Definition eqb x y : bool := if eq_dec x y then true else false.

(* Specifications: *)
Lemma eqb_alt : forall x y, eqb x y = match compare x y with EQ _ => true | _ => false end.
Lemma if_eq_dec x y (B:Type)(b b':B) : (if eq_dec x y then b else b') = (match compare x y with Eq => b | _ => b' end).
Lemma eqb_alt : forall x y, eqb x y = match compare x y with Eq => true | _ => false end.