--- a/CONTRIBUTORS Thu Jan 07 17:40:55 2016 +0000
+++ b/CONTRIBUTORS Thu Jan 07 17:42:01 2016 +0000
@@ -16,6 +16,10 @@
The Generalised Binomial Theorem.
The complex and real Gamma/log-Gamma/Digamma/Polygamma functions and their
most important properties.
+
+* Autumn 2015: Manuel Eberl, TUM
+ Proper definition of division (with remainder) for formal power series;
+ Euclidean Ring and GCD instance for formal power series.
* Autumn 2015: Florian Haftmann, TUM
Rewrite definitions for global interpretations and sublocale declarations.
--- a/NEWS Thu Jan 07 17:40:55 2016 +0000
+++ b/NEWS Thu Jan 07 17:42:01 2016 +0000
@@ -646,6 +646,9 @@
* Library/Periodic_Fun: a locale that provides convenient lemmas for
periodic functions.
+* Library/Formal_Power_Series: proper definition of division (with remainder)
+for formal power series; instances for Euclidean Ring and GCD.
+
* HOL-Imperative_HOL: obsolete theory Legacy_Mrec has been removed.
* HOL-Statespace: command 'statespace' uses mandatory qualifier for
--- a/src/HOL/Multivariate_Analysis/Gamma.thy Thu Jan 07 17:40:55 2016 +0000
+++ b/src/HOL/Multivariate_Analysis/Gamma.thy Thu Jan 07 17:42:01 2016 +0000
@@ -288,7 +288,7 @@
continuous.
This will later allow us to lift holomorphicity and continuity from the log-Gamma
- function to the inverse Gamma function, and from that to the Gamma function itself.
+ function to the inverse of the Gamma function, and from that to the Gamma function itself.
\<close>
definition ln_Gamma_series :: "('a :: {banach,real_normed_field,ln}) \<Rightarrow> nat \<Rightarrow> 'a" where
@@ -824,9 +824,8 @@
proof -
have "-Digamma (3/2 :: real) = -Digamma (of_nat 1 + 1/2)" by simp
also have "\<dots> = 2 * ln 2 + euler_mascheroni - 2" by (subst Digamma_half_integer) simp
- also from euler_mascheroni_approx have "euler_mascheroni \<le> (0.58::real)"
- by (simp add: abs_real_def split: split_if_asm)
- also from ln_2_bounds have "ln 2 < (0.7 :: real)" by simp
+ also note euler_mascheroni_less_13_over_22
+ also note ln2_le_25_over_36
finally show ?thesis by simp
qed
@@ -911,7 +910,7 @@
text \<open>
- We define a type class that captures all the fundamental properties of the inverse Gamma function
+ We define a type class that captures all the fundamental properties of the inverse of the Gamma function
and defines the Gamma function upon that. This allows us to instantiate the type class both for
the reals and for the complex numbers with a minimal amount of proof duplication.
\<close>
@@ -2267,7 +2266,7 @@
subsection \<open>Limits and residues\<close>
text \<open>
- The inverse Gamma function has simple zeros:
+ The inverse of the Gamma function has simple zeros:
\<close>
lemma rGamma_zeros:
@@ -2285,7 +2284,7 @@
text \<open>
- The simple zeros of the inverse Gamma function correspond to simple poles of the Gamma function,
+ The simple zeros of the inverse of the Gamma function correspond to simple poles of the Gamma function,
and their residues can easily be computed from the limit we have just proven:
\<close>
@@ -2452,6 +2451,60 @@
finally show ?thesis by (simp add: Gamma_def)
qed
+subsubsection \<open>Binomial coefficient form\<close>
+
+lemma Gamma_binomial:
+ "(\<lambda>n. ((z + of_nat n) gchoose n) * exp (-z * of_real (ln (of_nat n)))) \<longlonglongrightarrow> rGamma (z+1)"
+proof (cases "z = 0")
+ case False
+ show ?thesis
+ proof (rule Lim_transform_eventually)
+ let ?powr = "\<lambda>a b. exp (b * of_real (ln (of_nat a)))"
+ show "eventually (\<lambda>n. rGamma_series z n / z =
+ ((z + of_nat n) gchoose n) * ?powr n (-z)) sequentially"
+ proof (intro always_eventually allI)
+ fix n :: nat
+ from False have "((z + of_nat n) gchoose n) = pochhammer z (Suc n) / z / fact n"
+ by (simp add: gbinomial_pochhammer' pochhammer_rec)
+ also have "pochhammer z (Suc n) / z / fact n * ?powr n (-z) = rGamma_series z n / z"
+ by (simp add: rGamma_series_def divide_simps exp_minus)
+ finally show "rGamma_series z n / z = ((z + of_nat n) gchoose n) * ?powr n (-z)" ..
+ qed
+
+ from False have "(\<lambda>n. rGamma_series z n / z) \<longlonglongrightarrow> rGamma z / z" by (intro tendsto_intros)
+ also from False have "rGamma z / z = rGamma (z + 1)" using rGamma_plus1[of z]
+ by (simp add: field_simps)
+ finally show "(\<lambda>n. rGamma_series z n / z) \<longlonglongrightarrow> rGamma (z+1)" .
+ qed
+qed (simp_all add: binomial_gbinomial [symmetric])
+
+lemma fact_binomial_limit:
+ "(\<lambda>n. of_nat ((k + n) choose n) / of_nat (n ^ k) :: 'a :: Gamma) \<longlonglongrightarrow> 1 / fact k"
+proof (rule Lim_transform_eventually)
+ have "(\<lambda>n. of_nat ((k + n) choose n) / of_real (exp (of_nat k * ln (real_of_nat n))))
+ \<longlonglongrightarrow> 1 / Gamma (of_nat (Suc k) :: 'a)" (is "?f \<longlonglongrightarrow> _")
+ using Gamma_binomial[of "of_nat k :: 'a"]
+ by (simp add: binomial_gbinomial add_ac Gamma_def divide_simps exp_of_real [symmetric] exp_minus)
+ also have "Gamma (of_nat (Suc k)) = fact k" by (rule Gamma_fact)
+ finally show "?f \<longlonglongrightarrow> 1 / fact k" .
