(* Copyright 2012-2015 by Adam Petcher.				* * Use of this source code is governed by the license described	* * in the LICENSE file at the root of the source tree.		*)
(* Definitions of asymptotic notions such as polynomial and negligible functions, and related theory. *)
Set Implicit Arguments.
Definition polynomial (f : nat -> nat) :=
exists x c1 c2, forall n,
(f n <= c1 * expnat n x + c2)%nat.
Definition polynomial_nz(f : nat -> nat) :=
exists x c1 c2,
x > 0 /\ c1 > 0 /\ c2 > 0 /\
forall n,
(f n <= c1 * expnat n x + c2)%nat.

Theorem polynomial_nz_equiv :
forall f,
polynomial f ->
polynomial_nz f.