--- a/src/HOL/Library/Float.thy Wed Jun 17 10:57:11 2015 +0200
+++ b/src/HOL/Library/Float.thy Wed Jun 17 11:03:05 2015 +0200
@@ -3,7 +3,7 @@
Copyright 2012 TU München
*)
-section {* Floating-Point Numbers *}
+section \<open>Floating-Point Numbers\<close>
theory Float
imports Complex_Main Lattice_Algebras
@@ -43,7 +43,7 @@
lemma real_float[simp]: "x \<in> float \<Longrightarrow> real (float_of x) = x"
unfolding real_of_float_def by (rule float_of_inverse)
-subsection {* Real operations preserving the representation as floating point number *}
+subsection \<open>Real operations preserving the representation as floating point number\<close>
lemma floatI: fixes m e :: int shows "m * 2 powr e = x \<Longrightarrow> x \<in> float"
by (auto simp: float_def)
@@ -161,7 +161,7 @@
code_datatype Float
-subsection {* Arithmetic operations on floating point numbers *}
+subsection \<open>Arithmetic operations on floating point numbers\<close>
instantiation float :: "{ring_1, linorder, linordered_ring, linordered_idom, numeral, equal}"
begin
@@ -264,7 +264,7 @@
and float_of_neg_numeral[simp]: "- numeral k = float_of (- numeral k)"
unfolding real_of_float_eq by simp_all
-subsection {* Quickcheck *}
+subsection \<open>Quickcheck\<close>
instantiation float :: exhaustive
begin
@@ -304,7 +304,7 @@
end
-subsection {* Represent floats as unique mantissa and exponent *}
+subsection \<open>Represent floats as unique mantissa and exponent\<close>
lemma int_induct_abs[case_names less]:
fixes j :: int
@@ -320,7 +320,7 @@
case (less n)
{ fix m assume n: "n \<noteq> 0" "n = m * r"
then have "\<bar>m \<bar> < \<bar>n\<bar>"
- using `1 < r` by (simp add: abs_mult)
+ using \<open>1 < r\<close> by (simp add: abs_mult)
from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto }
then show ?case
by (metis dvd_def monoid_mult_class.mult.right_neutral mult.commute power_0)
@@ -333,7 +333,7 @@
proof
have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto
assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2"
- with `e1 \<le> e2` have "m1 = m2 * 2 powr nat (e2 - e1)"
+ with \<open>e1 \<le> e2\<close> have "m1 = m2 * 2 powr nat (e2 - e1)"
by (simp add: powr_divide2[symmetric] field_simps)
also have "\<dots> = m2 * 2^nat (e2 - e1)"
by (simp add: powr_realpow)
@@ -342,7 +342,7 @@
with m1 have "m1 = m2"
by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
then show "m1 = m2 \<and> e1 = e2"
- using eq `m1 \<noteq> 0` by (simp add: powr_inj)
+ using eq \<open>m1 \<noteq> 0\<close> by (simp add: powr_inj)
qed simp
lemma mult_powr_eq_mult_powr_iff:
@@ -359,9 +359,9 @@
proof atomize_elim
{ assume "x \<noteq> 0"
from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)
- with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
+ with \<open>x \<noteq> 0\<close> int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"
by auto
- with `\<not> 2 dvd k` x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m"
+ with \<open>\<not> 2 dvd k\<close> x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m"
by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])
(simp add: powr_add powr_realpow) }
then show "x = 0 \<or> (\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m \<and> x \<noteq> 0)"
@@ -434,7 +434,7 @@
by (auto simp: mult_powr_eq_mult_powr_iff)
qed
-subsection {* Compute arithmetic operations *}
+subsection \<open>Compute arithmetic operations\<close>
lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"
unfolding real_of_float_eq mantissa_exponent[of f] by simp
@@ -467,7 +467,7 @@
using eq by simp
then have "mantissa f = m * 2^nat (e - exponent f)"
unfolding real_of_int_inject by simp
- with `exponent f < e` have "2 dvd mantissa f"
+ with \<open>exponent f < e\<close> have "2 dvd mantissa f"
apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])
apply (cases "nat (e - exponent f)")
apply auto
@@ -476,7 +476,7 @@
qed
ultimately have "real m = mantissa f * 2^nat (exponent f - e)"
by (simp add: powr_realpow[symmetric])
- with `e \<le> exponent f`
+ with \<open>e \<le> exponent f\<close>
show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"
unfolding real_of_int_inject by auto
qed
@@ -564,7 +564,7 @@
hide_fact (open) compute_float_eq