+
+ show "eventually (\<lambda>n. ?f n = of_nat ((k + n) choose n) / of_nat (n ^ k)) sequentially"
+ using eventually_gt_at_top[of "0::nat"]
+ proof eventually_elim
+ fix n :: nat assume n: "n > 0"
+ from n have "exp (real_of_nat k * ln (real_of_nat n)) = real_of_nat (n^k)"
+ by (simp add: exp_of_nat_mult)
+ thus "?f n = of_nat ((k + n) choose n) / of_nat (n ^ k)" by simp
+ qed
+qed
+
+lemma binomial_asymptotic:
+ "(\<lambda>n. of_nat ((k + n) choose n) / (of_nat (n ^ k) / fact k) :: 'a :: Gamma) \<longlonglongrightarrow> 1"
+ using tendsto_mult[OF fact_binomial_limit[of k] tendsto_const[of "fact k :: 'a"]] by simp
+
+
+subsection \<open>The Weierstraß product formula for the sine\<close>
+
lemma sin_product_formula_complex:
fixes z :: complex
shows "(\<lambda>n. of_real pi * z * (\<Prod>k=1..n. 1 - z^2 / of_nat k^2)) \<longlonglongrightarrow> sin (of_real pi * z)"
@@ -2496,6 +2549,9 @@
using tendsto_divide[OF sin_product_formula_real[of x] tendsto_const[of "pi * x"]] assms
by simp
+
+subsection \<open>The Solution to the Basel problem\<close>
+
theorem inverse_squares_sums: "(\<lambda>n. 1 / (n + 1)\<^sup>2) sums (pi\<^sup>2 / 6)"
proof -
def P \<equiv> "\<lambda>x n. (\<Prod>k=1..n. 1 - x^2 / of_nat k^2 :: real)"
@@ -2589,56 +2645,4 @@
qed
-
-subsection \<open>Binomial coefficient form\<close>
-
-lemma Gamma_binomial:
- "(\<lambda>n. ((z + of_nat n) gchoose n) * exp (-z * of_real (ln (of_nat n)))) \<longlonglongrightarrow> rGamma (z+1)"
-proof (cases "z = 0")
- case False
- show ?thesis
- proof (rule Lim_transform_eventually)
- let ?powr = "\<lambda>a b. exp (b * of_real (ln (of_nat a)))"
- show "eventually (\<lambda>n. rGamma_series z n / z =
- ((z + of_nat n) gchoose n) * ?powr n (-z)) sequentially"
- proof (intro always_eventually allI)
- fix n :: nat
- from False have "((z + of_nat n) gchoose n) = pochhammer z (Suc n) / z / fact n"
- by (simp add: gbinomial_pochhammer' pochhammer_rec)
- also have "pochhammer z (Suc n) / z / fact n * ?powr n (-z) = rGamma_series z n / z"
- by (simp add: rGamma_series_def divide_simps exp_minus)
- finally show "rGamma_series z n / z = ((z + of_nat n) gchoose n) * ?powr n (-z)" ..
- qed
-
- from False have "(\<lambda>n. rGamma_series z n / z) \<longlonglongrightarrow> rGamma z / z" by (intro tendsto_intros)
- also from False have "rGamma z / z = rGamma (z + 1)" using rGamma_plus1[of z]
- by (simp add: field_simps)
- finally show "(\<lambda>n. rGamma_series z n / z) \<longlonglongrightarrow> rGamma (z+1)" .
- qed
-qed (simp_all add: binomial_gbinomial [symmetric])
-
-lemma fact_binomial_limit:
- "(\<lambda>n. of_nat ((k + n) choose n) / of_nat (n ^ k) :: 'a :: Gamma) \<longlonglongrightarrow> 1 / fact k"
-proof (rule Lim_transform_eventually)
- have "(\<lambda>n. of_nat ((k + n) choose n) / of_real (exp (of_nat k * ln (real_of_nat n))))
- \<longlonglongrightarrow> 1 / Gamma (of_nat (Suc k) :: 'a)" (is "?f \<longlonglongrightarrow> _")
- using Gamma_binomial[of "of_nat k :: 'a"]
- by (simp add: binomial_gbinomial add_ac Gamma_def divide_simps exp_of_real [symmetric] exp_minus)
- also have "Gamma (of_nat (Suc k)) = fact k" by (rule Gamma_fact)
- finally show "?f \<longlonglongrightarrow> 1 / fact k" .
-
- show "eventually (\<lambda>n. ?f n = of_nat ((k + n) choose n) / of_nat (n ^ k)) sequentially"
- using eventually_gt_at_top[of "0::nat"]
- proof eventually_elim
- fix n :: nat assume n: "n > 0"
- from n have "exp (real_of_nat k * ln (real_of_nat n)) = real_of_nat (n^k)"
- by (simp add: exp_of_nat_mult)
- thus "?f n = of_nat ((k + n) choose n) / of_nat (n ^ k)" by simp
- qed
-qed
-
-lemma binomial_asymptotic:
- "(\<lambda>n. of_nat ((k + n) choose n) / (of_nat (n ^ k) / fact k) :: 'a :: Gamma) \<longlonglongrightarrow> 1"
- using tendsto_mult[OF fact_binomial_limit[of k] tendsto_const[of "fact k :: 'a"]] by simp
-
end
--- a/src/HOL/Multivariate_Analysis/Harmonic_Numbers.thy Thu Jan 07 17:40:55 2016 +0000
+++ b/src/HOL/Multivariate_Analysis/Harmonic_Numbers.thy Thu Jan 07 17:42:01 2016 +0000
@@ -28,35 +28,9 @@
lemma setsum_Suc_diff':
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
assumes "m \<le> n"
- shows "(\<Sum>i = m..<n. f(Suc i) - f i) = f n - f m"
+ shows "(\<Sum>i = m..<n. f (Suc i) - f i) = f n - f m"
using assms by (induct n) (auto simp: le_Suc_eq)
-lemma eval_fact:
- "fact 0 = 1"
- "fact (Suc 0) = 1"
- "fact (numeral n) = numeral n * fact (pred_numeral n)"
- by (simp, simp, simp_all only: numeral_eq_Suc fact_Suc,
- simp only: numeral_eq_Suc [symmetric] of_nat_numeral)
-
-lemma setsum_poly_horner_expand:
- "(\<Sum>k<(numeral n::nat). f k * x^k) = f 0 + (\<Sum>k<pred_numeral n. f (k+1) * x^k) * x"
- "(\<Sum>k<Suc 0. f k * x^k) = (f 0 :: 'a :: semiring_1)"
- "(\<Sum>k<(0::nat). f k * x^k) = 0"
-proof -
- {
- fix m :: nat
- have "(\<Sum>k<Suc m. f k * x^k) = f 0 + (\<Sum>k=Suc 0..<Suc m. f k * x^k)"
- by (subst atLeast0LessThan [symmetric], subst setsum_head_upt_Suc) simp_all
- also have "(\<Sum>k=Suc 0..<Suc m. f k * x^k) = (\<Sum>k<m. f (k+1) * x^k) * x"
- by (subst setsum_shift_bounds_Suc_ivl)
- (simp add: setsum_left_distrib algebra_simps atLeast0LessThan power_commutes)
- finally have "(\<Sum>k<Suc m. f k * x ^ k) = f 0 + (\<Sum>k<m. f (k + 1) * x ^ k) * x" .
