Fix Coq proofs broken by the qualified-name changes.

examples/euler001/euler001_DivModHints_mod_div_unique_1.v

-(* This file is generated by Why3's Coq driver *)
+(* This file is generated by Why3's Coq 8.4 driver *)
 (* Beware! Only edit allowed sections below    *)
-Require Import ZArith.
-Require Import Rbase.
-Require Import ZOdiv.
-Axiom Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
-  (x <= y)%Z).
-
-(* YOU MAY EDIT THE CONTEXT BELOW *)
-
-(* DO NOT EDIT BELOW *)
+Require Import BuiltIn.
+Require BuiltIn.
+Require int.Int.
+Require int.Abs.
+Require int.ComputerDivision.
 
+(* Why3 goal *)
 Theorem mod_div_unique : forall (x:Z) (y:Z) (q:Z) (r:Z), ((0%Z <= x)%Z /\
-  ((0%Z <  y)%Z /\ ((x = ((q * y)%Z + r)%Z) /\ ((0%Z <= r)%Z /\
-  (r <  y)%Z)))) -> ((q = (ZOdiv x y)) /\ (r = (ZOmod x y))).
+  ((0%Z < y)%Z /\ ((x = ((q * y)%Z + r)%Z) /\ ((0%Z <= r)%Z /\
+  (r < y)%Z)))) -> ((q = (ZArith.BinInt.Z.quot x y)) /\
+  (r = (ZArith.BinInt.Z.rem x y))).
+(* Why3 intros x y q r (h1,(h2,(h3,(h4,h5)))). *)
 (* YOU MAY EDIT THE PROOF BELOW *)
 intros x y q r (H1,(H2,(H3,(H4,H5)))).
-apply ZOdiv_mod_unique_full.
+apply Zquot.Zquot_mod_unique_full.
 2: rewrite H3; ring.
 red.
 left.
 rewrite Zabs_eq; auto with zarith.
 Qed.
-(* DO NOT EDIT BELOW *)
-
 

examples/euler001/euler001_DivModHints_mod_succ_1_1.v

 (* This file is generated by Why3's Coq 8.4 driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import ZOdiv.
 Require BuiltIn.
 Require int.Int.
 Require int.Abs.
@@ -9,28 +8,29 @@ Require int.ComputerDivision.
 
 Axiom mod_div_unique : forall (x:Z) (y:Z) (q:Z) (r:Z), ((0%Z <= x)%Z /\
   ((0%Z < y)%Z /\ ((x = ((q * y)%Z + r)%Z) /\ ((0%Z <= r)%Z /\
-  (r < y)%Z)))) -> ((q = (ZOdiv x y)) /\ (r = (ZOmod x y))).
+  (r < y)%Z)))) -> ((q = (ZArith.BinInt.Z.quot x y)) /\
+  (r = (ZArith.BinInt.Z.rem x y))).
 
+Import Zquot.
 Open Scope Z_scope.
 
 (* Why3 goal *)
 Theorem mod_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
-  ((~ ((ZOmod (x + 1%Z)%Z y) = 0%Z)) ->
-  ((ZOmod (x + 1%Z)%Z y) = ((ZOmod x y) + 1%Z)%Z)).
+  ((~ ((ZArith.BinInt.Z.rem (x + 1%Z)%Z y) = 0%Z)) ->
+  ((ZArith.BinInt.Z.rem (x + 1%Z)%Z y) = ((ZArith.BinInt.Z.rem x y) + 1%Z)%Z)).
 (* Why3 intros x y (h1,h2) h3. *)
 intros x y (Hx,Hy) H.
 assert (h: y>0) by omega.
-generalize (ZO_div_mod_eq x y); intro h1.
-generalize (ZO_div_mod_eq (x+1) y); intro h2.
-assert (h3:x = y * (ZOdiv (x + 1) y) + (ZOmod (x + 1) y - 1)) by omega.
-generalize (mod_div_unique x y (ZOdiv (x + 1) y) (ZOmod (x + 1) y - 1)).
+generalize (Z_quot_rem_eq x y); intro h1.
+generalize (Z_quot_rem_eq (x+1) y); intro h2.
+assert (h3:x = y * (Z.quot (x + 1) y) + (Z.rem (x + 1) y - 1)) by omega.
+generalize (mod_div_unique x y (Z.quot (x + 1) y) (Z.rem (x + 1) y - 1)).
 intuition.
 destruct H1 ; auto with zarith.
 rewrite h3 at 1.
 ring.
-assert (0 <= ZOmod (x + 1) y < y).
-  apply Zquot.Zrem_lt_pos_pos; omega.
+assert (0 <= Z.rem (x + 1) y < y).
+  apply Zrem_lt_pos_pos; omega.
 omega.
 Qed.
 
