (Reals.Rbasic_fun.Rabs ((value x) - (exact x))%R).
 
 (* Why3 assumption *)
 Definition total_error (x:floating_point.SingleFormat.single): R :=
-  (Rabs ((value x) - (model x))%R).
+  (Reals.Rbasic_fun.Rabs ((value x) - (model x))%R).
 
 (* Why3 assumption *)
 Definition no_overflow (m:floating_point.Rounding.mode) (x:R): Prop :=
-  ((Rabs (round m x)) <= (33554430 * 10141204801825835211973625643008)%R)%R.
+  ((Reals.Rbasic_fun.Rabs (round m
+  x)) <= (33554430 * 10141204801825835211973625643008)%R)%R.
 
 Lemma max_single_eq: (33554430 * 10141204801825835211973625643008 = max 24 128)%R.
 unfold max, Fcore_defs.F2R; simpl.
@@ -51,7 +52,8 @@ Qed.
 
 (* Why3 goal *)
 Lemma Bounded_real_no_overflow : forall (m:floating_point.Rounding.mode)
-  (x:R), ((Rabs x) <= (33554430 * 10141204801825835211973625643008)%R)%R ->
+  (x:R),
+  ((Reals.Rbasic_fun.Rabs x) <= (33554430 * 10141204801825835211973625643008)%R)%R ->
   (no_overflow m x).
 intros m x Hx.
 unfold no_overflow.
@@ -81,7 +83,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma Bounded_value : forall (x:floating_point.SingleFormat.single),
-  ((Rabs (value x)) <= (33554430 * 10141204801825835211973625643008)%R)%R.
+  ((Reals.Rbasic_fun.Rabs (value x)) <= (33554430 * 10141204801825835211973625643008)%R)%R.
 rewrite max_single_eq.
 now apply Bounded_value.
 Qed.
@@ -89,7 +91,7 @@ Qed.
 (* Why3 goal *)
 Lemma Exact_rounding_for_integers : forall (m:floating_point.Rounding.mode)
   (i:Z), (((-16777216%Z)%Z <= i)%Z /\ (i <= 16777216%Z)%Z) -> ((round m
-  (IZR i)) = (IZR i)).
+  (Reals.Raxioms.IZR i)) = (Reals.Raxioms.IZR i)).
 intros m i Hi.
 now apply Exact_rounding_for_integers.
 Qed.
@@ -171,9 +173,9 @@ Definition div_post (m:floating_point.Rounding.mode)
   (x:floating_point.SingleFormat.single)
   (y:floating_point.SingleFormat.single)
   (res:floating_point.SingleFormat.single): Prop := ((value res) = (round m
-  (Rdiv (value x) (value y))%R)) /\
-  (((exact res) = (Rdiv (exact x) (exact y))%R) /\
-  ((model res) = (Rdiv (model x) (model y))%R)).
+  ((value x) / (value y))%R)) /\
+  (((exact res) = ((exact x) / (exact y))%R) /\
+  ((model res) = ((model x) / (model y))%R)).
 
 (* Why3 assumption *)
 Definition neg_post (x:floating_point.SingleFormat.single)

lib/coq/int/Abs.v

@@ -5,11 +5,12 @@ Require BuiltIn.
 Require int.Int.
 
 (* Why3 comment *)
-(* abs is replaced with (Zabs x) by the coq driver *)
+(* abs is replaced with (ZArith.BinInt.Z.abs x) by the coq driver *)
 
 (* Why3 goal *)
-Lemma abs_def : forall (x:Z), ((0%Z <= x)%Z -> ((Zabs x) = x)) /\
-  ((~ (0%Z <= x)%Z) -> ((Zabs x) = (-x)%Z)).
+Lemma abs_def : forall (x:Z), ((0%Z <= x)%Z ->
+  ((ZArith.BinInt.Z.abs x) = x)) /\ ((~ (0%Z <= x)%Z) ->
+  ((ZArith.BinInt.Z.abs x) = (-x)%Z)).
 intros x.
 split ; intros H.
 now apply Zabs_eq.
@@ -21,15 +22,15 @@ now apply Zgt_lt.
 Qed.
 
