(* Title: HOL/Library/Log_Nat.thy
Author: Johannes Hölzl, Fabian Immler
Copyright 2012 TU München
*)
section \<open>Logarithm of Natural Numbers\<close>
theory Log_Nat
imports Complex_Main
begin
subsection \<open>Preliminaries\<close>
lemma divide_nat_diff_div_nat_less_one:
"real x / real b - real (x div b) < 1" for x b :: nat
proof (cases "b = 0")
case True
then show ?thesis
by simp
next
case False
then have "real (x div b) + real (x mod b) / real b - real (x div b) < 1"
by (simp add: field_simps)
then show ?thesis
by (simp add: real_of_nat_div_aux [symmetric])
qed
subsection \<open>Floorlog\<close>
definition floorlog :: "nat \<Rightarrow> nat \<Rightarrow> nat"
where "floorlog b a = (if a > 0 \<and> b > 1 then nat \<lfloor>log b a\<rfloor> + 1 else 0)"
lemma floorlog_mono: "x \<le> y \<Longrightarrow> floorlog b x \<le> floorlog b y"
by (auto simp: floorlog_def floor_mono nat_mono)
lemma floorlog_bounds:
"b ^ (floorlog b x - 1) \<le> x \<and> x < b ^ (floorlog b x)" if "x > 0" "b > 1"
proof
show "b ^ (floorlog b x - 1) \<le> x"
proof -
have "b ^ nat \<lfloor>log b x\<rfloor> = b powr \<lfloor>log b x\<rfloor>"
using powr_realpow[symmetric, of b "nat \<lfloor>log b x\<rfloor>"] \<open>x > 0\<close> \<open>b > 1\<close>
by simp
also have "\<dots> \<le> b powr log b x" using \<open>b > 1\<close> by simp
also have "\<dots> = real_of_int x" using \<open>0 < x\<close> \<open>b > 1\<close> by simp
finally have "b ^ nat \<lfloor>log b x\<rfloor> \<le> real_of_int x" by simp
then show ?thesis
using \<open>0 < x\<close> \<open>b > 1\<close> of_nat_le_iff
by (fastforce simp add: floorlog_def)
qed
show "x < b ^ (floorlog b x)"
proof -
have "x \<le> b powr (log b x)" using \<open>x > 0\<close> \<open>b > 1\<close> by simp
also have "\<dots> < b powr (\<lfloor>log b x\<rfloor> + 1)"
using that by (intro powr_less_mono) auto
also have "\<dots> = b ^ nat (\<lfloor>log b (real_of_int x)\<rfloor> + 1)"
using that by (simp flip: powr_realpow)
finally
have "x < b ^ nat (\<lfloor>log b (int x)\<rfloor> + 1)"
by (rule of_nat_less_imp_less)
then show ?thesis
using \<open>x > 0\<close> \<open>b > 1\<close> by (simp add: floorlog_def nat_add_distrib)
qed
qed
lemma floorlog_power [simp]:
"floorlog b (a * b ^ c) = floorlog b a + c" if "a > 0" "b > 1"
proof -
have "\<lfloor>log b a + real c\<rfloor> = \<lfloor>log b a\<rfloor> + c" by arith
then show ?thesis using that
by (auto simp: floorlog_def log_mult powr_realpow[symmetric] nat_add_distrib)
qed
lemma floor_log_add_eqI:
"\<lfloor>log b (a + r)\<rfloor> = \<lfloor>log b a\<rfloor>" if "b > 1" "a \<ge> 1" "0 \<le> r" "r < 1"
for a b :: nat and r :: real
proof (rule floor_eq2)
have "log b a \<le> log b (a + r)" using that by force
then show "\<lfloor>log b a\<rfloor> \<le> log b (a + r)" by arith
next
define l::int where "l = int b ^ (nat \<lfloor>log b a\<rfloor> + 1)"
have l_def_real: "l = b powr (\<lfloor>log b a\<rfloor> + 1)"
using that by (simp add: l_def powr_add powr_real_of_int)
have "a < l"
proof -
have "a = b powr (log b a)" using that by simp
also have "\<dots> < b powr floor ((log b a) + 1)"
using that(1) by auto
also have "\<dots> = l"
using that by (simp add: l_def powr_real_of_int powr_add)
finally show ?thesis by simp
qed
then have "a + r < l" using that by simp
then have "log b (a + r) < log b l" using that by simp
also have "\<dots> = real_of_int \<lfloor>log b a\<rfloor> + 1"
using that by (simp add: l_def_real)
finally show "log b (a + r) < real_of_int \<lfloor>log b a\<rfloor> + 1" .