-

examples/euler001/euler001_DivModHints_mod_succ_2_1.v

 (* This file is generated by Why3's Coq 8.4 driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import ZOdiv.
 Require BuiltIn.
 Require int.Int.
 Require int.Abs.
@@ -9,27 +8,30 @@ Require int.ComputerDivision.
 
 Axiom mod_div_unique : forall (x:Z) (y:Z) (q:Z) (r:Z), ((0%Z <= x)%Z /\
   ((0%Z < y)%Z /\ ((x = ((q * y)%Z + r)%Z) /\ ((0%Z <= r)%Z /\
-  (r < y)%Z)))) -> ((q = (ZOdiv x y)) /\ (r = (ZOmod x y))).
+  (r < y)%Z)))) -> ((q = (ZArith.BinInt.Z.quot x y)) /\
+  (r = (ZArith.BinInt.Z.rem x y))).
 
 Axiom mod_succ_1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
-  ((~ ((ZOmod (x + 1%Z)%Z y) = 0%Z)) ->
-  ((ZOmod (x + 1%Z)%Z y) = ((ZOmod x y) + 1%Z)%Z)).
+  ((~ ((ZArith.BinInt.Z.rem (x + 1%Z)%Z y) = 0%Z)) ->
+  ((ZArith.BinInt.Z.rem (x + 1%Z)%Z y) = ((ZArith.BinInt.Z.rem x y) + 1%Z)%Z)).
+
+Import Zquot.
 
 (* Why3 goal *)
 Theorem mod_succ_2 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
-  (((ZOmod (x + 1%Z)%Z y) = 0%Z) -> ((ZOmod x y) = (y - 1%Z)%Z)).
+  (((ZArith.BinInt.Z.rem (x + 1%Z)%Z y) = 0%Z) ->
+  ((ZArith.BinInt.Z.rem x y) = (y - 1%Z)%Z)).
 (* Why3 intros x y (h1,h2) h3. *)
 intros x y (Hx,Hy) H.
-generalize (ZO_div_mod_eq x y); intro h1.
-generalize (ZO_div_mod_eq (x+1) y); intro h2.
-assert (h3: (x = y * (ZOdiv (x + 1) y - 1) + (ZOmod (x + 1) y + y - 1))%Z).
+generalize (Z_quot_rem_eq x y); intro h1.
+generalize (Z_quot_rem_eq (x+1) y); intro h2.
+assert (h3: (x = y * (Z.quot (x + 1) y - 1) + (Z.rem (x + 1) y + y - 1))%Z).
   ring_simplify; omega.
 rewrite H in h3.
-generalize (mod_div_unique x y (ZOdiv (x + 1) y - 1) (0 + y - 1)).
+generalize (mod_div_unique x y (Z.quot (x + 1) y - 1) (0 + y - 1)).
 intuition.
 destruct H1; auto with zarith.
 rewrite h3 at 1.
 ring.
 Qed.
 
-

examples/gcd/gcd_WP_EuclideanAlgorithm_WP_parameter_gcd_1.v

 (* This file is generated by Why3's Coq 8.4 driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import ZOdiv.
 Require BuiltIn.
 Require int.Int.
 Require int.Abs.
@@ -14,18 +13,23 @@ Require number.Gcd.
 (* Why3 assumption *)
 Definition unit := unit.
 
+Axiom qtmark : Type.
+Parameter qtmark_WhyType : WhyType qtmark.
+Existing Instance qtmark_WhyType.
+
 (* Why3 goal *)
 Theorem WP_parameter_euclid : forall (u:Z) (v:Z), ((0%Z <= u)%Z /\
-  (0%Z <= v)%Z) -> ((~ (v = 0%Z)) -> let o := (ZOmod u v) in
+  (0%Z <= v)%Z) -> ((~ (v = 0%Z)) -> let o := (ZArith.BinInt.Z.rem u v) in
   (((0%Z <= v)%Z /\ (0%Z <= o)%Z) -> ((number.Gcd.gcd v
   o) = (number.Gcd.gcd u v)))).
+(* Why3 intros u v (h1,h2) h3 o (h4,h5). *)
 Proof.
 intuition.
 symmetry.
 rewrite Gcd.Comm.
-rewrite Gcd.gcd_euclid with (q:=(ZOdiv u v)).
+rewrite Gcd.gcd_euclid with (q := Z.quot u v).
 apply f_equal.
-rewrite (ZO_div_mod_eq u v) at 1.
+rewrite (Zquot.Z_quot_rem_eq u v) at 1.
 ring.
 Qed.
 

