induction x1; intuition; simpl in *.
remember (split x1) as z.
destruct z.
simpl in *.
unfold collidesWith in *.
simpl in *.
case_eq (eqb b2 b1); intuition.
rewrite H1 in H.
rewrite H1 in H0.
inversion H; clear H; subst.
simpl in *.
case_eq (eqb a0 a); intuition.
rewrite H in H0.
simpl in *.
rewrite eqb_leibniz in H1.
rewrite eqb_leibniz in H.
subst.
unfold eqbPair.
simpl.
repeat rewrite eqb_refl.
simpl.
trivial.
rewrite H in H0.
simpl in *.
discriminate.
rewrite H1 in H0.
rewrite H1 in H.
simpl in *.
unfold eqbPair.
simpl.
rewrite H1.
case_eq (eqb a0 a); intuition.
simpl.
eapply IHx1; eauto.
simpl.
eapply IHx1; eauto.
Qed.

Theorem funcCollision_true_cons :
forall (A B : Set)(eqda : EqDec A)(eqdb : EqDec B) (ls : list (A * B)) a,
funcCollision _ _ ls = true ->
funcCollision _ _ (a :: ls) = true.