Proof using.
induction xs.
- auto.
- unfold terms_and_indices_from_one in *. simpl. intros. break_or_hyp; eauto.
Qed.

(* Specifications: *)
Lemma terms_and_indices_from_one_In :
forall (xs ys : list entry),
(forall x, In x xs -> In x ys) ->
terms_and_indices_from_one ys ->
terms_and_indices_from_one xs.