--- a/src/HOL/Data_Structures/Balance_List.thy Wed Aug 10 18:57:20 2016 +0200
+++ b/src/HOL/Data_Structures/Balance_List.thy Fri Aug 12 08:20:17 2016 +0200
@@ -5,7 +5,7 @@
theory Balance_List
imports
"~~/src/HOL/Library/Tree"
- "~~/src/HOL/Library/Float"
+ "~~/src/HOL/Library/Log_Nat"
begin
fun bal :: "'a list \<Rightarrow> nat \<Rightarrow> 'a tree * 'a list" where
@@ -73,7 +73,7 @@
hence le: "?log2 \<le> ?log1" by(simp add:floorlog_mono)
have "height t = max ?log1 ?log2 + 1" by (simp add: t IH1 IH2)
also have "\<dots> = ?log1 + 1" using le by (simp add: max_absorb1)
- also have "\<dots> = floorlog 2 n" by (simp add: Float.compute_floorlog)
+ also have "\<dots> = floorlog 2 n" by (simp add: compute_floorlog)
finally show ?thesis .
qed
qed
@@ -106,7 +106,7 @@
also have "\<dots> = floorlog 2 (n - n div 2)" by(simp add: floorlog_def)
also have "n - n div 2 = (n+1) div 2" by arith
also have "floorlog 2 \<dots> = floorlog 2 (n+1) - 1"
- by (simp add: Float.compute_floorlog)
+ by (simp add: compute_floorlog)
finally show ?thesis .
qed
qed
--- a/src/HOL/Library/Float.thy Wed Aug 10 18:57:20 2016 +0200
+++ b/src/HOL/Library/Float.thy Fri Aug 12 08:20:17 2016 +0200
@@ -6,7 +6,7 @@
section \<open>Floating-Point Numbers\<close>
theory Float
-imports Complex_Main Lattice_Algebras
+imports Log_Nat Lattice_Algebras
begin
definition "float = {m * 2 powr e | (m :: int) (e :: int). True}"
@@ -878,193 +878,6 @@
end
-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:
- assumes "x > 0" "b > 1"
- shows "b ^ (floorlog b x - 1) \<le> x \<and> x < b ^ (floorlog b x)"
-proof
- show "b ^ (floorlog b x - 1) \<le> x"
- proof -
- have "(b::real) ^ nat \<lfloor>log b (real_of_int x)\<rfloor> = b powr real_of_int \<lfloor>log b (real_of_int x)\<rfloor>"
- using powr_realpow[symmetric, of b "nat \<lfloor>log b (real_of_int x)\<rfloor>"] \<open>x > 0\<close> \<open>b > 1\<close>
- by simp
- also have "\<dots> \<le> b powr log b (real_of_int 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 (real_of_int 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 assms
- by (intro powr_less_mono) auto
- also have "\<dots> = b ^ nat (\<lfloor>log b (real_of_int x)\<rfloor> + 1)"
- using assms
- by (simp add: powr_realpow[symmetric])
- finally
- have "x < b ^ nat (\<lfloor>log (real b) (real_of_int (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]:
- assumes "a > 0" "b > 1"
- shows "floorlog b (a * b ^ c) = floorlog b a + c"
-proof -
- have "\<lfloor>log (real b) (real a) + real c\<rfloor> = \<lfloor>log (real b) (real a)\<rfloor> + c"
- by arith
- then show ?thesis using assms
- by (auto simp: floorlog_def log_mult powr_realpow[symmetric] nat_add_distrib)
-qed
-
-lemma floor_log_add_eqI:
- fixes a::nat and b::nat and r::real
- assumes "b > 1" "a \<ge> 1" "0 \<le> r" "r < 1"
- shows "\<lfloor>log b (a + r)\<rfloor> = \<lfloor>log b a\<rfloor>"
-proof (rule floor_eq2)
- have "log (real b) (real a) \<le> log (real b) (real a + r)"
- using assms by force
- then show "\<lfloor>log (real b) (real a)\<rfloor> \<le> log (real b) (real 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 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 "\<dots> < b powr floor ((log b a) + 1)"
- using assms(1) by auto
- also have "\<dots> = 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 "\<dots> = real_of_int \<lfloor>log b a\<rfloor> + 1"
- using assms by (simp add: l_def_real)
- finally show "log b (a + r) < real_of_int \<lfloor>log b a\<rfloor> + 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 \<noteq> \<lfloor>1::real\<rfloor>" 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 "\<lfloor>log b x\<rfloor> = \<lfloor>log b (x div b)\<rfloor> + 1"
-proof-
- have "\<lfloor>log (real b) (real x)\<rfloor> = \<lfloor>log (real b) (x / b * b)\<rfloor>"
- using assms by simp
- also have "\<dots> = \<lfloor>log b (x / b) + log b b\<rfloor>"
- using assms by (subst log_mult) auto
- also have "\<dots> = \<lfloor>log b (x / b)\<rfloor> + 1" using assms 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 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 \<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:
- fixes b x y :: nat
- assumes "x div b = y div b" "b > 1" "x > 0" "x div b \<ge> 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 \<ge> 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
