(* 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.
intuition.
unfold polynomial, polynomial_nz in *.
do 3 (destruct H).
exists (S x).
exists (S x0).
exists (x0 + S x1).
intuition.
rewrite H.
rewrite plus_assoc.
eapply plus_le_compat; intuition.
simpl.
destruct (eq_nat_dec x 0).
subst.
simpl.
repeat rewrite mult_1_r.
rewrite <- plus_0_l at 1.
eapply plus_le_compat; intuition.
rewrite <- plus_0_l at 1.
rewrite <- plus_0_r at 1.
eapply plus_le_compat; intuition.
eapply plus_le_compat; intuition.
eapply mult_le_compat; intuition.
destruct (eq_nat_dec n 0); subst.
rewrite expnat_0;
omega.
rewrite <- mult_1_l at 1.
eapply mult_le_compat; intuition.
Qed.
intuition.
apply polynomial_nz_equiv in H.
apply polynomial_nz_equiv in H0.
unfold polynomial, polynomial_nz in *.
Ltac des :=
match goal with
| [H : exists _, _ |- _] => destruct H
end.
repeat des.
intuition.
exists (max x2 x).
exists (x3 + x0).
exists (x4 + x1).
intuition.
eapply le_trans.
eapply plus_le_compat.
eapply H6; trivial.
eapply H7; trivial.
rewrite mult_plus_distr_r.
repeat rewrite plus_assoc.
eapply plus_le_compat; trivial.
rewrite plus_comm.
rewrite plus_assoc.
eapply plus_le_compat; trivial.
rewrite plus_comm.
eapply plus_le_compat;
eapply mult_le_compat; intuition.
eapply expnat_exp_le; intuition.
eapply expnat_exp_le; intuition.
Qed.
intuition.
unfold polynomial.
exists 0.
exists 0.
exists c.
intuition.
Qed.
unfold polynomial.
intuition.
exists 1.
exists 1.
exists 0.
intuition.
simpl.
omega.
Qed.
intros f1 f2 H H0.
apply polynomial_nz_equiv in H.
apply polynomial_nz_equiv in H0.
unfold polynomial, polynomial_nz in *.
repeat des.
intuition.
exists (x + x2).
exists (3 * (x3 * x0 * x4 * x1)).
exists (x4 * x1).
intros n.
eapply le_trans.
eapply mult_le_compat.
eapply H6; intuition.
eapply H7; intuition.
repeat rewrite mult_plus_distr_l.
repeat rewrite mult_plus_distr_r.
rewrite plus_assoc.
eapply plus_le_compat; trivial.
rewrite expnat_plus.
simpl.
rewrite plus_0_r.
repeat rewrite mult_plus_distr_r.
rewrite plus_assoc.
eapply plus_le_compat.
eapply plus_le_compat.
rewrite (mult_comm (expnat n x)).
repeat rewrite mult_assoc.
eapply mult_le_compat. 2: reflexivity.
rewrite mult_comm.
rewrite mult_assoc.
eapply mult_le_compat. 2: reflexivity.
rewrite <- mult_1_r at 1.
rewrite <- mult_1_r at 1.
eapply mult_le_compat. 2: auto with zarith.
eapply mult_le_compat. 2: auto with zarith.
rewrite mult_comm.
reflexivity.
destruct (eq_nat_dec n 0); subst.
repeat rewrite expnat_0.
simpl.
repeat rewrite mult_0_r.
reflexivity.
trivial.
trivial.
rewrite (mult_comm (expnat n x)).
repeat rewrite mult_assoc.
eapply mult_le_compat; trivial.
rewrite <- mult_1_r at 1.
eapply mult_le_compat; trivial.
rewrite <- mult_1_r at 1.
eapply mult_le_compat; trivial.
rewrite mult_comm.
eapply mult_le_compat; trivial.
rewrite <- mult_1_l at 1.
eapply mult_le_compat; trivial.
eapply expnat_ge_1.
omega.
destruct (eq_nat_dec n 0); subst.
repeat rewrite expnat_0.
simpl.
repeat rewrite mult_0_r.
intuition.
trivial.
trivial.
rewrite (mult_comm (expnat n x)).
repeat rewrite mult_assoc.
rewrite <- mult_1_r at 1.
eapply mult_le_compat; trivial.
rewrite <- mult_assoc.
rewrite (mult_comm (expnat n x2)).
rewrite mult_assoc.
eapply mult_le_compat; trivial.
eapply mult_le_compat; trivial.
rewrite <- mult_1_r at 1.
eapply mult_le_compat; trivial.
rewrite <- mult_1_r at 1.
eapply mult_le_compat; trivial.
eapply expnat_ge_1.
omega.
Qed.
Local Open Scope rat_scope.
Definition negligible(f : nat -> Rat) :=
forall c, exists n, forall x (pf_nz : nz x),
x > n ->
~ ((1 / expnat x c) <= f x)%rat.
intuition.
unfold negligible in *.
intuition.
edestruct H.
econstructor.
intuition.
eapply H1.
eauto.
rewrite H3.
rewrite <- H0.
intuition.
