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 poly_lt_exp_ge_6 : forall c x, x >= (Nat.pow 2 c) -> x >= (Nat.pow 2 6) -> Nat.pow x c < Nat.pow 2 x.