-subsection {* Lemmas for types @{typ real}, @{typ nat}, @{typ int}*}
+subsection \<open>Lemmas for types @{typ real}, @{typ nat}, @{typ int}\<close>
lemmas real_of_ints =
real_of_int_zero
@@ -588,7 +588,7 @@
lemmas nat_of_reals = real_of_nats[symmetric]
-subsection {* Rounding Real Numbers *}
+subsection \<open>Rounding Real Numbers\<close>
definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where
"round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"
@@ -663,9 +663,9 @@
proof -
have "x * 2 powr p < 1 / 2 * 2 powr p"
using assms by simp
- also have "\<dots> \<le> 2 powr p - 1" using `p > 0`
+ also have "\<dots> \<le> 2 powr p - 1" using \<open>p > 0\<close>
by (auto simp: powr_divide2[symmetric] powr_int field_simps self_le_power)
- finally show ?thesis using `p > 0`
+ finally show ?thesis using \<open>p > 0\<close>
by (simp add: round_up_def field_simps powr_minus powr_int ceiling_less_eq)
qed
@@ -705,7 +705,7 @@
by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)
-subsection {* Rounding Floats *}
+subsection \<open>Rounding Floats\<close>
definition div_twopow::"int \<Rightarrow> nat \<Rightarrow> int" where [simp]: "div_twopow x n = x div (2 ^ n)"
@@ -763,7 +763,7 @@
also have "... = 1 / 2 powr p / 2 powr e"
unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)
finally show ?thesis
- using `p + e < 0`
+ using \<open>p + e < 0\<close>
by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])
next
assume "\<not> p + e < 0"
@@ -771,7 +771,7 @@
have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p"
by (auto intro: exI[where x="m*2^nat (e+p)"]
simp add: ac_simps powr_add[symmetric] r powr_realpow)
- with `\<not> p + e < 0` show ?thesis
+ with \<open>\<not> p + e < 0\<close> show ?thesis
by transfer (auto simp add: round_down_def field_simps powr_add powr_minus)
qed
hide_fact (open) compute_float_down
@@ -791,16 +791,16 @@
proof cases
assume "\<not> b dvd a"
hence "a mod b \<noteq> 0" by auto
- hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto
+ hence ne: "real (a mod b) / real b \<noteq> 0" using \<open>b \<noteq> 0\<close> by auto
have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"
apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
proof -
have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp
moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"
- apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto
+ apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne \<open>b \<noteq> 0\<close> by auto
ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith
qed
- thus ?thesis using `\<not> b dvd a` by simp
+ thus ?thesis using \<open>\<not> b dvd a\<close> by simp
qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]
floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)
@@ -810,7 +810,7 @@
hide_fact (open) compute_float_up
-subsection {* Compute bitlen of integers *}
+subsection \<open>Compute bitlen of integers\<close>
definition bitlen :: "int \<Rightarrow> int" where
"bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"
@@ -820,7 +820,7 @@
{
assume "0 > x"
have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all
- also have "... < log 2 (-x)" using `0 > x` by auto
+ also have "... < log 2 (-x)" using \<open>0 > x\<close> by auto
finally have "-1 < log 2 (-x)" .
} thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
qed
@@ -830,22 +830,22 @@
shows "2 ^ nat (bitlen x - 1) \<le> x \<and> x < 2 ^ nat (bitlen x)"
proof
have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"
- using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] `x > 0`
+ using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] \<open>x > 0\<close>
using real_nat_eq_real[of "floor (log 2 (real x))"]
by simp
also have "... \<le> 2 powr log 2 (real x)"
by simp
also have "... = real x"
- using `0 < x` by simp
+ using \<open>0 < x\<close> by simp
finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp
- thus "2 ^ nat (bitlen x - 1) \<le> x" using `x > 0`
+ thus "2 ^ nat (bitlen x - 1) \<le> x" using \<open>x > 0\<close>
by (simp add: bitlen_def)
next
- have "x \<le> 2 powr (log 2 x)" using `x > 0` by simp
+ have "x \<le> 2 powr (log 2 x)" using \<open>x > 0\<close> by simp
