# Theory Log_Nat

theory Log_Nat
imports Complex_Main
```(*  Title:      HOL/Library/Log_Nat.thy
Author:     Johannes Hölzl, Fabian Immler
Copyright   2012  TU München
*)

section ‹Logarithm of Natural Numbers›

theory Log_Nat
imports Complex_Main
begin

definition floorlog :: "nat ⇒ nat ⇒ nat" where
"floorlog b a = (if a > 0 ∧ b > 1 then nat ⌊log b a⌋ + 1 else 0)"

lemma floorlog_mono: "x ≤ y ⟹ floorlog b x ≤ floorlog b y"
by(auto simp: floorlog_def floor_mono nat_mono)

lemma floorlog_bounds:
assumes "x > 0" "b > 1"
shows "b ^ (floorlog b x - 1) ≤ x ∧ x < b ^ (floorlog b x)"
proof
show "b ^ (floorlog b x - 1) ≤ x"
proof -
have "b ^ nat ⌊log b x⌋ = b powr ⌊log b x⌋"
using powr_realpow[symmetric, of b "nat ⌊log b x⌋"] ‹x > 0› ‹b > 1›
by simp
also have "… ≤ b powr log b x" using ‹b > 1› by simp
also have "… = real_of_int x" using ‹0 < x› ‹b > 1› by simp
finally have "b ^ nat ⌊log b x⌋ ≤ real_of_int x" by simp
then show ?thesis
using ‹0 < x› ‹b > 1› of_nat_le_iff
by (fastforce simp add: floorlog_def)
qed
show "x < b ^ (floorlog b x)"
proof -
have "x ≤ b powr (log b x)" using ‹x > 0› ‹b > 1› by simp
also have "… < b powr (⌊log b x⌋ + 1)"
using assms by (intro powr_less_mono) auto
also have "… = b ^ nat (⌊log b (real_of_int x)⌋ + 1)"
using assms by (simp flip: powr_realpow)
finally
have "x < b ^ nat (⌊log b (int x)⌋ + 1)"
by (rule of_nat_less_imp_less)
then show ?thesis
using ‹x > 0› ‹b > 1› by (simp add: floorlog_def nat_add_distrib)
qed
qed

lemma floorlog_power[simp]:
assumes "a > 0" "b > 1"
shows "floorlog b (a * b ^ c) = floorlog b a + c"
proof -
have "⌊log b a + real c⌋ = ⌊log b a⌋ + c" by arith
then show ?thesis using assms
by (auto simp: floorlog_def log_mult powr_realpow[symmetric] nat_add_distrib)
qed

fixes a::nat and b::nat and r::real
assumes "b > 1" "a ≥ 1" "0 ≤ r" "r < 1"
shows "⌊log b (a + r)⌋ = ⌊log b a⌋"
proof (rule floor_eq2)
have "log b a ≤ log b (a + r)" using assms by force
then show "⌊log b a⌋ ≤ log b (a + r)" by arith
next
define l::int where "l = int b ^ (nat ⌊log b a⌋ + 1)"
have l_def_real: "l = b powr (⌊log b a⌋ + 1)"
using assms by (simp add: l_def powr_add powr_real_of_int)
have "a < l"
proof -
have "a = b powr (log b a)" using assms by simp
also have "… < b powr floor ((log b a) + 1)"
using assms(1) by auto
also have "… = l"
using assms by (simp add: l_def powr_real_of_int powr_add)
finally show ?thesis by simp
qed
then have "a + r < l" using assms by simp
then have "log b (a + r) < log b l" using assms by simp
also have "… = real_of_int ⌊log b a⌋ + 1"
using assms by (simp add: l_def_real)
finally show "log b (a + r) < real_of_int ⌊log b a⌋ + 1" .