- }
- from this[of "pred_numeral n"]
- show "(\<Sum>k<numeral n. f k * x^k) = f 0 + (\<Sum>k<pred_numeral n. f (k+1) * x^k) * x"
- by (simp add: numeral_eq_Suc)
-qed simp_all
-
subsection \<open>The Harmonic numbers\<close>
@@ -133,7 +107,7 @@
qed (simp_all add: harm_def)
-subsection \<open>The Euler–Mascheroni constant\<close>
+subsection \<open>The Euler--Mascheroni constant\<close>
text \<open>
The limit of the difference between the partial harmonic sum and the natural logarithm
@@ -269,10 +243,9 @@
qed
-subsection \<open>Approximation of the Euler--Mascheroni constant\<close>
+subsection \<open>Bounds on the Euler--Mascheroni constant\<close>
-(* FIXME: ugly *)
-(* TODO: Move ? *)
+(* TODO: Move? *)
lemma ln_inverse_approx_le:
assumes "(x::real) > 0" "a > 0"
shows "ln (x + a) - ln x \<le> a * (inverse x + inverse (x + a))/2" (is "_ \<le> ?A")
@@ -401,7 +374,7 @@
qed
also from sums have "\<dots> = -inv (n+2) / 2" by (simp add: sums_iff)
finally have "euler_mascheroni \<ge> (\<Sum>k\<le>n. D k) + 1 / (of_nat (2 * (n+2)))"
- by (simp add: inv_def field_simps of_nat_mult)
+ by (simp add: inv_def field_simps)
also have "(\<Sum>k\<le>n. D k) = harm (Suc n) - (\<Sum>k\<le>n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1)))"
unfolding harm_altdef D_def by (subst lessThan_Suc_atMost) (simp add: setsum.distrib setsum_subtractf)
also have "(\<Sum>k\<le>n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1))) = ln (of_nat (n+2))"
@@ -447,7 +420,10 @@
lemma euler_mascheroni_pos: "euler_mascheroni > (0::real)"
using euler_mascheroni_lower[of 0] ln_2_less_1 by (simp add: harm_def)
-lemma ln_approx_aux:
+context
+begin
+
+private lemma ln_approx_aux:
fixes n :: nat and x :: real
defines "y \<equiv> (x-1)/(x+1)"
assumes x: "x > 0" "x \<noteq> 1"
@@ -514,65 +490,13 @@
thus "abs (ln x - (approx + d)) \<le> d" by auto
qed
-context
-begin
-
-qualified lemma ln_approx_abs':
- assumes "x > (1::real)"
- assumes "(x-1)/(x+1) = y"
- assumes "y^2 = ysqr"
- assumes "(\<Sum>k<n. inverse (of_nat (2*k+1)) * ysqr^k) = approx"
- assumes "y*ysqr^n / (1 - ysqr) / of_nat (2*n+1) = d"
- assumes "d \<le> e"
- shows "abs (ln x - (2*y*approx + d)) \<le> e"
-proof -
- note ln_approx_abs[OF assms(1), of n]
- also note assms(2)
- also have "y^(2*n+1) = y*ysqr^n" by (simp add: assms(3)[symmetric] power_mult)
- also note assms(3)
- also note assms(5)
- also note assms(5)
- also note assms(6)
- also have "(\<Sum>k<n. 2*y^(2*k+1) / real_of_nat (2 * k + 1)) = (2*y) * approx"
- apply (subst assms(4)[symmetric], subst setsum_right_distrib)
- apply (simp add: assms(3)[symmetric] power_mult)
- apply (simp add: mult_ac divide_simps)?
- done
- finally show ?thesis .
-qed
-
-lemma ln_2_approx: "\<bar>ln 2 - 0.69314718055\<bar> < inverse (2 ^ 36 :: real)" (is ?thesis1)
- and ln_2_bounds: "ln (2::real) \<in> {0.693147180549..0.693147180561}" (is ?thesis2)
-proof -
- def approx \<equiv> "0.69314718055 :: real" and approx' \<equiv> "4465284211343447 / 6442043387911560 :: real"
- def d \<equiv> "inverse (195259926456::real)"
- have "dist (ln 2) approx \<le> dist (ln 2) approx' + dist approx' approx" by (rule dist_triangle)
- also have "\<bar>ln (2::real) - (2 * (1/3) * (651187280816108 / 626309773824735) +
- inverse 195259926456)\<bar> \<le> inverse 195259926456"
- proof (rule ln_approx_abs'[where n = 10])
- show "(1/3::real)^2 = 1/9" by (simp add: power2_eq_square)
- qed (simp_all add: eval_nat_numeral)
- hence A: "dist (ln 2) approx' \<le> d" by (simp add: dist_real_def approx'_def d_def)
- hence "dist (ln 2) approx' + dist approx' approx \<le> \<dots> + dist approx' approx"
- by (rule add_right_mono)
- also have "\<dots> < inverse (2 ^ 36)" by (simp add: dist_real_def approx'_def approx_def d_def)
- finally show ?thesis1 unfolding dist_real_def approx_def .
-
- from A have "ln 2 \<in> {approx' - d..approx' + d}"
- by (simp add: dist_real_def abs_real_def split: split_if_asm)
- also have "\<dots> \<subseteq> {0.693147180549..0.693147180561}"
- by (subst atLeastatMost_subset_iff, rule disjI2) (simp add: approx'_def d_def)
- finally show ?thesis2 .
-qed
-
end
-
lemma euler_mascheroni_bounds:
fixes n :: nat assumes "n \<ge> 1" defines "t \<equiv> harm n - ln (of_nat (Suc n)) :: real"
shows "euler_mascheroni \<in> {t + inverse (of_nat (2*(n+1)))..t + inverse (of_nat (2*n))}"
using assms euler_mascheroni_upper[of "n-1"] euler_mascheroni_lower[of "n-1"]
- unfolding t_def by (cases n) (simp_all add: harm_Suc t_def inverse_eq_divide of_nat_mult)
+ unfolding t_def by (cases n) (simp_all add: harm_Suc t_def inverse_eq_divide)
lemma euler_mascheroni_bounds':
fixes n :: nat assumes "n \<ge> 1" "ln (real_of_nat (Suc n)) \<in> {l<..<u}"
@@ -580,28 +504,29 @@
{harm n - u + inverse (of_nat (2*(n+1)))<..<harm n - l + inverse (of_nat (2*n))}"
using euler_mascheroni_bounds[OF assms(1)] assms(2) by auto
-lemma euler_mascheroni_approx:
- defines "approx \<equiv> 0.577257 :: real" and "e \<equiv> 0.000063 :: real"
- shows "abs (euler_mascheroni - approx :: real) < e"
- (is "abs (_ - ?approx) < ?e")
+
+text \<open>
+ Approximation of @{term "ln 2"}. The lower bound is accurate to about 0.03; the upper
+ bound is accurate to about 0.0015.
+\<close>
+lemma ln2_ge_two_thirds: "2/3 \<le> ln (2::real)"
+ and ln2_le_25_over_36: "ln (2::real) \<le> 25/36"
+ using ln_approx_bounds[of 2 1, simplified, simplified eval_nat_numeral, simplified] by simp_all
+
+
+text \<open>
+ Approximation of the Euler--Mascheroni constant. The lower bound is accurate to about 0.0015;
+ the upper bound is accurate to about 0.015.