also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"
apply (simp add: powr_realpow[symmetric])
- using `x > 0` by simp
- finally show "x < 2 ^ nat (bitlen x)" using `x > 0`
+ using \<open>x > 0\<close> by simp
+ finally show "x < 2 ^ nat (bitlen x)" using \<open>x > 0\<close>
by (simp add: bitlen_def ac_simps)
qed
@@ -874,7 +874,7 @@
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 `f \<noteq> float_of 0`])
+ by (rule f_def[THEN denormalize_shift, OF \<open>f \<noteq> float_of 0\<close>])
ultimately show ?thesis by (simp add: abs_mult)
qed
@@ -890,28 +890,28 @@
next
def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"
then have "0 \<le> n"
- using `2 \<le> x` by simp
+ using \<open>2 \<le> x\<close> by simp
assume "x mod 2 \<noteq> 0"
- with `2 \<le> x` have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
- with `2 \<le> x` have "x \<noteq> 2^nat n" by (cases "nat n") auto
+ with \<open>2 \<le> x\<close> have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
+ with \<open>2 \<le> x\<close> have "x \<noteq> 2^nat n" by (cases "nat n") auto
moreover
{ have "real (2^nat n :: int) = 2 powr (nat n)"
by (simp add: powr_realpow)
also have "\<dots> \<le> 2 powr (log 2 x)"
- using `2 \<le> x` by (simp add: n_def del: powr_log_cancel)
- finally have "2^nat n \<le> x" using `2 \<le> x` by simp }
+ using \<open>2 \<le> x\<close> by (simp add: n_def del: powr_log_cancel)
+ finally have "2^nat n \<le> x" using \<open>2 \<le> x\<close> by simp }
ultimately have "2^nat n \<le> x - 1" by simp
then have "2^nat n \<le> real (x - 1)"
unfolding real_of_int_le_iff[symmetric] by simp
{ have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"
- using `0 \<le> n` by (simp add: log_nat_power)
+ using \<open>0 \<le> n\<close> by (simp add: log_nat_power)
also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"
- using `2^nat n \<le> real (x - 1)` `0 \<le> n` `2 \<le> x` by (auto intro: floor_mono)
+ using \<open>2^nat n \<le> real (x - 1)\<close> \<open>0 \<le> n\<close> \<open>2 \<le> x\<close> by (auto intro: floor_mono)
finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }
moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"
- using `2 \<le> x` by (auto simp add: n_def intro!: floor_mono)
+ using \<open>2 \<le> x\<close> by (auto simp add: n_def intro!: floor_mono)
ultimately show "\<lfloor>log 2 (x - x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>"
- unfolding n_def `x mod 2 = 1` by auto
+ unfolding n_def \<open>x mod 2 = 1\<close> by auto
qed
finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . }
moreover
@@ -934,7 +934,7 @@
hence "m \<noteq> 0" by auto
show ?thesis
proof (cases "0 \<le> e")
- case True thus ?thesis using `0 < m` by (simp add: bitlen_def)
+ case True thus ?thesis using \<open>0 < m\<close> by (simp add: bitlen_def)
next
have "(1::int) < 2" by simp
case False let ?S = "2^(nat (-e))"
@@ -945,8 +945,8 @@
hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
hence "?S \<le> real m" unfolding mult.assoc by auto
hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
- from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
- have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric]
+ from this bitlen_bounds[OF \<open>0 < m\<close>, THEN conjunct2]
+ have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF \<open>1 < 2\<close>, symmetric]
by (rule order_le_less_trans)
hence "-e < bitlen m" using False by auto
thus ?thesis by auto
@@ -959,22 +959,22 @@
proof -
let ?B = "2^nat(bitlen m - 1)"
- have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
+ have "?B \<le> m" using bitlen_bounds[OF \<open>0 <m\<close>] ..
hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
thus "1 \<le> real m / ?B" by auto
have "m \<noteq> 0" using assms by auto
- have "0 \<le> bitlen m - 1" using `0 < m` by (auto simp: bitlen_def)
+ have "0 \<le> bitlen m - 1" using \<open>0 < m\<close> by (auto simp: bitlen_def)
- have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
- also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using `0 < m` by (auto simp: bitlen_def)
- also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
+ have "m < 2^nat(bitlen m)" using bitlen_bounds[OF \<open>0 <m\<close>] ..
