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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** Base-2 Logarithm *)
Require Import NZAxioms NZMulOrder NZPow.
(** Interface of a log2 function, then its specification on naturals *)
Module Type Log2 (Import A : Typ).
Parameters Inline log2 : t -> t.
End Log2.
Module Type NZLog2Spec (A : NZOrdAxiomsSig')(B : Pow' A)(C : Log2 A).
Import A B C.
Declare Instance log2_wd : Proper (eq==>eq) log2.
Axiom log2_spec : forall a, 0<a -> 2^(log2 a) <= a < 2^(S (log2 a)).
Axiom log2_nonpos : forall a, a<=0 -> log2 a == 0.
End NZLog2Spec.
Module Type NZLog2 (A : NZOrdAxiomsSig)(B : Pow A) := Log2 A <+ NZLog2Spec A B.
(** Derived properties of logarithm *)
Module NZLog2Prop
(Import A : NZOrdAxiomsSig')
(Import B : NZPow' A)
(Import C : NZLog2 A B)
(Import D : NZMulOrderProp A)
(Import E : NZPowProp A B D).
(** log2 is always non-negative *)
Lemma log2_nonneg : forall a, 0 <= log2 a.
Proof.
intros a. destruct (le_gt_cases a 0) as [Ha|Ha].
now rewrite log2_nonpos.
destruct (log2_spec a Ha) as (_,LT).
apply lt_succ_r, (pow_gt_1 2). order'.
rewrite <- le_succ_l, <- one_succ in Ha. order.
Qed.
(** A tactic for proving positivity and non-negativity *)
Ltac order_pos :=
((apply add_pos_pos || apply add_nonneg_nonneg ||
apply mul_pos_pos || apply mul_nonneg_nonneg ||
apply pow_nonneg || apply log2_nonneg ||
apply (le_le_succ_r 0));
order_pos) (* in case of success of an apply, we recurse *)
|| order'. (* otherwise *)
(** The spec of log2 indeed determines it *)
Lemma log2_unique : forall a b, 0<=b -> 2^b<=a<2^(S b) -> log2 a == b.
Proof.
intros a b Hb (LEb,LTb).
assert (Ha : 0 < a).
apply lt_le_trans with (2^b); trivial.
apply pow_pos_nonneg; order'.
assert (Hc := log2_nonneg a).
destruct (log2_spec a Ha) as (LEc,LTc).
assert (log2 a <= b).
apply lt_succ_r, (pow_lt_mono_r_iff 2); try order'.
now apply le_le_succ_r.
assert (b <= log2 a).
apply lt_succ_r, (pow_lt_mono_r_iff 2); try order'.
now apply le_le_succ_r.
order.
Qed.
(** log2 is exact on powers of 2 *)
Lemma log2_pow2 : forall a, 0<=a -> log2 (2^a) == a.
Proof.
intros a Ha.
apply log2_unique; trivial.
split. order. apply pow_lt_mono_r. order'. split; trivial.
apply lt_succ_diag_r.
Qed.
(** log2 and basic constants *)
Lemma log2_1 : log2 1 == 0.
Proof.
rewrite <- (pow_0_r 2). now apply log2_pow2.
Qed.
Lemma log2_2 : log2 2 == 1.
Proof.
rewrite <- (pow_1_r 2). apply log2_pow2; order'.
Qed.
(** log2 n is strictly positive for 1<n *)
Lemma log2_pos : forall a, 1<a -> 0 < log2 a.
Proof.
intros a Ha.
assert (Ha' : 0 < a) by order'.
assert (H := log2_nonneg a). apply le_lteq in H.
destruct H; trivial.
generalize (log2_spec a Ha'). rewrite <- H in *. nzsimpl; try order.
intros (_,H'). rewrite two_succ in H'. apply lt_succ_r in H'; order.
Qed.
(** Said otherwise, log2 is null only below 1 *)
Lemma log2_null : forall a, log2 a == 0 <-> a <= 1.