+\<close>
+lemma euler_mascheroni_gt_19_over_33: "(euler_mascheroni :: real) > 19/33" (is ?th1)
+ and euler_mascheroni_less_13_over_22: "(euler_mascheroni :: real) < 13/22" (is ?th2)
proof -
- def l \<equiv> "47388813395531028639296492901910937/82101866951584879688289000000000000 :: real"
- def u \<equiv> "142196984054132045946501548559032969 / 246305600854754639064867000000000000 :: real"
- have impI: "P \<longrightarrow> Q" if Q for P Q using that by blast
- have hsum_63: "harm 63 = (310559566510213034489743057 / 65681493561267903750631200 ::real)"
- by (simp add: harm_expand)
- from harm_Suc[of 63] have hsum_64: "harm 64 =
- 623171679694215690971693339 / (131362987122535807501262400::real)"
- by (subst (asm) hsum_63) simp
- have "ln (64::real) = real (6::nat) * ln 2" by (subst ln_realpow[symmetric]) simp_all
- hence "ln (real_of_nat (Suc 63)) \<in> {4.158883083293<..<4.158883083367}" using ln_2_bounds by simp
- from euler_mascheroni_bounds'[OF _ this]
- have "(euler_mascheroni :: real) \<in> {l<..<u}"
- by (simp add: hsum_63 del: greaterThanLessThan_iff) (simp only: l_def u_def)
- also have "\<dots> \<subseteq> {approx - e<..<approx + e}"
- by (subst greaterThanLessThan_subseteq_greaterThanLessThan, rule impI)
- (simp add: approx_def e_def u_def l_def)
- finally show ?thesis by (simp add: abs_real_def)
+ have "ln (real (Suc 7)) = 3 * ln 2" by (simp add: ln_powr [symmetric] powr_numeral)
+ also from ln_approx_bounds[of 2 3] have "\<dots> \<in> {3*307/443<..<3*4615/6658}"
+ by (simp add: eval_nat_numeral)
+ finally have "ln (real (Suc 7)) \<in> \<dots>" .
+ from euler_mascheroni_bounds'[OF _ this] have "?th1 \<and> ?th2" by (simp_all add: harm_expand)
+ thus ?th1 ?th2 by blast+
qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Multivariate_Analysis/Summation.thy Thu Jan 07 17:40:55 2016 +0000
+++ b/src/HOL/Multivariate_Analysis/Summation.thy Thu Jan 07 17:42:01 2016 +0000
@@ -2,7 +2,7 @@
Author: Manuel Eberl, TU München
*)
-section \<open>Rounded dual logarithm\<close>
+section \<open>Radius of Convergence and Summation Tests\<close>
theory Summation
imports
@@ -16,6 +16,8 @@
various summability tests, lemmas to compute the radius of convergence etc.
\<close>
+subsection \<open>Rounded dual logarithm\<close>
+
(* This is required for the Cauchy condensation criterion *)
definition "natlog2 n = (if n = 0 then 0 else nat \<lfloor>log 2 (real_of_nat n)\<rfloor>)"
@@ -206,7 +208,7 @@
qed
-subsection \<open>Cauchy's condensation test\<close>
+subsubsection \<open>Cauchy's condensation test\<close>
context
fixes f :: "nat \<Rightarrow> real"
@@ -319,7 +321,7 @@
end
-subsection \<open>Summability of powers\<close>
+subsubsection \<open>Summability of powers\<close>
lemma abs_summable_complex_powr_iff:
"summable (\<lambda>n. norm (exp (of_real (ln (of_nat n)) * s))) \<longleftrightarrow> Re s < -1"
@@ -389,7 +391,7 @@
qed
-subsection \<open>Kummer's test\<close>
+subsubsection \<open>Kummer's test\<close>
lemma kummers_test_convergence:
fixes f p :: "nat \<Rightarrow> real"
@@ -480,7 +482,7 @@
qed
-subsection \<open>Ratio test\<close>
+subsubsection \<open>Ratio test\<close>
lemma ratio_test_convergence:
fixes f :: "nat \<Rightarrow> real"
@@ -511,7 +513,7 @@
qed (simp_all add: summable_const_iff)
-subsection \<open>Raabe's test\<close>
+subsubsection \<open>Raabe's test\<close>
lemma raabes_test_convergence:
fixes f :: "nat \<Rightarrow> real"
--- a/src/HOL/Multivariate_Analysis/ex/Approximations.thy Thu Jan 07 17:40:55 2016 +0000
+++ b/src/HOL/Multivariate_Analysis/ex/Approximations.thy Thu Jan 07 17:42:01 2016 +0000
@@ -1,38 +1,497 @@
-section \<open>Binary Approximations to Constants\<close>
+section \<open>Numeric approximations to Constants\<close>
theory Approximations
-imports Complex_Transcendental
+imports "../Complex_Transcendental" "../Harmonic_Numbers"
+begin
+
+lemma eval_fact:
+ "fact 0 = 1"
+ "fact (Suc 0) = 1"
+ "fact (numeral n) = numeral n * fact (pred_numeral n)"
+ by (simp, simp, simp_all only: numeral_eq_Suc fact_Suc,
+ simp only: numeral_eq_Suc [symmetric] of_nat_numeral)
+
+lemma setsum_poly_horner_expand:
+ "(\<Sum>k<(numeral n::nat). f k * x^k) = f 0 + (\<Sum>k<pred_numeral n. f (k+1) * x^k) * x"
+ "(\<Sum>k<Suc 0. f k * x^k) = (f 0 :: 'a :: semiring_1)"
+ "(\<Sum>k<(0::nat). f k * x^k) = 0"
+proof -
+ {
+ fix m :: nat
+ have "(\<Sum>k<Suc m. f k * x^k) = f 0 + (\<Sum>k=Suc 0..<Suc m. f k * x^k)"
+ by (subst atLeast0LessThan [symmetric], subst setsum_head_upt_Suc) simp_all
+ also have "(\<Sum>k=Suc 0..<Suc m. f k * x^k) = (\<Sum>k<m. f (k+1) * x^k) * x"
+ by (subst setsum_shift_bounds_Suc_ivl)
+ (simp add: setsum_left_distrib algebra_simps atLeast0LessThan power_commutes)
+ finally have "(\<Sum>k<Suc m. f k * x ^ k) = f 0 + (\<Sum>k<m. f (k + 1) * x ^ k) * x" .
