(** Start at 0 *)
Definition linear_search_from_0 (e : exists n, P n) : nat :=
    let b := let (n, p) := e in O_witness n (stop n p) in
    prog_linear_search 0 b.

Lemma linear_search_from_0_rel (e : exists n, P n) : rel_ls 0 (linear_search_from_0 e).