-
-
-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_nonneg: "0 \<le> bitlen x"
-by (simp add: bitlen_def)
-
-lemma bitlen_bounds:
- assumes "x > 0"
- shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
-proof -
- from assms have "bitlen x \<ge> 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 bitlen_Float:
- fixes m e
- defines "f \<equiv> Float m e"
- shows "bitlen \<bar>mantissa f\<bar> + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
-proof (cases "m = 0")
- case True
- then show ?thesis by (simp add: f_def bitlen_alt_def Float_def)
-next
- case False
- then have "f \<noteq> float_of 0"
- unfolding real_of_float_eq by (simp add: f_def)
- then have "mantissa f \<noteq> 0"
- by (simp add: mantissa_noteq_0)
- moreover
- obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"
- by (rule f_def[THEN denormalize_shift, OF \<open>f \<noteq> float_of 0\<close>])
- ultimately show ?thesis by (simp add: abs_mult)
-qed
-
-context
-begin
-
-qualified lemma compute_bitlen[code]: "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"
- unfolding bitlen_def
- by (subst Float.compute_floorlog) (simp add: nat_div_distrib)
-
-end
lemma float_gt1_scale:
assumes "1 \<le> Float m e"
@@ -1653,7 +1466,7 @@
lemma bitlen_eq_zero_iff: "bitlen x = 0 \<longleftrightarrow> x \<le> 0"
by (auto simp add: bitlen_alt_def)
- (metis Float.compute_bitlen add.commute bitlen_alt_def bitlen_nonneg less_add_same_cancel2
+ (metis compute_bitlen add.commute bitlen_alt_def bitlen_nonneg less_add_same_cancel2
not_less zero_less_one)
lemma sum_neq_zeroI:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Log_Nat.thy Fri Aug 12 08:20:17 2016 +0200
@@ -0,0 +1,161 @@
+(* 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
+
+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:
+ assumes "x > 0" "b > 1"
+ shows "b ^ (floorlog b x - 1) \<le> x \<and> x < b ^ (floorlog b x)"
+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 assms by (intro powr_less_mono) auto
+ also have "\<dots> = b ^ nat (\<lfloor>log b (real_of_int x)\<rfloor> + 1)"
+ using assms by (simp add: powr_realpow[symmetric])
+ 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]:
+ assumes "a > 0" "b > 1"
+ shows "floorlog b (a * b ^ c) = floorlog b a + c"
+proof -
+ have "\<lfloor>log b a + real c\<rfloor> = \<lfloor>log b a\<rfloor> + c" by arith
+ then show ?thesis using assms
+ by (auto simp: floorlog_def log_mult powr_realpow[symmetric] nat_add_distrib)
+qed
+
+lemma floor_log_add_eqI:
+ fixes a::nat and b::nat and r::real
+ assumes "b > 1" "a \<ge> 1" "0 \<le> r" "r < 1"
+ shows "\<lfloor>log b (a + r)\<rfloor> = \<lfloor>log b a\<rfloor>"
+proof (rule floor_eq2)
+ have "log b a \<le> log b (a + r)" using assms 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 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 "\<dots> < b powr floor ((log b a) + 1)"
+ using assms(1) by auto
+ also have "\<dots> = 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 "\<dots> = real_of_int \<lfloor>log b a\<rfloor> + 1"
+ using assms by (simp add: l_def_real)
+ finally show "log b (a + r) < real_of_int \<lfloor>log b a\<rfloor> + 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 \<noteq> \<lfloor>1::real\<rfloor>" 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 "\<lfloor>log b x\<rfloor> = \<lfloor>log b (x div b)\<rfloor> + 1"
+proof-
+ have "\<lfloor>log b x\<rfloor> = \<lfloor>log b (x / b * b)\<rfloor>" using assms by simp
+ also have "\<dots> = \<lfloor>log b (x / b) + log b b\<rfloor>"
+ using assms by (subst log_mult) auto
+ also have "\<dots> = \<lfloor>log b (x / b)\<rfloor> + 1" using assms 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 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 \<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:
+ fixes b x y :: nat
+ assumes "x div b = y div b" "b > 1" "x > 0" "x div b \<ge> 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 \<ge> 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
+
+
+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_nonneg: "0 \<le> bitlen x"
+by (simp add: bitlen_def)
+
+lemma bitlen_bounds:
+ assumes "x > 0"
+ shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
+proof -
+ from assms have "bitlen x \<ge> 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)
+
+end