examples/knuth_prime_numbers/knuth_prime_numbers_WP_PrimeNumbers_WP_parameter_prime_numbers_4.v

 (* This file is generated by Why3's Coq 8.4 driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import ZOdiv.
 Require BuiltIn.
 Require int.Int.
 Require int.Abs.
@@ -15,6 +14,10 @@ Require map.Map.
 (* Why3 assumption *)
 Definition unit := unit.
 
+Axiom qtmark : Type.
+Parameter qtmark_WhyType : WhyType qtmark.
+Existing Instance qtmark_WhyType.
+
 (* Why3 assumption *)
 Definition lt_nat (x:Z) (y:Z): Prop := (0%Z <= y)%Z /\ (x < y)%Z.
 
@@ -69,7 +72,7 @@ Definition set {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT)) (i:Z)
 
 (* Why3 assumption *)
 Definition make {a:Type} {a_WT:WhyType a} (n:Z) (v:a): (@array a a_WT) :=
-  (mk_array n (map.Map.const v:(@map.Map.map Z _ a a_WT))).
+  (mk_array n (map.Map.const v: (@map.Map.map Z _ a a_WT))).
 
 (* Why3 assumption *)
 Definition no_prime_in (l:Z) (u:Z): Prop := forall (x:Z), ((l < x)%Z /\
@@ -100,11 +103,13 @@ apply Zge_le in H'.
 now apply Zmult_le_compat.
 Qed.
 
