Definition negligible(f : nat -> Rat) :=
  forall c, exists n, forall x (pf_nz : nz x),
    x > n ->
    ~ ((1 / expnat x c) <= f x)%rat.

Theorem S_log_square_lt : forall x, Nat.pow 2 6 <= x-> S (Nat.log2 x) * S (Nat.log2 x) <= x.