+ }
+ from this[of "pred_numeral n"]
+ show "(\<Sum>k<numeral n. f k * x^k) = f 0 + (\<Sum>k<pred_numeral n. f (k+1) * x^k) * x"
+ by (simp add: numeral_eq_Suc)
+qed simp_all
+
+lemma power_less_one:
+ assumes "n > 0" "x \<ge> 0" "x < 1"
+ shows "x ^ n < (1::'a::linordered_semidom)"
+proof -
+ from assms consider "x > 0" | "x = 0" by force
+ thus ?thesis
+ proof cases
+ assume "x > 0"
+ with assms show ?thesis
+ by (cases n) (simp, hypsubst, rule power_Suc_less_one)
+ qed (insert assms, cases n, simp_all)
+qed
+
+lemma combine_bounds:
+ "x \<in> {a1..b1} \<Longrightarrow> y \<in> {a2..b2} \<Longrightarrow> a3 = a1 + a2 \<Longrightarrow> b3 = b1 + b2 \<Longrightarrow> x + y \<in> {a3..(b3::real)}"
+ "x \<in> {a1..b1} \<Longrightarrow> y \<in> {a2..b2} \<Longrightarrow> a3 = a1 - b2 \<Longrightarrow> b3 = b1 - a2 \<Longrightarrow> x - y \<in> {a3..(b3::real)}"
+ "c \<ge> 0 \<Longrightarrow> x \<in> {a..b} \<Longrightarrow> c * x \<in> {c*a..c*b}"
+ by (auto simp: mult_left_mono)
+
+lemma approx_coarsen:
+ "\<bar>x - a1\<bar> \<le> eps1 \<Longrightarrow> \<bar>a1 - a2\<bar> \<le> eps2 - eps1 \<Longrightarrow> \<bar>x - a2\<bar> \<le> (eps2 :: real)"
+ by simp
+
+
+subsection \<open>Approximation of $\ln 2$\<close>
+
+context
begin
-declare of_real_numeral [simp]
-
-subsection\<open>Approximation to pi\<close>
+qualified lemma ln_approx_abs':
+ assumes "x > (1::real)"
+ assumes "(x-1)/(x+1) = y"
+ assumes "y^2 = ysqr"
+ assumes "(\<Sum>k<n. inverse (of_nat (2*k+1)) * ysqr^k) = approx"
+ assumes "y*ysqr^n / (1 - ysqr) / of_nat (2*n+1) = d"
+ assumes "d \<le> e"
+ shows "abs (ln x - (2*y*approx + d)) \<le> e"
+proof -
+ note ln_approx_abs[OF assms(1), of n]
+ also note assms(2)
+ also have "y^(2*n+1) = y*ysqr^n" by (simp add: assms(3)[symmetric] power_mult)
+ also note assms(3)
+ also note assms(5)
+ also note assms(5)
+ also note assms(6)
+ also have "(\<Sum>k<n. 2*y^(2*k+1) / real_of_nat (2 * k + 1)) = (2*y) * approx"
+ apply (subst assms(4)[symmetric], subst setsum_right_distrib)
+ apply (simp add: assms(3)[symmetric] power_mult)
+ apply (simp add: mult_ac divide_simps)?
+ done
+ finally show ?thesis .
+qed
-lemma sin_pi6_straddle:
- assumes "0 \<le> a" "a \<le> b" "b \<le> 4" "sin(a/6) \<le> 1/2" "1/2 \<le> sin(b/6)"
- shows "a \<le> pi \<and> pi \<le> b"
+lemma ln_2_approx: "\<bar>ln 2 - 0.69314718055\<bar> < inverse (2 ^ 36 :: real)" (is ?thesis1)
+ and ln_2_bounds: "ln (2::real) \<in> {0.693147180549..0.693147180561}" (is ?thesis2)
+proof -
+ def approx \<equiv> "0.69314718055 :: real" and approx' \<equiv> "4465284211343447 / 6442043387911560 :: real"
+ def d \<equiv> "inverse (195259926456::real)"
+ have "dist (ln 2) approx \<le> dist (ln 2) approx' + dist approx' approx" by (rule dist_triangle)
+ also have "\<bar>ln (2::real) - (2 * (1/3) * (651187280816108 / 626309773824735) +
+ inverse 195259926456)\<bar> \<le> inverse 195259926456"
+ proof (rule ln_approx_abs'[where n = 10])
+ show "(1/3::real)^2 = 1/9" by (simp add: power2_eq_square)
+ qed (simp_all add: eval_nat_numeral)
+ hence A: "dist (ln 2) approx' \<le> d" by (simp add: dist_real_def approx'_def d_def)
+ hence "dist (ln 2) approx' + dist approx' approx \<le> \<dots> + dist approx' approx"
+ by (rule add_right_mono)
+ also have "\<dots> < inverse (2 ^ 36)" by (simp add: dist_real_def approx'_def approx_def d_def)
+ finally show ?thesis1 unfolding dist_real_def approx_def .
+
+ from A have "ln 2 \<in> {approx' - d..approx' + d}"
+ by (simp add: dist_real_def abs_real_def split: split_if_asm)
+ also have "\<dots> \<subseteq> {0.693147180549..0.693147180561}"
+ by (subst atLeastatMost_subset_iff, rule disjI2) (simp add: approx'_def d_def)
+ finally show ?thesis2 .
+qed
+
+end
+
+
+subsection \<open>Approximation of the Euler--Mascheroni constant\<close>
+
+lemma euler_mascheroni_approx:
+ defines "approx \<equiv> 0.577257 :: real" and "e \<equiv> 0.000063 :: real"
+ shows "abs (euler_mascheroni - approx :: real) < e"
+ (is "abs (_ - ?approx) < ?e")
proof -
- have *: "\<And>x::real. 0 < x & x < 7/5 \<Longrightarrow> 0 < sin x"
- using pi_ge_two
- by (auto intro: sin_gt_zero)
- have ab: "(b \<le> pi * 3 \<Longrightarrow> pi \<le> b)" "(a \<le> pi * 3 \<Longrightarrow> a \<le> pi)"
- using sin_mono_le_eq [of "pi/6" "b/6"] sin_mono_le_eq [of "a/6" "pi/6"] assms
- by (simp_all add: sin_30 power.power_Suc norm_divide)
- show ?thesis
- using assms Taylor_sin [of "a/6" 0] pi_ge_two
- by (auto simp: sin_30 power.power_Suc norm_divide intro: ab)
+ def l \<equiv> "47388813395531028639296492901910937/82101866951584879688289000000000000 :: real"
+ def u \<equiv> "142196984054132045946501548559032969 / 246305600854754639064867000000000000 :: real"
+ have impI: "P \<longrightarrow> Q" if Q for P Q using that by blast
+ have hsum_63: "harm 63 = (310559566510213034489743057 / 65681493561267903750631200 ::real)"
+ by (simp add: harm_expand)
+ from harm_Suc[of 63] have hsum_64: "harm 64 =
+ 623171679694215690971693339 / (131362987122535807501262400::real)"
+ by (subst (asm) hsum_63) simp
+ have "ln (64::real) = real (6::nat) * ln 2" by (subst ln_realpow[symmetric]) simp_all
+ hence "ln (real_of_nat (Suc 63)) \<in> {4.158883083293<..<4.158883083367}" using ln_2_bounds by simp
+ from euler_mascheroni_bounds'[OF _ this]
+ have "(euler_mascheroni :: real) \<in> {l<..<u}"
+ by (simp add: hsum_63 del: greaterThanLessThan_iff) (simp only: l_def u_def)
+ also have "\<dots> \<subseteq> {approx - e<..<approx + e}"
+ by (subst greaterThanLessThan_subseteq_greaterThanLessThan, rule impI)
+ (simp add: approx_def e_def u_def l_def)
+ finally show ?thesis by (simp add: abs_real_def)
+qed
+
+
+subsection \<open>Approximation to pi\<close>
+
+
+subsubsection \<open>Approximating the arctangent\<close>
+
+definition arctan_approx where
+ "arctan_approx n x = x * (\<Sum>k<n. (-(x^2))^k / real (2*k+1))"
+
+lemma arctan_series':
+ assumes "\<bar>x\<bar> \<le> 1"
+ shows "(\<lambda>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1))) sums arctan x"
+ using summable_arctan_series[OF assms] arctan_series[OF assms] by (simp add: sums_iff)
+
+lemma arctan_approx:
+ assumes x: "0 \<le> x" "x < 1" and n: "even n"
+ shows "arctan x - arctan_approx n x \<in> {0..x^(2*n+1) / (1-x^4)}"
+proof -
+ def c \<equiv> "\<lambda>k. 1 / (1+(4*real k + 2*real n)) - x\<^sup>2 / (3+(4*real k + 2*real n))"
+ from assms have "(\<lambda>k. (-1) ^ k * (1 / real (k * 2 + 1) * x^(k*2+1))) sums arctan x"
+ using arctan_series' by simp
+ also have "(\<lambda>k. (-1) ^ k * (1 / real (k * 2 + 1) * x^(k*2+1))) =
+ (\<lambda>k. x * ((- (x^2))^k / real (2*k+1)))"
+ by (simp add: power2_eq_square power_mult power_mult_distrib mult_ac power_minus')
+ finally have "(\<lambda>k. x * ((- x\<^sup>2) ^ k / real (2 * k + 1))) sums arctan x" .