+Import Zquot.
+
 (* Why3 goal *)
 Theorem WP_parameter_prime_numbers : forall (m:Z), (2%Z <= m)%Z ->
   ((0%Z <= m)%Z -> ((0%Z <= m)%Z -> (((0%Z <= 0%Z)%Z /\ (0%Z < m)%Z) ->
   forall (p:(@map.Map.map Z _ Z _)), ((0%Z <= m)%Z /\
-  (p = (map.Map.set (map.Map.const 0%Z:(@map.Map.map Z _ Z _)) 0%Z 2%Z))) ->
+  (p = (map.Map.set (map.Map.const 0%Z: (@map.Map.map Z _ Z _)) 0%Z 2%Z))) ->
   (((0%Z <= 1%Z)%Z /\ (1%Z < m)%Z) -> forall (p1:(@map.Map.map Z _ Z _)),
   ((0%Z <= m)%Z /\ (p1 = (map.Map.set p 1%Z 3%Z))) -> let o := (m - 1%Z)%Z in
   ((2%Z <= o)%Z -> forall (n:Z) (p2:(@map.Map.map Z _ Z _)), forall (j:Z),
@@ -118,10 +123,15 @@ Theorem WP_parameter_prime_numbers : forall (m:Z), (2%Z <= m)%Z ->
   ((number.Parity.odd n1) /\ ((no_prime_in (map.Map.get p3 (j - 1%Z)%Z)
   n1) /\ forall (i:Z), ((0%Z <= i)%Z /\ (i < k)%Z) ->
   ~ (number.Divisibility.divides (map.Map.get p3 i) n1))))))) ->
-  (((0%Z <= k)%Z /\ (k < m)%Z) -> ((~ ((ZOmod n1 (map.Map.get p3
-  k)) = 0%Z)) -> (((0%Z <= k)%Z /\ (k < m)%Z) -> (((0%Z <= k)%Z /\
-  (k < m)%Z) -> ((~ ((map.Map.get p3 k) < (ZOdiv n1 (map.Map.get p3
+  (((0%Z <= k)%Z /\ (k < m)%Z) ->
+  ((~ ((ZArith.BinInt.Z.rem n1 (map.Map.get p3 k)) = 0%Z)) ->
+  (((0%Z <= k)%Z /\ (k < m)%Z) -> (((0%Z <= k)%Z /\ (k < m)%Z) ->
+  ((~ ((map.Map.get p3 k) < (ZArith.BinInt.Z.quot n1 (map.Map.get p3
   k)))%Z) -> (number.Prime.prime n1)))))))))))).
+(* Why3 intros m h1 h2 h3 (h4,h5) p (h6,h7) (h8,h9) p1 (h10,h11) o h12 n p2 j
+        (h13,h14) (h15,((h16,h17),(h18,h19))) k n1 p3
+        (h20,((h21,h22),(h23,((h24,h25),(h26,(h27,h28)))))) (h29,h30) h31
+        (h32,h33) (h34,h35) h36. *)
 intros m h1 h2 h3 (h4,h5) p (h6,h7) (h8,h9) p1 (h10,h11) o h12 n p2 j
         (h13,h14) (h15,((h16,h17),(h18,h19))) k n1 p3
         (h20,((h21,h22),(h23,((h24,h25),(h26,(h27,h28)))))) (h29,h30) h31
@@ -133,21 +143,21 @@ rewrite <- p0. apply sorted; omega.
 apply Prime.small_divisors; auto.
 omega.
 intros.
-generalize (ZO_div_mod_eq n1 (Map.get p3 k)). intro div.
+generalize (Z_quot_rem_eq n1 (Map.get p3 k)). intro div.
 assert (ne1: (0 <= n1 /\ Map.get p3 k <> 0)%Z) by omega.
-assert (mod1: (0 <= ZOmod n1 (Map.get p3 k))%Z).
+assert (mod1: (0 <= Z.rem n1 (Map.get p3 k))%Z).
 destruct (not_Zeq_inf _ _ (proj2 ne1)) as [Zm|Zm].
-now apply ZOmod_lt_pos_neg.
-now apply ZOmod_lt_pos_pos.
-assert (mod2: (ZOmod n1 (Map.get p3 k) < Map.get p3 k)%Z).
-apply ZOmod_lt_pos_pos ; omega.
+now apply Zrem_lt_pos_neg.
+now apply Zrem_lt_pos_pos.
+assert (mod2: (Z.rem n1 (Map.get p3 k) < Map.get p3 k)%Z).
+apply Zrem_lt_pos_pos ; omega.
 assert (d <= Map.get p3 k)%Z.
 assert (d < Map.get p3 k+1)%Z. 2: omega.
 apply Zle_sqrt; try omega.
 assert (2 < Map.get p3 k)%Z. 
 rewrite <- p0. apply sorted; omega.
 apply Zle_lt_trans with n1; try omega.
-assert (Map.get p3 k * (ZOdiv n1 (Map.get p3 k)) <= Map.get p3 k * Map.get p3 k)%Z.
+assert (Map.get p3 k * (Z.quot n1 (Map.get p3 k)) <= Map.get p3 k * Map.get p3 k)%Z.
 apply Zmult_le_compat_l; try omega.
 replace ((Map.get p3 k + 1) * (Map.get p3 k + 1))%Z with (Map.get p3 k * Map.get p3 k + 2 * Map.get p3 k + 1)%Z by ring.
 omega.

examples/knuth_prime_numbers/knuth_prime_numbers_WP_PrimeNumbers_WP_parameter_prime_numbers_7.v

 (* This file is generated by Why3's Coq 8.4 driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import ZOdiv.
 Require BuiltIn.
 Require int.Int.
 Require int.Abs.
@@ -15,6 +14,10 @@ Require map.Map.
 (* Why3 assumption *)
 Definition unit := unit.
 
+Axiom qtmark : Type.
+Parameter qtmark_WhyType : WhyType qtmark.
+Existing Instance qtmark_WhyType.
+
 (* Why3 assumption *)
 Definition lt_nat (x:Z) (y:Z): Prop := (0%Z <= y)%Z /\ (x < y)%Z.
 
@@ -69,7 +72,7 @@ Definition set {a:Type} {a_WT:WhyType a} (a1:(@array a a_WT)) (i:Z)
 
 (* Why3 assumption *)
 Definition make {a:Type} {a_WT:WhyType a} (n:Z) (v:a): (@array a a_WT) :=
-  (mk_array n (map.Map.const v:(@map.Map.map Z _ a a_WT))).
+  (mk_array n (map.Map.const v: (@map.Map.map Z _ a a_WT))).
 
