simple induction 1.
apply good_idx.
intros.
apply (H0 (f (length ws0))).
apply is_prefix_cons.
reflexivity.
assumption.
Defined.

(* Specifications: *)
Theorem higman_idx :
forall f : nat -> word, {i : nat &  {j : nat | emb (f i) (f j) /\ i < j}}.