Definition rightI (d:nat)(x:TypeI) : nat :=
 match x with
 | I _ n => n
 | E _ => d
 end.

Lemma inf_list_ex : forall l e n s, In e l -> inf_list (map inf l) = I n s -> exists n', exists s', n' <= n /\ s' <= s /\ inf e = I n' s'.