Definition StrictCompSpec {A} (eq lt: A -> A -> Prop)
                          (cmp: A -> A -> comparison) :=
  StrictOrder lt /\ forall x y, CompSpec eq lt x y (cmp x y).Definition CompSpec' {A} (cmp: A -> A -> comparison) :=
   StrictCompSpec (@Logic.eq A) (fun x y => Lt = cmp x y) cmp.Ltac do_comp cspec x y := destruct (CompSpec2Type (proj2 cspec x y)).

Lemma eq_comp: forall {A} lt cmp, StrictCompSpec Logic.eq lt cmp -> forall x y: A, (x = y <-> Eq = cmp x y).