 (* Why3 assumption *)
 Definition no_prime_in (l:Z) (u:Z): Prop := forall (x:Z), ((l < x)%Z /\
@@ -91,11 +94,13 @@ Axiom exists_prime : forall (p:(@array Z _)) (u:Z), (1%Z <= u)%Z ->
 Axiom Bertrand_postulate : forall (p:Z), (number.Prime.prime p) ->
   ~ (no_prime_in p (2%Z * p)%Z).
 
+Import Zquot.
+
 (* Why3 goal *)
 Theorem WP_parameter_prime_numbers : forall (m:Z), (2%Z <= m)%Z ->
   ((0%Z <= m)%Z -> ((0%Z <= m)%Z -> (((0%Z <= 0%Z)%Z /\ (0%Z < m)%Z) ->
   forall (p:(@map.Map.map Z _ Z _)), ((0%Z <= m)%Z /\
-  (p = (map.Map.set (map.Map.const 0%Z:(@map.Map.map Z _ Z _)) 0%Z 2%Z))) ->
+  (p = (map.Map.set (map.Map.const 0%Z: (@map.Map.map Z _ Z _)) 0%Z 2%Z))) ->
   (((0%Z <= 1%Z)%Z /\ (1%Z < m)%Z) -> forall (p1:(@map.Map.map Z _ Z _)),
   ((0%Z <= m)%Z /\ (p1 = (map.Map.set p 1%Z 3%Z))) -> let o := (m - 1%Z)%Z in
   ((2%Z <= o)%Z -> forall (n:Z) (p2:(@map.Map.map Z _ Z _)), forall (j:Z),
@@ -109,9 +114,10 @@ Theorem WP_parameter_prime_numbers : forall (m:Z), (2%Z <= m)%Z ->
   ((number.Parity.odd n1) /\ ((no_prime_in (map.Map.get p3 (j - 1%Z)%Z)
   n1) /\ forall (i:Z), ((0%Z <= i)%Z /\ (i < k)%Z) ->
   ~ (number.Divisibility.divides (map.Map.get p3 i) n1))))))) ->
-  (((0%Z <= k)%Z /\ (k < m)%Z) -> ((~ ((ZOmod n1 (map.Map.get p3
-  k)) = 0%Z)) -> (((0%Z <= k)%Z /\ (k < m)%Z) -> (((0%Z <= k)%Z /\
-  (k < m)%Z) -> (((map.Map.get p3 k) < (ZOdiv n1 (map.Map.get p3 k)))%Z ->
+  (((0%Z <= k)%Z /\ (k < m)%Z) ->
+  ((~ ((ZArith.BinInt.Z.rem n1 (map.Map.get p3 k)) = 0%Z)) ->
+  (((0%Z <= k)%Z /\ (k < m)%Z) -> (((0%Z <= k)%Z /\ (k < m)%Z) ->
+  (((map.Map.get p3 k) < (ZArith.BinInt.Z.quot n1 (map.Map.get p3 k)))%Z ->
   ((k + 1%Z)%Z < j)%Z))))))))))).
 (* Why3 intros m h1 h2 h3 (h4,h5) p (h6,h7) (h8,h9) p1 (h10,h11) o h12 n p2 j
         (h13,h14) (h15,((h16,h17),(h18,h19))) k n1 p3
@@ -127,14 +133,14 @@ subst k.
 assert (2 < Map.get p3 (j-1))%Z.
 red in h23. destruct h23 as (hh1, (hh2, _)).
 rewrite <- hh1. apply hh2; omega.
-generalize (ZO_div_mod_eq n1 (Map.get p3 (j-1))%Z). intro div.
+generalize (Z_quot_rem_eq n1 (Map.get p3 (j-1))%Z). intro div.
 assert (ne1: (0 <= n1 /\ Map.get p3 (j-1) <> 0)%Z) by omega.
-assert (mod_: (0 <= ZOmod n1 (Map.get p3 (j-1)))%Z).
+assert (mod_: (0 <= Z.rem n1 (Map.get p3 (j-1)))%Z).
 destruct (not_Zeq_inf _ _ (proj2 ne1)) as [Zm|Zm].
-now apply ZOmod_lt_pos_neg.
-now apply ZOmod_lt_pos_pos.
+now apply Zrem_lt_pos_neg.
+now apply Zrem_lt_pos_pos.
 assert (n1 > Map.get p3 (j - 1) * Map.get p3 (j-1))%Z.
-assert (Map.get p3 (j - 1) * (ZOdiv n1 (Map.get p3 (j - 1))) > Map.get p3 (j - 1) * Map.get p3 (j-1))%Z.
+assert (Map.get p3 (j - 1) * (Z.quot n1 (Map.get p3 (j - 1))) > Map.get p3 (j - 1) * Map.get p3 (j-1))%Z.
 apply Zmult_gt_compat_l.
 omega.
 apply Zlt_gt.