+ from sums_split_initial_segment[OF this, of n]
+ have "(\<lambda>i. x * ((- x\<^sup>2) ^ (i + n) / real (2 * (i + n) + 1))) sums
+ (arctan x - arctan_approx n x)"
+ by (simp add: arctan_approx_def setsum_right_distrib)
+ from sums_group[OF this, of 2] assms
+ have sums: "(\<lambda>k. x * (x\<^sup>2)^n * (x^4)^k * c k) sums (arctan x - arctan_approx n x)"
+ by (simp add: algebra_simps power_add power_mult [symmetric] c_def)
+
+ from assms have "0 \<le> arctan x - arctan_approx n x"
+ by (intro sums_le[OF _ sums_zero sums] allI mult_nonneg_nonneg)
+ (auto intro!: frac_le power_le_one simp: c_def)
+ moreover {
+ from assms have "c k \<le> 1 - 0" for k unfolding c_def
+ by (intro diff_mono divide_nonneg_nonneg add_nonneg_nonneg) auto
+ with assms have "x * x\<^sup>2 ^ n * (x ^ 4) ^ k * c k \<le> x * x\<^sup>2 ^ n * (x ^ 4) ^ k * 1" for k
+ by (intro mult_left_mono mult_right_mono mult_nonneg_nonneg) simp_all
+ with assms have "arctan x - arctan_approx n x \<le> x * (x\<^sup>2)^n * (1 / (1 - x^4))"
+ by (intro sums_le[OF _ sums sums_mult[OF geometric_sums]] allI mult_left_mono)
+ (auto simp: power_less_one)
+ also have "x * (x^2)^n = x^(2*n+1)" by (simp add: power_mult power_add)
+ finally have "arctan x - arctan_approx n x \<le> x^(2*n+1) / (1 - x^4)" by simp
+ }
+ ultimately show ?thesis by simp
+qed
+
+lemma arctan_approx_def': "arctan_approx n (1/x) =
+ (\<Sum>k<n. inverse (real (2 * k + 1) * (- x\<^sup>2) ^ k)) / x"
+proof -
+ have "(-1)^k / b = 1 / ((-1)^k * b)" for k :: nat and b :: real
+ by (cases "even k") auto
+ thus ?thesis by (simp add: arctan_approx_def field_simps power_minus')
+qed
+
+lemma expand_arctan_approx:
+ "(\<Sum>k<(numeral n::nat). inverse (f k) * inverse (x ^ k)) =
+ inverse (f 0) + (\<Sum>k<pred_numeral n. inverse (f (k+1)) * inverse (x^k)) / x"
+ "(\<Sum>k<Suc 0. inverse (f k) * inverse (x^k)) = inverse (f 0 :: 'a :: field)"
+ "(\<Sum>k<(0::nat). inverse (f k) * inverse (x^k)) = 0"
+proof -
+ {
+ fix m :: nat
+ have "(\<Sum>k<Suc m. inverse (f k * x^k)) =
+ inverse (f 0) + (\<Sum>k=Suc 0..<Suc m. inverse (f k * x^k))"
+ by (subst atLeast0LessThan [symmetric], subst setsum_head_upt_Suc) simp_all
+ also have "(\<Sum>k=Suc 0..<Suc m. inverse (f k * x^k)) = (\<Sum>k<m. inverse (f (k+1) * x^k)) / x"
+ by (subst setsum_shift_bounds_Suc_ivl)
+ (simp add: setsum_right_distrib divide_inverse algebra_simps
+ atLeast0LessThan power_commutes)
+ finally have "(\<Sum>k<Suc m. inverse (f k) * inverse (x ^ k)) =
+ inverse (f 0) + (\<Sum>k<m. inverse (f (k + 1)) * inverse (x ^ k)) / x" by simp
+ }
+ from this[of "pred_numeral n"]
+ show "(\<Sum>k<numeral n. inverse (f k) * inverse (x^k)) =
+ inverse (f 0) + (\<Sum>k<pred_numeral n. inverse (f (k+1)) * inverse (x^k)) / x"
+ by (simp add: numeral_eq_Suc)
+qed simp_all
+
+lemma arctan_diff_small:
+ assumes "\<bar>x*y::real\<bar> < 1"
+ shows "arctan x - arctan y = arctan ((x - y) / (1 + x * y))"
+proof -
+ have "arctan x - arctan y = arctan x + arctan (-y)" by (simp add: arctan_minus)
+ also from assms have "\<dots> = arctan ((x - y) / (1 + x * y))" by (subst arctan_add_small) simp_all
+ finally show ?thesis .
qed
-(*32-bit approximation. SLOW simplification steps: big calculations with the rewriting engine*)
-lemma pi_approx_32: "\<bar>pi - 13493037705/4294967296\<bar> \<le> inverse(2 ^ 32)"
- apply (simp only: abs_diff_le_iff)
- apply (rule sin_pi6_straddle, simp_all)
- using Taylor_sin [of "1686629713/3221225472" 11]
- apply (simp add: in_Reals_norm sin_coeff_def Re_sin atMost_nat_numeral fact_numeral power_divide)
- apply (simp only: pos_le_divide_eq [symmetric])
- using Taylor_sin [of "6746518853/12884901888" 11]
- apply (simp add: in_Reals_norm sin_coeff_def Re_sin atMost_nat_numeral fact_numeral power_divide)
- apply (simp only: pos_le_divide_eq [symmetric] pos_divide_le_eq [symmetric])
- done
+
+subsubsection \<open>Machin-like formulae for pi\<close>
+
+text \<open>
+ We first define a small proof method that can prove Machin-like formulae for @{term "pi"}
+ automatically. Unfortunately, this takes far too much time for larger formulae because
+ the numbers involved become too large.