Qed.
intuition.
unfold negligible in *.
intuition.
edestruct H0.
econstructor.
intuition.
eapply H1.
eauto.
rewrite H3.
eauto.
Qed.
unfold negligible in *.
intuition.
destruct (H (S c)).
destruct (H0 (S c)).
exists (max 1 (max x x0)).
intuition.
apply Nat.max_lub_lt_iff in H3.
intuition.
assert (1 / expnat x1 c <= 2/1 * (maxRat (f1 x1) (f2 x1)))%rat.
eapply leRat_trans.
eapply H4.
eapply ratAdd_2_ratMax.
assert (1 / expnat x1 (S c) <= RatIntro 1 (posnatMult (pos 2) (pos (expnat x1 c))))%rat.
eapply leRat_terms; intuition.
unfold natToPosnat, posnatToNat, posnatMult.
eapply expnat_double_le.
omega.
unfold maxRat in *.
case_eq (bleRat (f1 x1) (f2 x1)); intuition.
rewrite H8 in H3.
eapply H2.
eapply le_lt_trans.
eapply Max.le_max_r.
eauto.
rewrite H7.
rewrite rat_mult_den.
rewrite H3.
rewrite <- ratMult_assoc.
rewrite <- ratMult_num_den.
rewrite num_dem_same_rat1.
rewrite ratMult_1_l.
intuition.
unfold posnatMult, natToPosnat, posnatToNat.
omega.
rewrite H8 in H3.
eapply H1.
eapply le_lt_trans.
eapply Max.le_max_l.
eauto.
rewrite H7.
rewrite rat_mult_den.
rewrite H3.
rewrite <- ratMult_assoc.
rewrite <- ratMult_num_den.
rewrite num_dem_same_rat1.
rewrite ratMult_1_l.
intuition.
unfold posnatMult, natToPosnat, posnatToNat.
omega.
Qed.
(* We need several facts about arithmetic to show that an inverse exponential function is negligible. *)
Local Open Scope nat_scope.
induction y; intuition; simpl in *.
rewrite plus_0_r in *.
assert (S (y + S y + 3)  =
(y + y + 3) + 2).
omega.
rewrite H1.
destruct (eq_nat_dec y 3).
subst.
assert (x >= Nat.pow 2 4).
rewrite H.
eapply Nat.log2_spec.
destruct (eq_nat_dec x 0).
subst.
rewrite log2_0 in H.
omega.
omega.
eapply le_trans.
2:{
eapply H2.
}
simpl.
omega.
assert ( y + y + 3 <= div2 x).
eapply IHy.
symmetry.
apply log2_div2.
trivial.
omega.
assert (2 <= div2 x).
assert (2 = div2 4).
trivial.
rewrite H3.
eapply div2_le_mono.
eapply le_trans.
2:{
eapply Nat.log2_le_lin.
destruct (eq_nat_dec 0 x); subst.
rewrite log2_0 in H.
omega.
omega.
}
omega.
eapply le_trans.
eapply plus_le_compat.
eapply H2.
eapply H3.
eapply div2_ge_double.
Qed.
induction y; intuition; simpl in *.
assert (y = Nat.log2 (div2 x)).
symmetry.
eapply log2_div2.
trivial.
rewrite (mult_comm _ (S (S y))).
simpl.
rewrite (mult_comm _ (S y)) in IHy.
simpl in *.
assert ( S (S (y + S (S (y + (y + (y + y * y)))))) =
(S (y + (y + y * y))) + (2 * y + 3)).
omega.
rewrite H2.
clear H2.
destruct (eq_nat_dec (Nat.log2 x) 6).
assert (y = 5).
assert (S y = 6).
rewrite H.
trivial.
omega.
rewrite H2.
simpl.
assert (x >= Nat.pow 2 6).
rewrite <- e.
eapply Nat.log2_spec.
destruct (eq_nat_dec x 0).
subst.
rewrite log2_0 in H.
omega.
omega.
eapply le_trans.
2:{
eapply H3.
}
simpl.
omega.
assert ( S (y + (y + y * y)) <= div2 x).
eapply IHy.
trivial.
omega.
assert (2 * y + 3 <= div2 x).
eapply double_log_plus_3_le_h; trivial.
omega.
eapply le_trans.
eapply plus_le_compat.
eapply H2.
eapply H3.
eapply div2_ge_double.
Qed.
intuition.
eapply S_log_square_lt_h; trivial.
eapply le_trans.
2:{
eapply Nat.log2_le_mono.
eapply H.
}
rewrite Nat.log2_pow2; omega.
Qed.
intuition.
assert (Nat.log2 x < S (Nat.log2 x)).
omega.
eapply lt_le_trans.
eapply mult_lt_compat.
eapply H0.
eapply H0.
eapply S_log_square_lt.
trivial.