+ also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using \<open>0 < m\<close> 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 m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
thus "real m / ?B < 2" by auto
qed
-subsection {* Truncating Real Numbers*}
+subsection \<open>Truncating Real Numbers\<close>
definition truncate_down::"nat \<Rightarrow> real \<Rightarrow> real" where
"truncate_down prec x = round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"
@@ -1051,7 +1051,7 @@
} ultimately show ?thesis by arith
qed
-subsection {* Truncating Floats*}
+subsection \<open>Truncating Floats\<close>
lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_up
by (simp add: truncate_up_def)
@@ -1093,7 +1093,7 @@
hide_fact (open) compute_float_round_up
-subsection {* Approximation of positive rationals *}
+subsection \<open>Approximation of positive rationals\<close>
lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"
by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)
@@ -1146,21 +1146,21 @@
def x' \<equiv> "x * 2 ^ nat l"
have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)
moreover have "real x * 2 powr real l = real x'"
- by (simp add: powr_realpow[symmetric] `0 \<le> l` x'_def)
+ by (simp add: powr_realpow[symmetric] \<open>0 \<le> l\<close> x'_def)
ultimately show ?thesis
- using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0`
+ using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] \<open>0 \<le> l\<close> \<open>y \<noteq> 0\<close>
l_def[symmetric, THEN meta_eq_to_obj_eq]
by transfer (auto simp add: floor_divide_eq_div [symmetric] round_up_def)
next
assume "\<not> 0 \<le> l"
def y' \<equiv> "y * 2 ^ nat (- l)"
- from `y \<noteq> 0` have "y' \<noteq> 0" by (simp add: y'_def)
+ from \<open>y \<noteq> 0\<close> have "y' \<noteq> 0" by (simp add: y'_def)
have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)
moreover have "real x * real (2::int) powr real l / real y = x / real y'"
- using `\<not> 0 \<le> l`
+ using \<open>\<not> 0 \<le> l\<close>
by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps)
ultimately show ?thesis
- using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0`
+ using ceil_divide_floor_conv[of y' x] \<open>\<not> 0 \<le> l\<close> \<open>y' \<noteq> 0\<close> \<open>y \<noteq> 0\<close>
l_def[symmetric, THEN meta_eq_to_obj_eq]
by transfer
(auto simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div [symmetric])
@@ -1214,7 +1214,7 @@
by transfer (simp add: round_down_uminus_eq)
hide_fact (open) compute_rapprox_rat
-subsection {* Division *}
+subsection \<open>Division\<close>
definition "real_divl prec a b = round_down (int prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)"
@@ -1250,7 +1250,7 @@
hide_fact (open) compute_float_divr
-subsection {* Approximate Power *}
+subsection \<open>Approximate Power\<close>
lemma div2_less_self[termination_simp]: fixes n::nat shows "odd n \<Longrightarrow> n div 2 < n"
by (simp add: odd_pos)
@@ -1306,9 +1306,9 @@
also have "\<dots> = x ^ (Suc n div 2 * 2)"
by (simp add: power_mult[symmetric])
also have "Suc n div 2 * 2 = Suc n"
- using `odd n` by presburger
+ using \<open>odd n\<close> by presburger
finally have ?case
- using `odd n`
+ using \<open>odd n\<close>
by (auto intro!: truncate_down_le simp del: odd_Suc_div_two)
} thus ?case
by (auto intro!: truncate_down_le mult_left_mono 2 mult_nonneg_nonneg power_down_nonneg)
@@ -1320,14 +1320,14 @@
{
assume "odd n"
hence "Suc n = Suc n div 2 * 2"
- using `odd n` even_Suc by presburger
+ using \<open>odd n\<close> even_Suc by presburger
hence "x ^ Suc n \<le> (x ^ (Suc n div 2))\<^sup>2"
by (simp add: power_mult[symmetric])
also have "\<dots> \<le> (power_up p x (Suc n div 2))\<^sup>2"
- using 2 `odd n`
+ using 2 \<open>odd n\<close>
by (auto intro: power_mono simp del: odd_Suc_div_two )
finally have ?case
- using `odd n`
+ using \<open>odd n\<close>
by (auto intro!: truncate_up_le simp del: odd_Suc_div_two )
} thus ?case
by (auto intro!: truncate_up_le mult_left_mono 2)
@@ -1350,7 +1350,7 @@
by transfer simp
-subsection {* Approximate Addition *}
+subsection \<open>Approximate Addition\<close>
definition "plus_down prec x y = truncate_down prec (x + y)"
@@ -1432,7 +1432,7 @@
also note b_le_1
finally have b_less_1: "b * 2 powr real p < 1" .