 (* Why3 goal *)
-Lemma Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
-  (x <= y)%Z).
+Lemma Abs_le : forall (x:Z) (y:Z), ((ZArith.BinInt.Z.abs x) <= y)%Z <->
+  (((-y)%Z <= x)%Z /\ (x <= y)%Z).
 intros x y.
 zify.
 omega.
 Qed.
 
 (* Why3 goal *)
-Lemma Abs_pos : forall (x:Z), (0%Z <= (Zabs x))%Z.
+Lemma Abs_pos : forall (x:Z), (0%Z <= (ZArith.BinInt.Z.abs x))%Z.
 exact Zabs_pos.
 Qed.
 

lib/coq/int/EuclideanDivision.v

@@ -49,7 +49,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma Mod_bound : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> ((0%Z <= (mod1 x
-  y))%Z /\ ((mod1 x y) < (Zabs y))%Z).
+  y))%Z /\ ((mod1 x y) < (ZArith.BinInt.Z.abs y))%Z).
 intros x y Zy.
 zify.
 assert (H1 := Z_mod_neg x y).

lib/coq/int/MinMax.v

@@ -5,21 +5,22 @@ Require BuiltIn.
 Require int.Int.
 
 (* Why3 comment *)
-(* min is replaced with (Zmin x x1) by the coq driver *)
+(* min is replaced with (ZArith.BinInt.Z.min x x1) by the coq driver *)
 
 (* Why3 comment *)
-(* max is replaced with (Zmax x x1) by the coq driver *)
+(* max is replaced with (ZArith.BinInt.Z.max x x1) by the coq driver *)
 
 (* Why3 goal *)
-Lemma Max_is_ge : forall (x:Z) (y:Z), (x <= (Zmax x y))%Z /\
-  (y <= (Zmax x y))%Z.
+Lemma Max_is_ge : forall (x:Z) (y:Z), (x <= (ZArith.BinInt.Z.max x y))%Z /\
+  (y <= (ZArith.BinInt.Z.max x y))%Z.
 split.
 apply Zle_max_l.
 apply Zle_max_r.
 Qed.
 
 (* Why3 goal *)
-Lemma Max_is_some : forall (x:Z) (y:Z), ((Zmax x y) = x) \/ ((Zmax x y) = y).
+Lemma Max_is_some : forall (x:Z) (y:Z), ((ZArith.BinInt.Z.max x y) = x) \/
+  ((ZArith.BinInt.Z.max x y) = y).
 intros x y.
 unfold Zmax.
 case Zcompare.
@@ -29,15 +30,16 @@ now left.
 Qed.
 
 (* Why3 goal *)
-Lemma Min_is_le : forall (x:Z) (y:Z), ((Zmin x y) <= x)%Z /\
-  ((Zmin x y) <= y)%Z.
+Lemma Min_is_le : forall (x:Z) (y:Z), ((ZArith.BinInt.Z.min x y) <= x)%Z /\
+  ((ZArith.BinInt.Z.min x y) <= y)%Z.
 split.
 apply Zle_min_l.
 apply Zle_min_r.
 Qed.
 
 (* Why3 goal *)
-Lemma Min_is_some : forall (x:Z) (y:Z), ((Zmin x y) = x) \/ ((Zmin x y) = y).
+Lemma Min_is_some : forall (x:Z) (y:Z), ((ZArith.BinInt.Z.min x y) = x) \/
+  ((ZArith.BinInt.Z.min x y) = y).
 intros x y.
 unfold Zmin.
 case Zcompare.
@@ -47,33 +49,39 @@ now right.
 Qed.
 
 (* Why3 goal *)
-Lemma Max_x : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmax x y) = x).
+Lemma Max_x : forall (x:Z) (y:Z), (y <= x)%Z ->
+  ((ZArith.BinInt.Z.max x y) = x).
 exact Zmax_l.
 Qed.
 
 (* Why3 goal *)
-Lemma Max_y : forall (x:Z) (y:Z), (x <= y)%Z -> ((Zmax x y) = y).
+Lemma Max_y : forall (x:Z) (y:Z), (x <= y)%Z ->
+  ((ZArith.BinInt.Z.max x y) = y).
 exact Zmax_r.
 Qed.
 
 (* Why3 goal *)
-Lemma Min_x : forall (x:Z) (y:Z), (x <= y)%Z -> ((Zmin x y) = x).
+Lemma Min_x : forall (x:Z) (y:Z), (x <= y)%Z ->
+  ((ZArith.BinInt.Z.min x y) = x).
 exact Zmin_l.
 Qed.
 
