eapply Coq.Logic.Eqdep_dec.UIP_dec. apply EquivDec.equiv_dec.
Qed.

Theorem UIP_equal : forall {x y : A} (p1 p2 : x = y), p1 = p2.
eapply Coq.Logic.Eqdep_dec.UIP_dec. apply EquivDec.equiv_dec.