- from b_less_1 `b > 0` have floor_eq: "\<lfloor>b * 2 powr real p\<rfloor> = 0" "\<lfloor>sgn b / 2\<rfloor> = 0"
+ from b_less_1 \<open>b > 0\<close> have floor_eq: "\<lfloor>b * 2 powr real p\<rfloor> = 0" "\<lfloor>sgn b / 2\<rfloor> = 0"
by (simp_all add: floor_eq_iff)
have "\<lfloor>(a + b) * 2 powr q\<rfloor> = \<lfloor>(a + b) * 2 powr p * 2 powr (q - p)\<rfloor>"
@@ -1474,12 +1474,12 @@
also have "\<dots> = \<lfloor>(2 * ai + b * 2 powr (p + 1)) / real ((2::int) ^ nat (p - q + 1))\<rfloor>"
using assms by (simp add: algebra_simps powr_realpow[symmetric])
also have "\<dots> = \<lfloor>(2 * ai - 1) / real ((2::int) ^ nat (p - q + 1))\<rfloor>"
- using `b < 0` assms
+ using \<open>b < 0\<close> assms
by (simp add: floor_divide_eq_div floor_eq floor_divide_real_eq_div
del: real_of_int_mult real_of_int_power real_of_int_diff)
also have "\<dots> = \<lfloor>(2 * ai - 1) * 2 powr (q - p - 1)\<rfloor>"
using assms by (simp add: algebra_simps divide_powr_uminus powr_realpow[symmetric])
- finally have ?thesis using `b < 0` by simp
+ finally have ?thesis using \<open>b < 0\<close> by simp
} ultimately show ?thesis by arith
qed
@@ -1495,37 +1495,37 @@
def k \<equiv> "\<lfloor>log 2 \<bar>ai\<bar>\<rfloor>"
hence "\<lfloor>log 2 \<bar>ai\<bar>\<rfloor> = k" by simp
hence k: "2 powr k \<le> \<bar>ai\<bar>" "\<bar>ai\<bar> < 2 powr (k + 1)"
- by (simp_all add: floor_log_eq_powr_iff `ai \<noteq> 0`)
+ by (simp_all add: floor_log_eq_powr_iff \<open>ai \<noteq> 0\<close>)
have "k \<ge> 0"
using assms by (auto simp: k_def)
def r \<equiv> "\<bar>ai\<bar> - 2 ^ nat k"
have r: "0 \<le> r" "r < 2 powr k"
- using `k \<ge> 0` k
+ using \<open>k \<ge> 0\<close> k
by (auto simp: r_def k_def algebra_simps powr_add abs_if powr_int)
hence "r \<le> (2::int) ^ nat k - 1"
- using `k \<ge> 0` by (auto simp: powr_int)
- from this[simplified real_of_int_le_iff[symmetric]] `0 \<le> k`
+ using \<open>k \<ge> 0\<close> by (auto simp: powr_int)
+ from this[simplified real_of_int_le_iff[symmetric]] \<open>0 \<le> k\<close>
have r_le: "r \<le> 2 powr k - 1"
by (auto simp: algebra_simps powr_int simp del: real_of_int_le_iff)
have "\<bar>ai\<bar> = 2 powr k + r"
- using `k \<ge> 0` by (auto simp: k_def r_def powr_realpow[symmetric])
+ using \<open>k \<ge> 0\<close> by (auto simp: k_def r_def powr_realpow[symmetric])
have pos: "\<And>b::real. abs b < 1 \<Longrightarrow> 0 < 2 powr k + (r + b)"
- using `0 \<le> k` `ai \<noteq> 0`
+ using \<open>0 \<le> k\<close> \<open>ai \<noteq> 0\<close>
by (auto simp add: r_def powr_realpow[symmetric] abs_if sgn_if algebra_simps
split: split_if_asm)
have less: "\<bar>sgn ai * b\<bar> < 1"
and less': "\<bar>sgn (sgn ai * b) / 2\<bar> < 1"
- using `abs b \<le> _` by (auto simp: abs_if sgn_if split: split_if_asm)
+ using \<open>abs b \<le> _\<close> by (auto simp: abs_if sgn_if split: split_if_asm)
have floor_eq: "\<And>b::real. abs b \<le> 1 / 2 \<Longrightarrow>
\<lfloor>log 2 (1 + (r + b) / 2 powr k)\<rfloor> = (if r = 0 \<and> b < 0 then -1 else 0)"
- using `k \<ge> 0` r r_le
+ using \<open>k \<ge> 0\<close> r r_le
by (auto simp: floor_log_eq_powr_iff powr_minus_divide field_simps sgn_if)
- from `real \<bar>ai\<bar> = _` have "\<bar>ai + b\<bar> = 2 powr k + (r + sgn ai * b)"
- using `abs b <= _` `0 \<le> k` r
+ from \<open>real \<bar>ai\<bar> = _\<close> have "\<bar>ai + b\<bar> = 2 powr k + (r + sgn ai * b)"
+ using \<open>abs b <= _\<close> \<open>0 \<le> k\<close> r
by (auto simp add: sgn_if abs_if)
also have "\<lfloor>log 2 \<dots>\<rfloor> = \<lfloor>log 2 (2 powr k + r + sgn (sgn ai * b) / 2)\<rfloor>"
proof -
@@ -1537,14 +1537,14 @@
also
let ?if = "if r = 0 \<and> sgn ai * b < 0 then -1 else 0"
have "\<lfloor>log 2 (1 + (r + sgn ai * b) / 2 powr k)\<rfloor> = ?if"
- using `abs b <= _`
+ using \<open>abs b <= _\<close>
by (intro floor_eq) (auto simp: abs_mult sgn_if)
also
have "\<dots> = \<lfloor>log 2 (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k)\<rfloor>"
by (subst floor_eq) (auto simp: sgn_if)
also have "k + \<dots> = \<lfloor>log 2 (2 powr k * (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k))\<rfloor>"
unfolding floor_add2[symmetric]
- using pos[OF less'] `abs b \<le> _`
+ using pos[OF less'] \<open>abs b \<le> _\<close>
by (simp add: field_simps add_log_eq_powr)
also have "2 powr k * (1 + (r + sgn (sgn ai * b) / 2) / 2 powr k) =
2 powr k + r + sgn (sgn ai * b) / 2"
@@ -1552,7 +1552,7 @@
finally show ?thesis .