Qed.
intuition.
specialize (Nat.log2_spec_alt); intuition.
destruct (H1 x).
eapply lt_le_trans.
2:{
eapply H.
}
eapply (expnat_2_ge_1 c).
intuition.
(* This case split probably isn't necessary *)
destruct (eq_nat_dec x0 0).
rewrite e in H3.
rewrite plus_0_r in *.
rewrite H3.
rewrite <- Nat.pow_mul_r.
eapply Nat.pow_lt_mono_r.
omega.
rewrite <- H3.
assert (c <= Nat.log2 x).
eapply (@Nat.pow_le_mono_r_iff 2).
omega.
rewrite <- H3.
trivial.
eapply le_lt_trans.
eapply mult_le_compat.
eapply le_refl.
eapply H4.
eapply log_square_lt.
eapply le_trans.
2:{
eapply H0.
}
eapply Nat.pow_le_mono_r.
omega.
omega.
destruct (eq_nat_dec c 0).
rewrite e.
simpl.
eapply le_lt_trans.
assert (1 <= expnat 2 0).
trivial.
eapply H4.
eapply Nat.pow_lt_mono_r.
omega.
omega.
assert (expnat x c < expnat (2 ^ S (Nat.log2 x)) c).
eapply Nat.pow_lt_mono_l.
omega.
eapply Nat.log2_spec.
omega.
eapply lt_le_trans.
eapply H4.
rewrite <- Nat.pow_mul_r.
eapply Nat.pow_le_mono_r.
omega.
assert (c <= S (Nat.log2 x)).
eapply (@Nat.pow_le_mono_r_iff 2).
omega.
eapply le_trans.
eapply H.
eapply lt_le_weak.
eapply Nat.log2_spec.
omega.
eapply le_trans.
eapply mult_le_compat.
eapply le_refl.
eapply H6.
eapply S_log_square_lt.
eapply le_trans.
2:{
eapply H0.
}
eapply Nat.pow_le_mono_r.
omega.
omega.
Qed.
intuition.
exists (expnat 2 (max c 6)).
intuition.
eapply poly_lt_exp_ge_6.
eapply le_trans.
2:{
eapply H.
}
eapply Nat.pow_le_mono_r.
omega.
eapply Max.le_max_l.
eapply le_trans.
2:{
eapply H.
}
eapply Nat.pow_le_mono_r.
omega.
eapply Max.le_max_r.
Qed.
unfold negligible in *.
intuition.
destruct (poly_lt_exp c).
exists x.
intuition.
eapply (rat_num_not_le).
eapply H1.
unfold posnatToNat, natToPosnat.
eapply H.
omega.
Qed.
unfold negligible in *.
intuition.
destruct (eq_nat_dec n 0).
subst.
exists 1.
intuition.
assert ((1 / expnat x c) == 0)%rat.
eapply leRat_0_eq.
rewrite H1.
rewrite rat_num_0.
rewrite ratMult_0_l.
intuition.
eapply rat_num_nz; [idtac | eauto].
omega.
destruct (H (c + n)).
exists (x + n)%nat.
intuition.
eapply H0.
omega.
assert (1 / expnat x0 (c + n) == RatIntro 1 (pos  (expnat x0 c)) * RatIntro 1 (pos (expnat x0 n)))%rat.
rewrite <- ratMult_num_den.
eapply eqRat_terms.
symmetry.
eapply mult_1_r.
unfold natToPosnat, posnatToNat, posnatMult.
rewrite expnat_plus.
trivial.
rewrite H3.
rewrite H2.
rewrite ratMult_comm.
rewrite <- ratMult_assoc.
eapply leRat_trans.
2:{
eapply eqRat_impl_leRat.
rewrite <- ratMult_1_l.
eapply eqRat_refl.
}
eapply ratMult_leRat_compat; intuition.
eapply rat_le_1.
destruct d.
unfold natToPosnat, posnatToNat, posnatMult.
rewrite mult_comm.
eapply mult_le_compat.
eapply le_trans.
eapply le_expnat_2.
eapply expnat_base_le.
omega.
omega.
Qed.
unfold negligible in *.
intuition.
destruct (H (S c)).
exists x.
intuition.
eapply H0.
omega.
simpl.
assert ( (RatIntro 1 (natToPosnat (expnat_nz (S c) pf_nz)))  ==
RatIntro 1 (posnatMult (pos x0) (pos expnat x0 c)) )%rat.
eapply eqRat_terms; trivial.
simpl in *.
rewrite H3.
rewrite ratMult_denom.
rewrite H2.
rewrite <- ratMult_assoc.
rewrite <- ratMult_num_den.
rewrite num_dem_same_rat1.
rewrite ratMult_1_l.
intuition.
unfold natToPosnat, posnatToNat, posnatMult.
omega.
Qed.

Theorem negligible_exp :
forall z,
negligible (fun n => expnat n z / expnat 2 n)%rat.