Section Prefix.
Variable A : Type.
Fixpoint onth (s : seq A) n : option A :=
if s is x :: s' then
if n is n'.+1 then onth s' n' else Some x
else None.
Definition prefix s1 s2 :=
forall n x, onth s1 n = some x -> onth s2 n = some x.

Lemma prefix_onth (s t : seq A) x : x < size s -> prefix s t -> onth s x = onth t x.