qed
also have "2 powr k + r + sgn (sgn ai * b) / 2 = \<bar>ai + sgn b / 2\<bar>"
- unfolding `real \<bar>ai\<bar> = _`[symmetric] using `ai \<noteq> 0`
+ unfolding \<open>real \<bar>ai\<bar> = _\<close>[symmetric] using \<open>ai \<noteq> 0\<close>
by (auto simp: abs_if sgn_if algebra_simps)
finally show ?thesis .
qed
@@ -1590,7 +1590,7 @@
by simp
finally have b_less_quarter: "\<bar>?b\<bar> < 1/4 * 2 powr real e1"
by (simp add: powr_add field_simps powr_divide2[symmetric] powr_numeral abs_mult)
- also have "1/4 < \<bar>real m1\<bar> / 2" using `m1 \<noteq> 0` by simp
+ also have "1/4 < \<bar>real m1\<bar> / 2" using \<open>m1 \<noteq> 0\<close> by simp
finally have b_less_half_a: "\<bar>?b\<bar> < 1/2 * \<bar>?a\<bar>"
by (simp add: algebra_simps powr_mult_base abs_mult)
hence a_half_less_sum: "\<bar>?a\<bar> / 2 < \<bar>?sum\<bar>"
@@ -1600,7 +1600,7 @@
by simp_all
have "\<bar>real (Float m1 e1)\<bar> \<ge> 1/4 * 2 powr real e1"
- using `m1 \<noteq> 0`
+ using \<open>m1 \<noteq> 0\<close>
by (auto simp: powr_add powr_int bitlen_nonneg divide_right_mono abs_mult)
hence "?sum \<noteq> 0" using b_less_quarter
by (rule sum_neq_zeroI)
@@ -1608,16 +1608,16 @@
unfolding sum_eq by (simp add: abs_mult zero_less_mult_iff)
have "\<bar>real ?m1\<bar> \<ge> 2 ^ Suc k1" "\<bar>?m2'\<bar> < 2 ^ Suc k1"
- using `m1 \<noteq> 0` `m2 \<noteq> 0` by (auto simp: sgn_if less_1_mult abs_mult simp del: power.simps)
+ using \<open>m1 \<noteq> 0\<close> \<open>m2 \<noteq> 0\<close> by (auto simp: sgn_if less_1_mult abs_mult simp del: power.simps)
hence sum'_nz: "?m1 + ?m2' \<noteq> 0"
by (intro sum_neq_zeroI)
have "\<lfloor>log 2 \<bar>real (Float m1 e1) + real (Float m2 e2)\<bar>\<rfloor> = \<lfloor>log 2 \<bar>?m1 + ?m2\<bar>\<rfloor> + ?e"
- using `?m1 + ?m2 \<noteq> 0`
+ using \<open>?m1 + ?m2 \<noteq> 0\<close>
unfolding floor_add[symmetric] sum_eq
by (simp add: abs_mult log_mult)
also have "\<lfloor>log 2 \<bar>?m1 + ?m2\<bar>\<rfloor> = \<lfloor>log 2 \<bar>?m1 + sgn (real m2 * 2 powr ?shift) / 2\<bar>\<rfloor>"
- using abs_m2_less_half `m1 \<noteq> 0`
+ using abs_m2_less_half \<open>m1 \<noteq> 0\<close>
by (intro log2_abs_int_add_less_half_sgn_eq) (auto simp: abs_mult)
also have "sgn (real m2 * 2 powr ?shift) = sgn m2"
by (auto simp: sgn_if zero_less_mult_iff less_not_sym)
@@ -1625,7 +1625,7 @@
have "\<bar>?m1 + ?m2'\<bar> * 2 powr ?e = \<bar>?m1 * 2 + sgn m2\<bar> * 2 powr (?e - 1)"
by (auto simp: field_simps powr_minus[symmetric] powr_divide2[symmetric] powr_mult_base)
hence "\<lfloor>log 2 \<bar>?m1 + ?m2'\<bar>\<rfloor> + ?e = \<lfloor>log 2 \<bar>real (Float (?m1 * 2 + sgn m2) (?e - 1))\<bar>\<rfloor>"
- using `?m1 + ?m2' \<noteq> 0`
+ using \<open>?m1 + ?m2' \<noteq> 0\<close>
unfolding floor_add[symmetric]
by (simp add: log_add_eq_powr abs_mult_pos)
finally
@@ -1645,16 +1645,16 @@
by (simp add: abs_mult powr_add[symmetric] algebra_simps powr_mult_base)
next
have "e1 + \<lfloor>log 2 \<bar>real m1\<bar>\<rfloor> - 1 = \<lfloor>log 2 \<bar>?a\<bar>\<rfloor> - 1"
- using `m1 \<noteq> 0`
+ using \<open>m1 \<noteq> 0\<close>
by (simp add: floor_add2[symmetric] algebra_simps log_mult abs_mult del: floor_add2)
also have "\<dots> \<le> \<lfloor>log 2 \<bar>?a + ?b\<bar>\<rfloor>"
- using a_half_less_sum `m1 \<noteq> 0` `?sum \<noteq> 0`
+ using a_half_less_sum \<open>m1 \<noteq> 0\<close> \<open>?sum \<noteq> 0\<close>
unfolding floor_subtract[symmetric]
by (auto simp add: log_minus_eq_powr powr_minus_divide