Proof.
intros a. split; intros H.
destruct (le_gt_cases a 1) as [Ha|Ha]; trivial.
generalize (log2_pos a Ha); order.
apply le_lteq in H; destruct H.
apply log2_nonpos. apply lt_succ_r. now rewrite <- one_succ.
rewrite H. apply log2_1.
Qed.
(** log2 is a monotone function (but not a strict one) *)
Lemma log2_le_mono : forall a b, a<=b -> log2 a <= log2 b.
Proof.
intros a b H.
destruct (le_gt_cases a 0) as [Ha|Ha].
rewrite log2_nonpos; order_pos.
assert (Hb : 0 < b) by order.
destruct (log2_spec a Ha) as (LEa,_).
destruct (log2_spec b Hb) as (_,LTb).
apply lt_succ_r, (pow_lt_mono_r_iff 2); order_pos.
Qed.
(** No reverse result for <=, consider for instance log2 3 <= log2 2 *)
Lemma log2_lt_cancel : forall a b, log2 a < log2 b -> a < b.
Proof.
intros a b H.
destruct (le_gt_cases b 0) as [Hb|Hb].
rewrite (log2_nonpos b) in H; trivial.
generalize (log2_nonneg a); order.
destruct (le_gt_cases a 0) as [Ha|Ha]. order.
destruct (log2_spec a Ha) as (_,LTa).
destruct (log2_spec b Hb) as (LEb,_).
apply le_succ_l in H.
apply (pow_le_mono_r_iff 2) in H; order_pos.
Qed.
(** When left side is a power of 2, we have an equivalence for <= *)
Lemma log2_le_pow2 : forall a b, 0<a -> 0<=b -> (2^b<=a <-> b <= log2 a).
Proof.
intros a b Ha Hb.
split; intros H.
rewrite <- (log2_pow2 b); trivial. now apply log2_le_mono.
transitivity (2^(log2 a)).
apply pow_le_mono_r; order'.
now destruct (log2_spec a Ha).
Qed.
(** When right side is a square, we have an equivalence for < *)
Lemma log2_lt_pow2 : forall a b, 0<a -> 0<=b -> (a<2^b <-> log2 a < b).
Proof.
intros a b Ha Hb.
split; intros H.
apply (pow_lt_mono_r_iff 2); try order_pos.
apply le_lt_trans with a; trivial.
now destruct (log2_spec a Ha).
apply log2_lt_cancel; try order.
now rewrite log2_pow2.
Qed.
(** Comparing log2 and identity *)
Lemma log2_lt_lin : forall a, 0<a -> log2 a < a.
Proof.
intros a Ha.
apply (pow_lt_mono_r_iff 2); try order_pos.
apply le_lt_trans with a.
now destruct (log2_spec a Ha).
apply pow_gt_lin_r; order'.
Qed.
Lemma log2_le_lin : forall a, 0<=a -> log2 a <= a.
Proof.
intros a Ha.
apply le_lteq in Ha; destruct Ha as [Ha|Ha].
now apply lt_le_incl, log2_lt_lin.
rewrite <- Ha, log2_nonpos; order.
Qed.
(** Log2 and multiplication. *)
(** Due to rounding error, we don't have the usual
[log2 (a*b) = log2 a + log2 b] but we may be off by 1 at most *)
Lemma log2_mul_below : forall a b, 0<a -> 0<b ->
log2 a + log2 b <= log2 (a*b).
Proof.
intros a b Ha Hb.
apply log2_le_pow2; try order_pos.
rewrite pow_add_r by order_pos.
apply mul_le_mono_nonneg; try apply log2_spec; order_pos.
Qed.
Lemma log2_mul_above : forall a b, 0<=a -> 0<=b ->
log2 (a*b) <= log2 a + log2 b + 1.
