Proof with simpl; try solve[exfalso; auto]; auto.
induction l... intro C; split; [ solve [apply (C a); left; auto] | ].
apply IHl...
Qed.

Lemma listd_In_inv_pred a l s : listd f inter TT l s -> In a l -> f a s.
Proof with try solve[exfalso; auto]; auto.