Admitted.
Definition monotonic (f : nat -> nat) :=
forall x y, x <= y -> f x <= f y.

(* Specifications: *)
Lemma solve_cost_ineqs_clean :
exists (f: nat -> nat),
1 <= f 0 /\
monotonic f /\
forall n, 0 < n -> 2 + f (n / 2) <= f n.
Lemma solve_cost_ineq :
exists (f: nat -> nat),
forall n,
1 + (if zerop n then 0 else 1 + max (f (n / 2)) (f (n - (n / 2) - 1))) <= f n.