examples/logic/lagrange_inequality/lagrange_inequality_CauchySchwarzInequality_CauchySchwarz_1.v

-(* This file is generated by Why3's Coq driver *)
+(* This file is generated by Why3's Coq 8.4 driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import R_sqrt.
+Require Reals.R_sqrt.
 Require BuiltIn.
 Require real.Real.
 Require real.Square.
@@ -11,24 +11,26 @@ Definition dot (x1:R) (x2:R) (y1:R) (y2:R): R :=
   ((x1 * y1)%R + (x2 * y2)%R)%R.
 
 (* Why3 assumption *)
-Definition norm2 (x1:R) (x2:R): R := ((Rsqr x1) + (Rsqr x2))%R.
+Definition norm2 (x1:R) (x2:R): R :=
+  ((Reals.RIneq.Rsqr x1) + (Reals.RIneq.Rsqr x2))%R.
 
 Axiom norm2_pos : forall (x1:R) (x2:R), (0%R <= (norm2 x1 x2))%R.
 
 Axiom Lagrange : forall (a1:R) (a2:R) (b1:R) (b2:R), (((norm2 a1
-  a2) * (norm2 b1 b2))%R = ((Rsqr (dot a1 a2 b1
-  b2)) + (Rsqr ((a1 * b2)%R - (a2 * b1)%R)%R))%R).
+  a2) * (norm2 b1 b2))%R = ((Reals.RIneq.Rsqr (dot a1 a2 b1
+  b2)) + (Reals.RIneq.Rsqr ((a1 * b2)%R - (a2 * b1)%R)%R))%R).
 
-Axiom CauchySchwarz_aux : forall (x1:R) (x2:R) (y1:R) (y2:R), ((Rsqr (dot x1
-  x2 y1 y2)) <= ((norm2 x1 x2) * (norm2 y1 y2))%R)%R.
+Axiom CauchySchwarz_aux : forall (x1:R) (x2:R) (y1:R) (y2:R),
+  ((Reals.RIneq.Rsqr (dot x1 x2 y1 y2)) <= ((norm2 x1 x2) * (norm2 y1
+  y2))%R)%R.
 
 (* Why3 assumption *)
-Definition norm (x1:R) (x2:R): R := (sqrt (norm2 x1 x2)).
+Definition norm (x1:R) (x2:R): R := (Reals.R_sqrt.sqrt (norm2 x1 x2)).
 
 Axiom norm_pos : forall (x1:R) (x2:R), (0%R <= (norm x1 x2))%R.
 
-Axiom sqr_le_sqrt : forall (x:R) (y:R), ((Rsqr x) <= y)%R ->
-  (x <= (sqrt y))%R.
+Axiom sqr_le_sqrt : forall (x:R) (y:R), ((Reals.RIneq.Rsqr x) <= y)%R ->
+  (x <= (Reals.R_sqrt.sqrt y))%R.
 
 Require Import Why3.
 Ltac ae := why3 "Alt-Ergo,0.95.1," timelimit 3.
@@ -36,14 +38,13 @@ Ltac ae := why3 "Alt-Ergo,0.95.1," timelimit 3.
 (* Why3 goal *)
 Theorem CauchySchwarz : forall (x1:R) (x2:R) (y1:R) (y2:R), ((dot x1 x2 y1
   y2) <= ((norm x1 x2) * (norm y1 y2))%R)%R.
-(* intros x1 x2 y1 y2. *)
+(* Why3 intros x1 x2 y1 y2. *)
 intros x1 x2 y1 y2.
 unfold norm.
-rewrite <- sqrt_mult.
+rewrite <- R_sqrt.sqrt_mult.
 apply sqr_le_sqrt.
 ae.
 ae.
 ae.
 Qed.
 
-

examples/logic/lagrange_inequality/lagrange_inequality_CauchySchwarzInequality_sqr_le_sqrt_1.v

-(* This file is generated by Why3's Coq driver *)
+(* This file is generated by Why3's Coq 8.4 driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import R_sqrt.
+Require Reals.R_sqrt.
 Require BuiltIn.
 Require real.Real.
 Require real.Square.
@@ -11,30 +11,34 @@ Definition dot (x1:R) (x2:R) (y1:R) (y2:R): R :=
   ((x1 * y1)%R + (x2 * y2)%R)%R.
 
