(**************)
simple induction v. simpl in |- *. trivial with arith.
intros. inversion H0. rewrite trunc_all. trivial with arith.
simpl in |- *. rewrite H. trivial with arith.
apply le_Sn_le. generalize H1. simpl in |- *. trivial with arith.
Qed.

Lemma trunc_O : forall v : list A, trunc v 0 = nil.
simple induction v; auto with arith.