intro!: floor_mono)
finally
have "int p - \<lfloor>log 2 \<bar>?a + ?b\<bar>\<rfloor> \<le> p - (bitlen \<bar>m1\<bar>) - e1 + 2"
- by (auto simp: algebra_simps bitlen_def `m1 \<noteq> 0`)
+ by (auto simp: algebra_simps bitlen_def \<open>m1 \<noteq> 0\<close>)
also have "\<dots> \<le> 1 - ?e"
using bitlen_nonneg[of "\<bar>m1\<bar>"] by (simp add: k1_def)
finally show "?f \<le> - ?e" by simp
@@ -1707,7 +1707,7 @@
by (metis mantissa_0 zero_float.abs_eq)
-subsection {* Lemmas needed by Approximate *}
+subsection \<open>Lemmas needed by Approximate\<close>
lemma Float_num[simp]: shows
"real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and
@@ -1804,7 +1804,7 @@
have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp
also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"
- using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg `mantissa x \<noteq> 0`
+ using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg \<open>mantissa x \<noteq> 0\<close>
by (auto simp del: real_of_int_abs simp add: powr_int)
finally show ?thesis by (simp add: powr_add)
qed
@@ -1813,7 +1813,7 @@
assumes "0 < x" "x \<le> 1" "prec \<ge> 1"
shows "1 \<le> real_divl prec 1 x"
proof -
- have "log 2 x \<le> real prec + real \<lfloor>log 2 x\<rfloor>" using `prec \<ge> 1` by arith
+ have "log 2 x \<le> real prec + real \<lfloor>log 2 x\<rfloor>" using \<open>prec \<ge> 1\<close> by arith
from this assms show ?thesis
by (simp add: real_divl_def log_divide round_down_ge1)
qed
@@ -1827,7 +1827,7 @@
lemma real_divr_pos_less1_lower_bound: assumes "0 < x" and "x \<le> 1" shows "1 \<le> real_divr prec 1 x"
proof -
- have "1 \<le> 1 / x" using `0 < x` and `x <= 1` by auto
+ have "1 \<le> 1 / x" using \<open>0 < x\<close> and \<open>x <= 1\<close> by auto
also have "\<dots> \<le> real_divr prec 1 x" using real_divr[where x=1 and y=x] by auto
finally show ?thesis by auto
qed
@@ -1877,7 +1877,7 @@
using real_of_int_floor_add_one_ge[of "log 2 x"] assms
by (auto simp add: algebra_simps powr_divide2 intro!: mult_left_mono)
thus "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real ((2::int) ^ prec)"
- using `0 < x` by (simp add: powr_realpow)
+ using \<open>0 < x\<close> by (simp add: powr_realpow)
qed
hence "real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> 2 powr int prec"
by (auto simp: powr_realpow)
@@ -1885,14 +1885,14 @@
have "2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)"
using logless flogless by (auto intro!: floor_mono)
also have "2 powr real (int prec) \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>))"
- using assms `0 < x`
+ using assms \<open>0 < x\<close>
by (auto simp: algebra_simps)
finally have "truncate_up prec x \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>)) * 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)"
by simp
also have "\<dots> = 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>) - real (int prec - \<lfloor>log 2 y\<rfloor>))"
by (subst powr_add[symmetric]) simp
also have "\<dots> = y"
- using `0 < x` assms
+ using \<open>0 < x\<close> assms
by (simp add: powr_add)
also have "\<dots> \<le> truncate_up prec y"
by (rule truncate_up)
@@ -1910,8 +1910,8 @@
assumes "x \<le> 0" "0 \<le> y"
shows "truncate_up prec x \<le> truncate_up prec y"
proof -
- note truncate_up_nonpos[OF `x \<le> 0`]
- also note truncate_up_le[OF `0 \<le> y`]
+ note truncate_up_nonpos[OF \<open>x \<le> 0\<close>]
+ also note truncate_up_le[OF \<open>0 \<le> y\<close>]
finally show ?thesis .