 (* Why3 assumption *)
-Definition norm2 (x1:R) (x2:R): R := ((Rsqr x1) + (Rsqr x2))%R.
+Definition norm2 (x1:R) (x2:R): R :=
+  ((Reals.RIneq.Rsqr x1) + (Reals.RIneq.Rsqr x2))%R.
 
 Axiom norm2_pos : forall (x1:R) (x2:R), (0%R <= (norm2 x1 x2))%R.
 
 Axiom Lagrange : forall (a1:R) (a2:R) (b1:R) (b2:R), (((norm2 a1
-  a2) * (norm2 b1 b2))%R = ((Rsqr (dot a1 a2 b1
-  b2)) + (Rsqr ((a1 * b2)%R - (a2 * b1)%R)%R))%R).
+  a2) * (norm2 b1 b2))%R = ((Reals.RIneq.Rsqr (dot a1 a2 b1
+  b2)) + (Reals.RIneq.Rsqr ((a1 * b2)%R - (a2 * b1)%R)%R))%R).
 
-Axiom CauchySchwarz_aux : forall (x1:R) (x2:R) (y1:R) (y2:R), ((Rsqr (dot x1
-  x2 y1 y2)) <= ((norm2 x1 x2) * (norm2 y1 y2))%R)%R.
+Axiom CauchySchwarz_aux : forall (x1:R) (x2:R) (y1:R) (y2:R),
+  ((Reals.RIneq.Rsqr (dot x1 x2 y1 y2)) <= ((norm2 x1 x2) * (norm2 y1
+  y2))%R)%R.
 
 (* Why3 assumption *)
-Definition norm (x1:R) (x2:R): R := (sqrt (norm2 x1 x2)).
+Definition norm (x1:R) (x2:R): R := (Reals.R_sqrt.sqrt (norm2 x1 x2)).
 
 Axiom norm_pos : forall (x1:R) (x2:R), (0%R <= (norm x1 x2))%R.
 
 Require Import Why3.
 Ltac ae := why3 "Alt-Ergo,0.95.1," timelimit 3.
+Import R_sqrt.
 
 Open Scope R_scope.
 
 (* Why3 goal *)
-Theorem sqr_le_sqrt : forall (x:R) (y:R), ((Rsqr x) <= y)%R ->
-  (x <= (sqrt y))%R.
+Theorem sqr_le_sqrt : forall (x:R) (y:R), ((Reals.RIneq.Rsqr x) <= y)%R ->
+  (x <= (Reals.R_sqrt.sqrt y))%R.
+(* Why3 intros x y h1. *)
 intros x y h1.
 assert (0 <= Rsqr x) by ae.
 assert (0 <= y) by ae.
@@ -50,4 +54,3 @@ ae.
 ae.
 Qed.
 
-

examples/logic/triangle_inequality/triangle_inequality_CauchySchwarzInequality_CauchySchwarz_1.v

-(* This file is generated by Why3's Coq driver *)
+(* This file is generated by Why3's Coq 8.4 driver *)
 (* Beware! Only edit allowed sections below    *)
 Require Import BuiltIn.
-Require Import R_sqrt.
+Require Reals.R_sqrt.
 Require BuiltIn.
 Require real.Real.
 Require real.Square.
@@ -11,14 +11,15 @@ Definition dot (x1:R) (x2:R) (y1:R) (y2:R): R :=
   ((x1 * y1)%R + (x2 * y2)%R)%R.
 
 (* Why3 assumption *)
-Definition norm2 (x1:R) (x2:R): R := ((Rsqr x1) + (Rsqr x2))%R.
+Definition norm2 (x1:R) (x2:R): R :=
+  ((Reals.RIneq.Rsqr x1) + (Reals.RIneq.Rsqr x2))%R.
 
 Axiom norm2_pos : forall (x1:R) (x2:R), (0%R <= (norm2 x1 x2))%R.
 
 (* Why3 assumption *)
 Definition p (x1:R) (x2:R) (y1:R) (y2:R) (t:R): R := (((norm2 x1
-  x2) + ((2%R * t)%R * (dot x1 x2 y1 y2))%R)%R + ((Rsqr t) * (norm2 y1
-  y2))%R)%R.
+  x2) + ((2%R * t)%R * (dot x1 x2 y1
+  y2))%R)%R + ((Reals.RIneq.Rsqr t) * (norm2 y1 y2))%R)%R.
 