 (* Why3 goal *)
-Lemma Min_y : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmin x y) = y).
+Lemma Min_y : forall (x:Z) (y:Z), (y <= x)%Z ->
+  ((ZArith.BinInt.Z.min x y) = y).
 exact Zmin_r.
 Qed.
 
 (* Why3 goal *)
-Lemma Max_sym : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmax x y) = (Zmax y x)).
+Lemma Max_sym : forall (x:Z) (y:Z), (y <= x)%Z ->
+  ((ZArith.BinInt.Z.max x y) = (ZArith.BinInt.Z.max y x)).
 intros x y _.
 apply Zmax_comm.
 Qed.
 
 (* Why3 goal *)
-Lemma Min_sym : forall (x:Z) (y:Z), (y <= x)%Z -> ((Zmin x y) = (Zmin y x)).
+Lemma Min_sym : forall (x:Z) (y:Z), (y <= x)%Z ->
+  ((ZArith.BinInt.Z.min x y) = (ZArith.BinInt.Z.min y x)).
 intros x y _.
 apply Zmin_comm.
 Qed.

lib/coq/list/Append.v

@@ -11,8 +11,8 @@ Require list.Mem.
 Lemma infix_plpl_def  {a:Type} {a_WT:WhyType a}: forall (l1:(list a))
   (l2:(list a)),
   ((List.app l1 l2) = match l1 with
-  | nil => l2
-  | (cons x1 r1) => (cons x1 (List.app r1 l2))
+  | Init.Datatypes.nil => l2
+  | (Init.Datatypes.cons x1 r1) => (Init.Datatypes.cons x1 (List.app r1 l2))
   end).
 Proof.
 now intros [|h1 q1] l2.
@@ -31,7 +31,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma Append_l_nil : forall {a:Type} {a_WT:WhyType a}, forall (l:(list a)),
-  ((List.app l nil) = l).
+  ((List.app l Init.Datatypes.nil) = l).
 Proof.
 intros a a_WT l.
 apply app_nil_r.
@@ -75,7 +75,7 @@ Qed.
 (* Why3 goal *)
 Lemma mem_decomp : forall {a:Type} {a_WT:WhyType a}, forall (x:a)
   (l:(list a)), (list.Mem.mem x l) -> exists l1:(list a), exists l2:(list a),
-  (l = (List.app l1 (cons x l2))).
+  (l = (List.app l1 (Init.Datatypes.cons x l2))).
 Proof.
 intros a a_WT x l h1.
 apply in_split.

lib/coq/list/HdTl.v

@@ -8,14 +8,14 @@ Require option.Option.
 (* Why3 assumption *)
 Definition hd {a:Type} {a_WT:WhyType a} (l:(list a)): (option a) :=
   match l with
-  | nil => None
-  | (cons h _) => (Some h)
+  | Init.Datatypes.nil => Init.Datatypes.None
+  | (Init.Datatypes.cons h _) => (Init.Datatypes.Some h)
   end.
 
 (* Why3 assumption *)
 Definition tl {a:Type} {a_WT:WhyType a} (l:(list a)): (option (list a)) :=
   match l with
-  | nil => None
-  | (cons _ t) => (Some t)
+  | Init.Datatypes.nil => Init.Datatypes.None
+  | (Init.Datatypes.cons _ t) => (Init.Datatypes.Some t)
   end.
 

lib/coq/list/Length.v

@@ -8,8 +8,8 @@ Require list.List.
 (* Why3 assumption *)
 Fixpoint length {a:Type} {a_WT:WhyType a} (l:(list a)) {struct l}: Z :=
   match l with
-  | nil => 0%Z
-  | (cons _ r) => (1%Z + (length r))%Z
+  | Init.Datatypes.nil => 0%Z
+  | (Init.Datatypes.cons _ r) => (1%Z + (length r))%Z
   end.
 