qed
@@ -1922,7 +1922,7 @@
have "x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1) = x * inverse (2 powr ((real \<lfloor>log 2 x\<rfloor> + 1)))"
by (simp add: powr_divide2[symmetric] powr_add powr_minus inverse_eq_divide)
also have "\<dots> = 2 powr (log 2 x - (real \<lfloor>log 2 x\<rfloor>) - 1)"
- using `0 < x`
+ using \<open>0 < x\<close>
by (auto simp: field_simps powr_add powr_divide2[symmetric])
also have "\<dots> < 2 powr 0"
using real_of_int_floor_add_one_gt
@@ -1933,7 +1933,7 @@
by simp
moreover
have "0 \<le> \<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor>"
- using `x > 0` by auto
+ using \<open>x > 0\<close> by auto
ultimately have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> \<in> {0 ..< 1}"
by simp
also have "\<dots> \<subseteq> {0}" by auto
@@ -1947,8 +1947,8 @@
assumes "x \<le> y"
shows "truncate_down prec x \<le> truncate_down prec y"
proof -
- note truncate_down_le[OF `x \<le> 0`]
- also note truncate_down_nonneg[OF `0 \<le> y`]
+ note truncate_down_le[OF \<open>x \<le> 0\<close>]
+ also note truncate_down_nonneg[OF \<open>0 \<le> y\<close>]
finally show ?thesis .
qed
@@ -1976,33 +1976,33 @@
moreover
assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"
ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"
- unfolding atomize_conj abs_of_pos[OF `0 < x`] abs_of_pos[OF `0 < y`]
+ unfolding atomize_conj abs_of_pos[OF \<open>0 < x\<close>] abs_of_pos[OF \<open>0 < y\<close>]
by (metis floor_less_cancel linorder_cases not_le)
assume "prec \<noteq> 0" hence [simp]: "prec \<ge> Suc 0" by auto
have "2 powr (prec - 1) \<le> y * 2 powr real (prec - 1) / (2 powr log 2 y)"
- using `0 < y`
+ using \<open>0 < y\<close>
by simp
also have "\<dots> \<le> y * 2 powr real prec / (2 powr (real \<lfloor>log 2 y\<rfloor> + 1))"
- using `0 \<le> y` `0 \<le> x` assms(2)
+ using \<open>0 \<le> y\<close> \<open>0 \<le> x\<close> assms(2)
by (auto intro!: powr_mono divide_left_mono
simp: real_of_nat_diff powr_add
powr_divide2[symmetric])
also have "\<dots> = y * 2 powr real prec / (2 powr real \<lfloor>log 2 y\<rfloor> * 2)"
by (auto simp: powr_add)
finally have "(2 ^ (prec - 1)) \<le> \<lfloor>y * 2 powr real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)\<rfloor>"
- using `0 \<le> y`
+ using \<open>0 \<le> y\<close>
by (auto simp: powr_divide2[symmetric] le_floor_eq powr_realpow)
hence "(2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1) \<le> truncate_down prec y"
by (auto simp: truncate_down_def round_down_def)
moreover
{
- have "x = 2 powr (log 2 \<bar>x\<bar>)" using `0 < x` by simp
+ have "x = 2 powr (log 2 \<bar>x\<bar>)" using \<open>0 < x\<close> by simp
also have "\<dots> \<le> (2 ^ (prec )) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)"
using real_of_int_floor_add_one_ge[of "log 2 \<bar>x\<bar>"]
by (auto simp: powr_realpow[symmetric] powr_add[symmetric] algebra_simps)
also
have "2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor>)"
- using logless flogless `x > 0` `y > 0`
+ using logless flogless \<open>x > 0\<close> \<open>y > 0\<close>
by (auto intro!: floor_mono)
finally have "x \<le> (2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)"
by (auto simp: powr_realpow[symmetric] powr_divide2[symmetric] assms real_of_nat_diff)