 Axiom p_expr : forall (x1:R) (x2:R) (y1:R) (y2:R) (t:R), ((p x1 x2 y1 y2
   t) = (norm2 (x1 + (t * y1)%R)%R (x2 + (t * y2)%R)%R)).
@@ -27,47 +28,51 @@ Axiom p_pos : forall (x1:R) (x2:R) (y1:R) (y2:R) (t:R), (0%R <= (p x1 x2 y1
   y2 t))%R.
 
 Axiom mul_div_simpl : forall (x:R) (y:R), (~ (y = 0%R)) ->
-  (((Rdiv x y)%R * y)%R = x).
+  (((x / y)%R * y)%R = x).
 
 Axiom p_val_part : forall (x1:R) (x2:R) (y1:R) (y2:R), (0%R < (norm2 y1
-  y2))%R -> ((p x1 x2 y1 y2 (-(Rdiv (dot x1 x2 y1 y2) (norm2 y1
-  y2))%R)%R) = ((norm2 x1 x2) - (Rdiv (Rsqr (dot x1 x2 y1 y2)) (norm2 y1
-  y2))%R)%R).
+  y2))%R -> ((p x1 x2 y1 y2 (-((dot x1 x2 y1 y2) / (norm2 y1
+  y2))%R)%R) = ((norm2 x1 x2) - ((Reals.RIneq.Rsqr (dot x1 x2 y1
+  y2)) / (norm2 y1 y2))%R)%R).
 
 Axiom p_val_part_pos : forall (x1:R) (x2:R) (y1:R) (y2:R), (0%R < (norm2 y1
-  y2))%R -> (0%R <= ((norm2 x1 x2) - (Rdiv (Rsqr (dot x1 x2 y1 y2)) (norm2 y1
-  y2))%R)%R)%R.
+  y2))%R -> (0%R <= ((norm2 x1 x2) - ((Reals.RIneq.Rsqr (dot x1 x2 y1
+  y2)) / (norm2 y1 y2))%R)%R)%R.
 
 Axiom p_val_part_pos_aux : forall (x1:R) (x2:R) (y1:R) (y2:R),
   (0%R < (norm2 y1 y2))%R -> (0%R <= ((norm2 y1 y2) * (p x1 x2 y1 y2
-  (Rdiv (-(dot x1 x2 y1 y2))%R (norm2 y1 y2))%R))%R)%R.
+  ((-(dot x1 x2 y1 y2))%R / (norm2 y1 y2))%R))%R)%R.
 
 Axiom CauchySchwarz_aux_non_null : forall (x1:R) (x2:R) (y1:R) (y2:R),
-  (0%R < (norm2 y1 y2))%R -> ((Rsqr (dot x1 x2 y1 y2)) <= ((norm2 x1
-  x2) * (norm2 y1 y2))%R)%R.
+  (0%R < (norm2 y1 y2))%R -> ((Reals.RIneq.Rsqr (dot x1 x2 y1
+  y2)) <= ((norm2 x1 x2) * (norm2 y1 y2))%R)%R.
 
 Axiom norm_null : forall (y1:R) (y2:R), ((norm2 y1 y2) = 0%R) ->
   ((y1 = 0%R) \/ (y2 = 0%R)).
 
 Axiom CauchySchwarz_aux_null : forall (x1:R) (x2:R) (y1:R) (y2:R), ((norm2 y1
-  y2) = 0%R) -> ((Rsqr (dot x1 x2 y1 y2)) <= ((norm2 x1 x2) * (norm2 y1
-  y2))%R)%R.
+  y2) = 0%R) -> ((Reals.RIneq.Rsqr (dot x1 x2 y1 y2)) <= ((norm2 x1
+  x2) * (norm2 y1 y2))%R)%R.
 
-Axiom CauchySchwarz_aux : forall (x1:R) (x2:R) (y1:R) (y2:R), ((Rsqr (dot x1
-  x2 y1 y2)) <= ((norm2 x1 x2) * (norm2 y1 y2))%R)%R.
+Axiom CauchySchwarz_aux : forall (x1:R) (x2:R) (y1:R) (y2:R),
+  ((Reals.RIneq.Rsqr (dot x1 x2 y1 y2)) <= ((norm2 x1 x2) * (norm2 y1