qed
lemma floor_log_div:
"\<lfloor>log b x\<rfloor> = \<lfloor>log b (x div b)\<rfloor> + 1" if "b > 1" "x > 0" "x div b > 0"
for b x :: nat
proof-
have "\<lfloor>log b x\<rfloor> = \<lfloor>log b (x / b * b)\<rfloor>" using that by simp
also have "\<dots> = \<lfloor>log b (x / b) + log b b\<rfloor>"
using that by (subst log_mult) auto
also have "\<dots> = \<lfloor>log b (x / b)\<rfloor> + 1" using that by simp
also have "\<lfloor>log b (x / b)\<rfloor> = \<lfloor>log b (x div b + (x / b - x div b))\<rfloor>" by simp
also have "\<dots> = \<lfloor>log b (x div b)\<rfloor>"
using that real_of_nat_div4 divide_nat_diff_div_nat_less_one
by (intro floor_log_add_eqI) auto
finally show ?thesis .
qed
lemma compute_floorlog [code]:
"floorlog b x = (if x > 0 \<and> b > 1 then floorlog b (x div b) + 1 else 0)"
by (auto simp: floorlog_def floor_log_div[of b x] div_eq_0_iff nat_add_distrib
intro!: floor_eq2)
lemma floor_log_eq_if:
"\<lfloor>log b x\<rfloor> = \<lfloor>log b y\<rfloor>" if "x div b = y div b" "b > 1" "x > 0" "x div b \<ge> 1"
for b x y :: nat
proof -
have "y > 0" using that by (auto intro: ccontr)
thus ?thesis using that by (simp add: floor_log_div)
qed
lemma floorlog_eq_if:
"floorlog b x = floorlog b y" if "x div b = y div b" "b > 1" "x > 0" "x div b \<ge> 1"
for b x y :: nat
proof -
have "y > 0" using that by (auto intro: ccontr)
then show ?thesis using that
by (auto simp add: floorlog_def eq_nat_nat_iff intro: floor_log_eq_if)
qed
lemma floorlog_leD:
"floorlog b x \<le> w \<Longrightarrow> b > 1 \<Longrightarrow> x < b ^ w"
by (metis floorlog_bounds leD linorder_neqE_nat order.strict_trans power_strict_increasing_iff
zero_less_one zero_less_power)
lemma floorlog_leI:
"x < b ^ w \<Longrightarrow> 0 \<le> w \<Longrightarrow> b > 1 \<Longrightarrow> floorlog b x \<le> w"
by (drule less_imp_of_nat_less[where 'a=real])
(auto simp: floorlog_def Suc_le_eq nat_less_iff floor_less_iff log_of_power_less)
lemma floorlog_eq_zero_iff:
"floorlog b x = 0 \<longleftrightarrow> b \<le> 1 \<or> x \<le> 0"
by (auto simp: floorlog_def)
lemma floorlog_le_iff:
"floorlog b x \<le> w \<longleftrightarrow> b \<le> 1 \<or> b > 1 \<and> 0 \<le> w \<and> x < b ^ w"
using floorlog_leD[of b x w] floorlog_leI[of x b w]
by (auto simp: floorlog_eq_zero_iff[THEN iffD2])
lemma floorlog_ge_SucI:
"Suc w \<le> floorlog b x" if "b ^ w \<le> x" "b > 1"
using that le_log_of_power[of b w x] power_not_zero
by (force simp: floorlog_def Suc_le_eq powr_realpow not_less Suc_nat_eq_nat_zadd1
zless_nat_eq_int_zless int_add_floor less_floor_iff
simp del: floor_add2)
lemma floorlog_geI:
"w \<le> floorlog b x" if "b ^ (w - 1) \<le> x" "b > 1"
using floorlog_ge_SucI[of b "w - 1" x] that
by auto
lemma floorlog_geD:
"b ^ (w - 1) \<le> x" if "w \<le> floorlog b x" "w > 0"
proof -
have "b > 1" "0 < x"
using that by (auto simp: floorlog_def split: if_splits)
have "b ^ (w - 1) \<le> x" if "b ^ w \<le> x"
proof -
have "b ^ (w - 1) \<le> b ^ w"
using \<open>b > 1\<close>
by (auto intro!: power_increasing)
also note that
finally show ?thesis .
