Introduce the concept of decidable equality into Coq realizations.

lib/coq/BuiltIn.v

 Require Export ZArith.
 Require Export Rbase.
 
-Class WhyType T := why_inhabitant : T.
+Class WhyType T := {
+  why_inhabitant : T ;
+  why_decidable_eq : forall x y : T, { x = y } + { x <> y }
+}.
 
 Notation int := Z.
 
 Global Instance int_WhyType : WhyType int.
+Proof.
+split.
 exact Z0.
+exact Z_eq_dec.
 Qed.
 
 Notation real := R.
 
 Global Instance real_WhyType : WhyType real.
+Proof.
+split.
 exact R0.
+intros x y.
+destruct (total_order_T x y) as [[H|H]|H] ;
+  try (left ; exact H) ; right.
+now apply Rlt_not_eq.
+now apply Rgt_not_eq.
 Qed.
 
 Global Instance tuple_WhyType : forall T {T' : WhyType T} U {U' : WhyType U}, WhyType (T * U).
-now split.
+Proof.
+intros T WT U WU.
+split.
+split ; apply why_inhabitant.
+intros (x1,x2) (y1,y2).
+destruct (why_decidable_eq x1 y1) as [H1|H1].
+destruct (why_decidable_eq x2 y2) as [H2|H2].
+left.
+now apply f_equal2.
+right.
+now injection.
+right.
+now injection.
 Qed.
 
 Global Instance unit_WhyType : WhyType unit.
+Proof.
+split.
 exact tt.
+intros [] [].
+now left.
 Qed.
 
 Global Instance bool_WhyType : WhyType bool.
+Proof.
+split.
 exact false.
+exact Bool.bool_dec.
 Qed.

lib/coq/set/Set.v

@@ -20,7 +20,10 @@ Defined.
 Global Instance set_WhyType : forall (a:Type) {a_WT:WhyType a}, WhyType (set a).
 Proof.
 intros.
+split.
 exact (fun _ => False).
+intros x y.
+apply excluded_middle_informative.
 Qed.
 
 (* Why3 goal *)
@@ -176,7 +179,7 @@ Qed.
 (* Why3 goal *)
 Definition choose: forall {a:Type} {a_WT:WhyType a}, (set a) -> a.
 intros a a_WT s.
-assert (i: inhabited a) by (apply inhabits; assumption).
+assert (i: inhabited a) by (apply inhabits, why_inhabitant).
 exact (epsilon i (fun x => mem x s)).
 Defined.