 Lemma length_std :
@@ -34,7 +34,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma Length_nil : forall {a:Type} {a_WT:WhyType a}, forall (l:(list a)),
-  ((length l) = 0%Z) <-> (l = nil).
+  ((length l) = 0%Z) <-> (l = Init.Datatypes.nil).
 Proof.
 intros a a_WT [|h t] ; split ; try easy.
 unfold length. fold length.

lib/coq/list/Mem.v

@@ -7,8 +7,8 @@ Require list.List.
 (* Why3 assumption *)
 Fixpoint mem {a:Type} {a_WT:WhyType a} (x:a) (l:(list a)) {struct l}: Prop :=
   match l with
-  | nil => False
-  | (cons y r) => (x = y) \/ (mem x r)
+  | Init.Datatypes.nil => False
+  | (Init.Datatypes.cons y r) => (x = y) \/ (mem x r)
   end.
 
 Lemma mem_std :

lib/coq/list/Nth.v

@@ -16,9 +16,10 @@ Defined.
 (* Why3 goal *)
 Lemma nth_def : forall {a:Type} {a_WT:WhyType a}, forall (n:Z) (l:(list a)),
   match l with
-  | nil => ((nth n l) = None)
-  | (cons x r) => ((n = 0%Z) -> ((nth n l) = (Some x))) /\ ((~ (n = 0%Z)) ->
-      ((nth n l) = (nth (n - 1%Z)%Z r)))
+  | Init.Datatypes.nil => ((nth n l) = Init.Datatypes.None)
+  | (Init.Datatypes.cons x r) => ((n = 0%Z) -> ((nth n
+      l) = (Init.Datatypes.Some x))) /\ ((~ (n = 0%Z)) -> ((nth n
+      l) = (nth (n - 1%Z)%Z r)))
   end.
 Proof.
 intros a a_WT n l.

lib/coq/list/NthHdTl.v

@@ -10,9 +10,9 @@ Require list.HdTl.
 
 (* Why3 goal *)
 Lemma Nth_tl : forall {a:Type} {a_WT:WhyType a}, forall (l1:(list a))
-  (l2:(list a)), ((list.HdTl.tl l1) = (Some l2)) -> forall (i:Z),
-  (~ (i = (-1%Z)%Z)) -> ((list.Nth.nth i l2) = (list.Nth.nth (i + 1%Z)%Z
-  l1)).
+  (l2:(list a)), ((list.HdTl.tl l1) = (Init.Datatypes.Some l2)) ->
+  forall (i:Z), (~ (i = (-1%Z)%Z)) -> ((list.Nth.nth i
+  l2) = (list.Nth.nth (i + 1%Z)%Z l1)).
 Proof.
 intros a a_WT [|x1 l1] l2 h1 i h2.
 easy.

lib/coq/list/NthLength.v

@@ -10,7 +10,7 @@ Require option.Option.
 
 (* Why3 goal *)
 Lemma nth_none_1 : forall {a:Type} {a_WT:WhyType a}, forall (l:(list a))
-  (i:Z), (i < 0%Z)%Z -> ((list.Nth.nth i l) = None).
+  (i:Z), (i < 0%Z)%Z -> ((list.Nth.nth i l) = Init.Datatypes.None).
 Proof.
 intros a a_WT l.
 induction l as [|h q].
@@ -28,7 +28,8 @@ Qed.
 
 (* Why3 goal *)
 Lemma nth_none_2 : forall {a:Type} {a_WT:WhyType a}, forall (l:(list a))
-  (i:Z), ((list.Length.length l) <= i)%Z -> ((list.Nth.nth i l) = None).
+  (i:Z), ((list.Length.length l) <= i)%Z -> ((list.Nth.nth i
+  l) = Init.Datatypes.None).
 Proof.
 intros a a_WT l.
 induction l as [|h q].
@@ -51,7 +52,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma nth_none_3 : forall {a:Type} {a_WT:WhyType a}, forall (l:(list a))
-  (i:Z), ((list.Nth.nth i l) = None) -> ((i < 0%Z)%Z \/
+  (i:Z), ((list.Nth.nth i l) = Init.Datatypes.None) -> ((i < 0%Z)%Z \/
   ((list.Length.length l) <= i)%Z).
 Proof.
 intros a a_WT l.