qed

lemma divide_nat_diff_div_nat_less_one:
fixes x b::nat shows "x / b - x div b < 1"
proof -
have "int 0 ≠ ⌊1::real⌋" by simp
thus ?thesis
by (metis add_diff_cancel_left' floor_divide_of_nat_eq less_eq_real_def
mod_div_trivial real_of_nat_div3 real_of_nat_div_aux)
qed

lemma floor_log_div:
fixes b x :: nat assumes "b > 1" "x > 0" "x div b > 0"
shows "⌊log b x⌋ = ⌊log b (x div b)⌋ + 1"
proof-
have "⌊log b x⌋ = ⌊log b (x / b * b)⌋" using assms by simp
also have "… = ⌊log b (x / b) + log b b⌋"
using assms by (subst log_mult) auto
also have "… = ⌊log b (x / b)⌋ + 1" using assms by simp
also have "⌊log b (x / b)⌋ = ⌊log b (x div b + (x / b - x div b))⌋" by simp
also have "… = ⌊log b (x div b)⌋"
using assms 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 ∧ 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:
fixes b x y :: nat
assumes "x div b = y div b" "b > 1" "x > 0" "x div b ≥ 1"
shows "floor(log b x) = floor(log b y)"
proof -
have "y > 0" using assms by(auto intro: ccontr)
thus ?thesis using assms by (simp add: floor_log_div)
qed

lemma floorlog_eq_if:
fixes b x y :: nat
assumes "x div b = y div b" "b > 1" "x > 0" "x div b ≥ 1"
shows "floorlog b x = floorlog b y"
proof -
have "y > 0" using assms by(auto intro: ccontr)
thus ?thesis using assms
by(auto simp add: floorlog_def eq_nat_nat_iff intro: floor_log_eq_if)
qed

lemma powr_eq_one_iff[simp]: "a powr x = 1 ⟷ (x = 0)"
if "a > 1"
for a x::real
using that
by (auto simp: powr_def split: if_splits)

lemma floorlog_leD: "floorlog b x ≤ w ⟹ b > 1 ⟹ 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 ⟹ 0 ≤ w ⟹ b > 1 ⟹ floorlog b x ≤ 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 ⟷ (b ≤ 1 ∨ x ≤ 0)"
by (auto simp: floorlog_def)

lemma floorlog_le_iff: "floorlog b x ≤ w ⟷ b ≤ 1 ∨ b > 1 ∧ 0 ≤ w ∧ 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 ≤ floorlog b x" if "b ^ w ≤ 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

lemma floorlog_geI: "w ≤ floorlog b x" if "b ^ (w - 1) ≤ x" "b > 1"
using floorlog_ge_SucI[of b "w - 1" x] that
by auto

lemma floorlog_geD: "b ^ (w - 1) ≤ x" if "w ≤ 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) ≤ x" if "b ^ w ≤ x"
proof -
have "b ^ (w - 1) ≤ b ^ w"
using ‹b > 1›
by (auto intro!: power_increasing)
also note that
finally show ?thesis .
qed
moreover have "b ^ nat ⌊log (real b) (real x)⌋ ≤ x" (is "?l ≤ _")
proof -
have "0 ≤ log (real b) (real x)"
using ‹b > 1› ‹0 < x›
by (auto simp: )
then have "?l ≤ b powr log (real b) (real x)"
using ‹b > 1›
by (auto simp flip: powr_realpow intro!: powr_mono of_nat_floor)
also have "… = x" using ‹b > 1› ‹0 < x›
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

definition bitlen :: "int ⇒ int" where "bitlen a = floorlog 2 (nat a)"

lemma bitlen_alt_def: "bitlen a = (if a > 0 then ⌊log 2 a⌋ + 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 ≤ bitlen x"
by (simp add: bitlen_def)

lemma bitlen_bounds:
assumes "x > 0"
shows "2 ^ nat (bitlen x - 1) ≤ x ∧ x < 2 ^ nat (bitlen x)"
proof -
from assms have "bitlen x ≥ 1" by (auto simp: bitlen_alt_def)
with assms 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]:
assumes "b > 0"
shows "bitlen (b * 2 ^ c) = bitlen b + c"
using assms
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 ⟷ x ≤ 0"
by (auto simp add: bitlen_alt_def)
not_less zero_less_one)

lemma bitlen_div:
assumes "0 < m"
shows "1 ≤ real_of_int m / 2^nat (bitlen m - 1)"
and "real_of_int m / 2^nat (bitlen m - 1) < 2"
proof -
let ?B = "2^nat (bitlen m - 1)"

have "?B ≤ m" using bitlen_bounds[OF ‹0 <m›] ..
then have "1 * ?B ≤ real_of_int m"
unfolding of_int_le_iff[symmetric] by auto
then show "1 ≤ real_of_int m / ?B" by auto

from assms have "m ≠ 0" by auto
from assms have "0 ≤ bitlen m - 1" by (auto simp: bitlen_alt_def)

have "m < 2^nat(bitlen m)" using bitlen_bounds[OF assms] ..
also from assms have "… = 2^nat(bitlen m - 1 + 1)"
by (auto simp: bitlen_def)
also have "… = ?B * 2"
unfolding nat_add_distrib[OF ‹0 ≤ bitlen m - 1› 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 ≤ w ⟷ w ≥ 0 ∧ floorlog 2 (nat x) ≤ nat w"
by (auto simp: bitlen_def)

lemma bitlen_le_iff_power: "bitlen x ≤ w ⟷ w ≥ 0 ∧ x < 2 ^ nat w"
by (auto simp: bitlen_le_iff_floorlog floorlog_le_iff)

lemma less_power_nat_iff_bitlen: "x < 2 ^ w ⟷ bitlen (int x) ≤ w"
using bitlen_le_iff_power[of x w]
by auto

lemma bitlen_ge_iff_power: "w ≤ bitlen x ⟷ w ≤ 0 ∨ 2 ^ (nat w - 1) ≤ 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 ≤ b" "b < 2 ^ w"
by (auto simp: that nat_add_distrib bitlen_le_iff_power bitlen_ge_iff_power intro!: antisym)

end
```