+\<close>
+
+definition "MACHIN_TAG a b \<equiv> a * (b::real)"
+
+lemma numeral_horner_MACHIN_TAG:
+ "MACHIN_TAG Numeral1 x = x"
+ "MACHIN_TAG (numeral (Num.Bit0 (Num.Bit0 n))) x =
+ MACHIN_TAG 2 (MACHIN_TAG (numeral (Num.Bit0 n)) x)"
+ "MACHIN_TAG (numeral (Num.Bit0 (Num.Bit1 n))) x =
+ MACHIN_TAG 2 (MACHIN_TAG (numeral (Num.Bit1 n)) x)"
+ "MACHIN_TAG (numeral (Num.Bit1 n)) x =
+ MACHIN_TAG 2 (MACHIN_TAG (numeral n) x) + x"
+ unfolding numeral_Bit0 numeral_Bit1 ring_distribs one_add_one[symmetric] MACHIN_TAG_def
+ by (simp_all add: algebra_simps)
+
+lemma tag_machin: "a * arctan b = MACHIN_TAG a (arctan b)" by (simp add: MACHIN_TAG_def)
+
+lemma arctan_double': "\<bar>a::real\<bar> < 1 \<Longrightarrow> MACHIN_TAG 2 (arctan a) = arctan (2 * a / (1 - a*a))"
+ unfolding MACHIN_TAG_def by (simp add: arctan_double power2_eq_square)
+
+ML \<open>
+ fun machin_term_conv ctxt ct =
+ let
+ val ctxt' = ctxt addsimps @{thms arctan_double' arctan_add_small}
+ in
+ case Thm.term_of ct of
+ Const (@{const_name MACHIN_TAG}, _) $ _ $
+ (Const (@{const_name "Transcendental.arctan"}, _) $ _) =>
+ Simplifier.rewrite ctxt' ct
+ |
+ Const (@{const_name MACHIN_TAG}, _) $ _ $
+ (Const (@{const_name "Groups.plus"}, _) $
+ (Const (@{const_name "Transcendental.arctan"}, _) $ _) $
+ (Const (@{const_name "Transcendental.arctan"}, _) $ _)) =>
+ Simplifier.rewrite ctxt' ct
+ | _ => raise CTERM ("machin_conv", [ct])
+ end
+
+ fun machin_tac ctxt =
+ let val conv = Conv.top_conv (Conv.try_conv o machin_term_conv) ctxt
+ in
+ SELECT_GOAL (
+ Local_Defs.unfold_tac ctxt
+ @{thms tag_machin[THEN eq_reflection] numeral_horner_MACHIN_TAG[THEN eq_reflection]}
+ THEN REPEAT (CHANGED (HEADGOAL (CONVERSION conv))))
+ THEN' Simplifier.simp_tac (ctxt addsimps @{thms arctan_add_small arctan_diff_small})
+ end
+\<close>
+
+method_setup machin = \<open>Scan.succeed (SIMPLE_METHOD' o machin_tac)\<close>
+
+text \<open>
+ We can now prove the ``standard'' Machin formula, which was already proven manually
+ in Isabelle, automatically.
+}\<close>
+lemma "pi / 4 = (4::real) * arctan (1 / 5) - arctan (1 / 239)"
+ by machin
+
+text \<open>
+ We can also prove the following more complicated formula:
+\<close>
+lemma machin': "pi/4 = (12::real) * arctan (1/18) + 8 * arctan (1/57) - 5 * arctan (1/239)"
+ by machin
+
+
+
+subsubsection \<open>Simple approximation of pi\<close>
+
+text \<open>
+ We can use the simple Machin formula and the Taylor series expansion of the arctangent
+ to approximate pi. For a given even natural number $n$, we expand @{term "arctan (1/5)"}
+ to $3n$ summands and @{term "arctan (1/239)"} to $n$ summands. This gives us at least
+ $13n-2$ bits of precision.
+\<close>
+
+definition "pi_approx n = 16 * arctan_approx (3*n) (1/5) - 4 * arctan_approx n (1/239)"
+
+lemma pi_approx:
+ fixes n :: nat assumes n: "even n" and "n > 0"
+ shows "\<bar>pi - pi_approx n\<bar> \<le> inverse (2^(13*n - 2))"
+proof -
+ from n have n': "even (3*n)" by simp
+ -- \<open>We apply the Machin formula\<close>
+ from machin have "pi = 16 * arctan (1/5) - 4 * arctan (1/239::real)" by simp
+ -- \<open>Taylor series expansion of the arctangent\<close>
+ also from arctan_approx[OF _ _ n', of "1/5"] arctan_approx[OF _ _ n, of "1/239"]
+ have "\<dots> - pi_approx n \<in> {-4*((1/239)^(2*n+1) / (1-(1/239)^4))..16*(1/5)^(6*n+1) / (1-(1/5)^4)}"
+ by (simp add: pi_approx_def)
+ -- \<open>Coarsening the bounds to make them a bit nicer\<close>
+ also have "-4*((1/239::real)^(2*n+1) / (1-(1/239)^4)) = -((13651919 / 815702160) / 57121^n)"
+ by (simp add: power_mult power2_eq_square) (simp add: field_simps)
+ also have "16*(1/5)^(6*n+1) / (1-(1/5::real)^4) = (125/39) / 15625^n"
+ by (simp add: power_mult power2_eq_square) (simp add: field_simps)
+ also have "{-((13651919 / 815702160) / 57121^n) .. (125 / 39) / 15625^n} \<subseteq>
+ {- (4 / 2^(13*n)) .. 4 / (2^(13*n)::real)}"
+ by (subst atLeastatMost_subset_iff, intro disjI2 conjI le_imp_neg_le)
+ (rule frac_le; simp add: power_mult power_mono)+
+ finally have "abs (pi - pi_approx n) \<le> 4 / 2^(13*n)" by auto
+ also from \<open>n > 0\<close> have "4 / 2^(13*n) = 1 / (2^(13*n - 2) :: real)"
+ by (cases n) (simp_all add: power_add)
+ finally show ?thesis by (simp add: divide_inverse)
+qed
+
+lemma pi_approx':
+ fixes n :: nat assumes n: "even n" and "n > 0" and "k \<le> 13*n - 2"
+ shows "\<bar>pi - pi_approx n\<bar> \<le> inverse (2^k)"
+ using assms(3) by (intro order.trans[OF pi_approx[OF assms(1,2)]]) (simp_all add: field_simps)
+
+text \<open>We can now approximate pi to 22 decimals within a fraction of a second.\<close>
+lemma pi_approx_75: "abs (pi - 3.1415926535897932384626 :: real) \<le> inverse (10^22)"
+proof -
+ def a \<equiv> "8295936325956147794769600190539918304 / 2626685325478320010006427764892578125 :: real"
+ def b \<equiv> "8428294561696506782041394632 / 503593538783547230635598424135 :: real"