qed
moreover have "b ^ nat \<lfloor>log (real b) (real x)\<rfloor> \<le> x" (is "?l \<le> _")
proof -
have "0 \<le> log (real b) (real x)"
using \<open>b > 1\<close> \<open>0 < x\<close>
by (auto simp: )
then have "?l \<le> b powr log (real b) (real x)"
using \<open>b > 1\<close>
by (auto simp flip: powr_realpow intro!: powr_mono of_nat_floor)
also have "\<dots> = x" using \<open>b > 1\<close> \<open>0 < x\<close>
by auto
finally show ?thesis
unfolding of_nat_le_iff .
qed
ultimately show ?thesis
using that
by (auto simp: floorlog_def le_nat_iff le_floor_iff le_log_iff powr_realpow
split: if_splits elim!: le_SucE)
qed
subsection \<open>Bitlen\<close>
definition bitlen :: "int \<Rightarrow> int"
where "bitlen a = floorlog 2 (nat a)"
lemma bitlen_alt_def:
"bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"
by (simp add: bitlen_def floorlog_def)
lemma bitlen_zero [simp]:
"bitlen 0 = 0"
by (auto simp: bitlen_def floorlog_def)
lemma bitlen_nonneg:
"0 \<le> bitlen x"
by (simp add: bitlen_def)
lemma bitlen_bounds:
"2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)" if "x > 0"
proof -
from that have "bitlen x \<ge> 1" by (auto simp: bitlen_alt_def)
with that floorlog_bounds[of "nat x" 2] show ?thesis
by (auto simp add: bitlen_def le_nat_iff nat_less_iff nat_diff_distrib)
qed
lemma bitlen_pow2 [simp]:
"bitlen (b * 2 ^ c) = bitlen b + c" if "b > 0"
using that by (simp add: bitlen_def nat_mult_distrib nat_power_eq)
lemma compute_bitlen [code]:
"bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
by (simp add: bitlen_def nat_div_distrib compute_floorlog)
lemma bitlen_eq_zero_iff:
"bitlen x = 0 \<longleftrightarrow> x \<le> 0"
by (auto simp add: bitlen_alt_def)
(metis compute_bitlen add.commute bitlen_alt_def bitlen_nonneg less_add_same_cancel2
not_less zero_less_one)
lemma bitlen_div:
"1 \<le> real_of_int m / 2^nat (bitlen m - 1)"
and "real_of_int m / 2^nat (bitlen m - 1) < 2" if "0 < m"
proof -
let ?B = "2^nat (bitlen m - 1)"
have "?B \<le> m" using bitlen_bounds[OF \<open>0 <m\<close>] ..
then have "1 * ?B \<le> real_of_int m"
unfolding of_int_le_iff[symmetric] by auto
then show "1 \<le> real_of_int m / ?B" by auto
from that have "0 \<le> bitlen m - 1" by (auto simp: bitlen_alt_def)
have "m < 2^nat(bitlen m)" using bitlen_bounds[OF that] ..
also from that have "\<dots> = 2^nat(bitlen m - 1 + 1)"
by (auto simp: bitlen_def)
also have "\<dots> = ?B * 2"
unfolding nat_add_distrib[OF \<open>0 \<le> bitlen m - 1\<close> zero_le_one] by auto
finally have "real_of_int m < 2 * ?B"
by (metis (full_types) mult.commute power.simps(2) of_int_less_numeral_power_cancel_iff)
then have "real_of_int m / ?B < 2 * ?B / ?B"
by (rule divide_strict_right_mono) auto
then show "real_of_int m / ?B < 2" by auto
qed
lemma bitlen_le_iff_floorlog:
"bitlen x \<le> w \<longleftrightarrow> w \<ge> 0 \<and> floorlog 2 (nat x) \<le> nat w"
by (auto simp: bitlen_def)
lemma bitlen_le_iff_power:
"bitlen x \<le> w \<longleftrightarrow> w \<ge> 0 \<and> x < 2 ^ nat w"
by (auto simp: bitlen_le_iff_floorlog floorlog_le_iff)
lemma less_power_nat_iff_bitlen:
"x < 2 ^ w \<longleftrightarrow> bitlen (int x) \<le> w"
using bitlen_le_iff_power[of x w]
by auto
lemma bitlen_ge_iff_power:
"w \<le> bitlen x \<longleftrightarrow> w \<le> 0 \<or> 2 ^ (nat w - 1) \<le> x"
unfolding bitlen_def
by (auto simp flip: nat_le_iff intro: floorlog_geI dest: floorlog_geD)
lemma bitlen_twopow_add_eq:
"bitlen (2 ^ w + b) = w + 1" if "0 \<le> b" "b < 2 ^ w"
by (auto simp: that nat_add_distrib bitlen_le_iff_power bitlen_ge_iff_power intro!: antisym)
end