lib/coq/list/Reverse.v

@@ -11,8 +11,9 @@ Require list.Append.
 (* Why3 goal *)
 Lemma reverse_def  {a:Type} {a_WT:WhyType a}: forall (l:(list a)),
   ((List.rev l) = match l with
-  | nil => nil
-  | (cons x r) => (List.app (List.rev r) (cons x nil))
+  | Init.Datatypes.nil => Init.Datatypes.nil
+  | (Init.Datatypes.cons x r) =>
+      (List.app (List.rev r) (Init.Datatypes.cons x Init.Datatypes.nil))
   end).
 Proof.
 now intros [|x l].
@@ -21,7 +22,7 @@ Qed.
 (* Why3 goal *)
 Lemma reverse_append : forall {a:Type} {a_WT:WhyType a}, forall (l1:(list a))
   (l2:(list a)) (x:a),
-  ((List.app (List.rev (cons x l1)) l2) = (List.app (List.rev l1) (cons x l2))).
+  ((List.app (List.rev (Init.Datatypes.cons x l1)) l2) = (List.app (List.rev l1) (Init.Datatypes.cons x l2))).
 Proof.
 intros a a_WT l1 l2 x.
 simpl.

lib/coq/number/Coprime.v

 (* This file is generated by Why3's Coq-realize driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import ZOdiv.
 Require BuiltIn.
 Require int.Int.
 Require int.Abs.

lib/coq/number/Divisibility.v

 (* This file is generated by Why3's Coq-realize driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import ZOdiv.
 Require BuiltIn.
 Require int.Int.
 Require int.Abs.
@@ -141,7 +140,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma divides_bounds : forall (a:Z) (b:Z), (divides a b) -> ((~ (b = 0%Z)) ->
-  ((Zabs a) <= (Zabs b))%Z).
+  ((ZArith.BinInt.Z.abs a) <= (ZArith.BinInt.Z.abs b))%Z).
 Proof.
 exact Zdivide_bounds.
 Qed.
@@ -176,7 +175,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma mod_divides_computer : forall (a:Z) (b:Z), (~ (b = 0%Z)) ->
-  (((ZOmod a b) = 0%Z) -> (divides b a)).
+  (((ZArith.BinInt.Z.rem a b) = 0%Z) -> (divides b a)).
 Proof.
 intros a b Zb H.
 exists (Z.quot a b).
@@ -186,7 +185,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma divides_mod_computer : forall (a:Z) (b:Z), (~ (b = 0%Z)) -> ((divides b
-  a) -> ((ZOmod a b) = 0%Z)).
+  a) -> ((ZArith.BinInt.Z.rem a b) = 0%Z)).
 Proof.
 intros a b Zb (q,H).
 rewrite H.

lib/coq/number/Gcd.v

 (* This file is generated by Why3's Coq-realize driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import ZOdiv.
 Require BuiltIn.
 Require int.Int.
 Require int.Abs.
@@ -115,7 +114,7 @@ Qed.
 
 (* Why3 goal *)
 Lemma Gcd_computer_mod : forall (a:Z) (b:Z), (~ (b = 0%Z)) -> ((gcd b
-  (ZOmod a b)) = (gcd a b)).
+  (ZArith.BinInt.Z.rem a b)) = (gcd a b)).
 Proof.
 intros a b _.
 rewrite (Zgcd_comm a b).

lib/coq/number/Prime.v

 (* This file is generated by Why3's Coq-realize driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import ZOdiv.
 Require BuiltIn.
 Require int.Int.
 Require int.Abs.

lib/coq/real/Abs.v

 (* This file is generated by Why3's Coq-realize driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import Rbasic_fun.
+Require Reals.Rbasic_fun.
 Require BuiltIn.
 Require real.Real.
 
 Import Rbasic_fun.
 
 (* Why3 comment *)
-(* abs is replaced with (Rabs x) by the coq driver *)
+(* abs is replaced with (Reals.Rbasic_fun.Rabs x) by the coq driver *)
 
 (* Why3 goal *)
-Lemma abs_def : forall (x:R), ((0%R <= x)%R -> ((Rabs x) = x)) /\
-  ((~ (0%R <= x)%R) -> ((Rabs x) = (-x)%R)).
+Lemma abs_def : forall (x:R), ((0%R <= x)%R ->
+  ((Reals.Rbasic_fun.Rabs x) = x)) /\ ((~ (0%R <= x)%R) ->
+  ((Reals.Rbasic_fun.Rabs x) = (-x)%R)).