unfold SPECIF in |- *.
apply exist with (leavemult t); auto.
Defined.
(* Here we  define a predicate "Has an occurrence of O" *)
Fixpoint Has_Zero (t : nat_tree) : Prop :=
match t return Prop with
| leaf n1 =>
(* (nat_leaf n1) *) n1 = 0 (* (nat_cons t1 t2) *)
| cons t1 t2 => Has_Zero t1 \/ Has_Zero t2
end.
(* If some tree t has an occurence of 0, then (leavmult t)=0 *)

Lemma zero_occ : forall t : nat_tree, Has_Zero t -> leavemult t = 0.
simple induction t.