Proof.
intros a b Ha Hb.
apply le_lteq in Ha. destruct Ha as [Ha|Ha].
apply le_lteq in Hb. destruct Hb as [Hb|Hb].
apply lt_succ_r.
rewrite add_1_r, <- add_succ_r, <- add_succ_l.
apply log2_lt_pow2; try order_pos.
rewrite pow_add_r by order_pos.
apply mul_lt_mono_nonneg; try order; now apply log2_spec.
rewrite <- Hb. nzsimpl. rewrite log2_nonpos; order_pos.
rewrite <- Ha. nzsimpl. rewrite log2_nonpos; order_pos.
Qed.
(** And we can't find better approximations in general.
- The lower bound is exact for powers of 2.
- Concerning the upper bound, for any c>0, take a=b=2^c-1,
then log2 (a*b) = c+c -1 while (log2 a) = (log2 b) = c-1
*)
(** Log2 and addition *)
Lemma log2_add_le : forall a b, a~=1 -> b~=1 -> log2 (a+b) <= log2 a + log2 b.
Proof.
intros a b Ha Hb.
destruct (lt_trichotomy a 1) as [Ha'|[Ha'|Ha']]; [|order|].
rewrite one_succ, lt_succ_r in Ha'.
rewrite (log2_nonpos a); trivial. nzsimpl. apply log2_le_mono.
rewrite <- (add_0_l b) at 2. now apply add_le_mono.
destruct (lt_trichotomy b 1) as [Hb'|[Hb'|Hb']]; [|order|].
rewrite one_succ, lt_succ_r in Hb'.
rewrite (log2_nonpos b); trivial. nzsimpl. apply log2_le_mono.
rewrite <- (add_0_r a) at 2. now apply add_le_mono.
clear Ha Hb.
apply lt_succ_r.
apply log2_lt_pow2; try order_pos.
rewrite pow_succ_r by order_pos.
rewrite two_succ, one_succ at 1. nzsimpl.
apply add_lt_mono.
apply lt_le_trans with (2^(S (log2 a))). apply log2_spec; order'.
apply pow_le_mono_r. order'. rewrite <- add_1_r. apply add_le_mono_l.
rewrite one_succ; now apply le_succ_l, log2_pos.
apply lt_le_trans with (2^(S (log2 b))). apply log2_spec; order'.
apply pow_le_mono_r. order'. rewrite <- add_1_l. apply add_le_mono_r.
rewrite one_succ; now apply le_succ_l, log2_pos.
Qed.
(** The sum of two log2 is less than twice the log2 of the sum.
This can almost be seen as a convexity inequality. *)
Lemma quadmul_le_squareadd : forall a b, 0<=a -> 0<=b ->
2*2*a*b <= (a+b)*(a+b).
Proof.
intros.
nzsimpl'.
rewrite !mul_add_distr_l, !mul_add_distr_r.
rewrite (add_comm _ (b*b)), add_assoc.
apply add_le_mono_r.
rewrite (add_shuffle0 (a*a)), (mul_comm b a).
apply add_le_mono_r.
rewrite (mul_comm a b) at 1.
now apply crossmul_le_addsquare.
Qed.
Lemma add_log2_lt : forall a b, 0<a -> 0<b ->
log2 a + log2 b < 2 * log2 (a+b).
Proof.
intros a b Ha Hb.
apply (pow_lt_mono_r_iff 2); try order_pos.
rewrite !pow_add_r by order_pos.
apply le_lt_trans with (a*b).
apply mul_le_mono_nonneg; try order_pos; now apply log2_spec.
apply (mul_lt_mono_pos_r (2^2)).
apply pow_pos_nonneg; order'.
rewrite (mul_comm 2).
rewrite <- pow_add_r by order_pos.
rewrite <- mul_succ_l.
rewrite pow_mul_r by order_pos.
apply le_lt_trans with ((a+b)^2).
rewrite !pow_2_r, (mul_comm (a*b)), mul_assoc.
apply quadmul_le_squareadd; order.
apply pow_lt_mono_l. order'. split. order_pos.
apply log2_spec; order_pos.
Qed.
(** We can't be more precise : consider for instance a=2 b=4, then 1+2<2*2 *)
End NZLog2Prop.
|