+ -- \<open>The introduction of this constant prevents the simplifier from applying solvers that
+ we don't want. We want it to simply evaluate the terms to rational constants.}\<close>
+ def eq \<equiv> "op = :: real \<Rightarrow> real \<Rightarrow> bool"
+
+ -- \<open>Splitting the computation into several steps has the advantage that simplification can
+ be done in parallel\<close>
+ have "abs (pi - pi_approx 6) \<le> inverse (2^76)" by (rule pi_approx') simp_all
+ also have "pi_approx 6 = 16 * arctan_approx (3 * 6) (1 / 5) - 4 * arctan_approx 6 (1 / 239)"
+ unfolding pi_approx_def by simp
+ also have [unfolded eq_def]: "eq (16 * arctan_approx (3 * 6) (1 / 5)) a"
+ by (simp add: arctan_approx_def' power2_eq_square,
+ simp add: expand_arctan_approx, unfold a_def eq_def, rule refl)
+ also have [unfolded eq_def]: "eq (4 * arctan_approx 6 (1 / 239::real)) b"
+ by (simp add: arctan_approx_def' power2_eq_square,
+ simp add: expand_arctan_approx, unfold b_def eq_def, rule refl)
+ also have [unfolded eq_def]:
+ "eq (a - b) (171331331860120333586637094112743033554946184594977368554649608 /
+ 54536456744112171868276045488779391002026386559009552001953125)"
+ by (unfold a_def b_def, simp, unfold eq_def, rule refl)
+ finally show ?thesis by (rule approx_coarsen) simp
+qed
+
+text \<open>
+ The previous estimate of pi in this file was based on approximating the root of the
+ $\sin(\pi/6)$ in the interval $[0;4]$ using the Taylor series expansion of the sine to
+ verify that it is between two given bounds.
+ This was much slower and much less precise. We can easily recover this coarser estimate from
+ the newer, precise estimate:
+\<close>
+lemma pi_approx_32: "\<bar>pi - 13493037705/4294967296 :: real\<bar> \<le> inverse(2 ^ 32)"
+ by (rule approx_coarsen[OF pi_approx_75]) simp
+
+
+subsection \<open>A more complicated approximation of pi\<close>
+
+text \<open>
+ There are more complicated Machin-like formulae that have more terms with larger
+ denominators. Although they have more terms, each term requires fewer summands of the
+ Taylor series for the same precision, since it is evaluated closer to $0$.
+
+ Using a good formula, one can therefore obtain the same precision with fewer operations.
+ The big formulae used for computations of pi in practice are too complicated for us to
+ prove here, but we can use the three-term Machin-like formula @{thm machin'}.
+\<close>
+
+definition "pi_approx2 n = 48 * arctan_approx (6*n) (1/18::real) +
+ 32 * arctan_approx (4*n) (1/57) - 20 * arctan_approx (3*n) (1/239)"
+
+lemma pi_approx2:
+ fixes n :: nat assumes n: "even n" and "n > 0"
+ shows "\<bar>pi - pi_approx2 n\<bar> \<le> inverse (2^(46*n - 1))"
+proof -
+ from n have n': "even (6*n)" "even (4*n)" "even (3*n)" by simp_all
+ from machin' have "pi = 48 * arctan (1/18) + 32 * arctan (1/57) - 20 * arctan (1/239::real)"
+ by simp
+ hence "pi - pi_approx2 n = 48 * (arctan (1/18) - arctan_approx (6*n) (1/18)) +
+ 32 * (arctan (1/57) - arctan_approx (4*n) (1/57)) -
+ 20 * (arctan (1/239) - arctan_approx (3*n) (1/239))"
+ by (simp add: pi_approx2_def)
+ also have "\<dots> \<in> {-((20/239/(1-(1/239)^4)) * (1/239)^(6*n))..
+ (48/18 / (1-(1/18)^4))*(1/18)^(12*n) + (32/57/(1-(1/57)^4)) * (1/57)^(8*n)}"
+ (is "_ \<in> {-?l..?u1 + ?u2}")
+ apply ((rule combine_bounds(1,2))+; (rule combine_bounds(3); (rule arctan_approx)?)?)
+ apply (simp_all add: n)
+ apply (simp_all add: divide_simps)?
+ done
+ also {
+ have "?l \<le> (1/8) * (1/2)^(46*n)"
+ unfolding power_mult by (intro mult_mono power_mono) (simp_all add: divide_simps)
+ also have "\<dots> \<le> (1/2) ^ (46 * n - 1)"
+ by (cases n; simp_all add: power_add divide_simps)
+ finally have "?l \<le> (1/2) ^ (46 * n - 1)" .
+ moreover {
+ have "?u1 + ?u2 \<le> 4 * (1/2)^(48*n) + 1 * (1/2)^(46*n)"
+ unfolding power_mult by (intro add_mono mult_mono power_mono) (simp_all add: divide_simps)
+ also from \<open>n > 0\<close> have "4 * (1/2::real)^(48*n) \<le> (1/2)^(46*n)"
+ by (cases n) (simp_all add: field_simps power_add)
+ also from \<open>n > 0\<close> have "(1/2::real) ^ (46 * n) + 1 * (1 / 2) ^ (46 * n) = (1/2) ^ (46 * n - 1)"
+ by (cases n; simp_all add: power_add power_divide)
+ finally have "?u1 + ?u2 \<le> (1/2) ^ (46 * n - 1)" by - simp
+ }
+ ultimately have "{-?l..?u1 + ?u2} \<subseteq> {-((1/2)^(46*n-1))..(1/2)^(46*n-1)}"
+ by (subst atLeastatMost_subset_iff) simp_all
+ }
+ finally have "\<bar>pi - pi_approx2 n\<bar> \<le> ((1/2) ^ (46 * n - 1))" by auto
+ thus ?thesis by (simp add: divide_simps)
+qed
+
+lemma pi_approx2':
+ fixes n :: nat assumes n: "even n" and "n > 0" and "k \<le> 46*n - 1"
+ shows "\<bar>pi - pi_approx2 n\<bar> \<le> inverse (2^k)"
+ using assms(3) by (intro order.trans[OF pi_approx2[OF assms(1,2)]]) (simp_all add: field_simps)
+
+text \<open>
+ We can now approximate pi to 54 decimals using this formula. The computations are much
+ slower now; this is mostly because we use arbitrary-precision rational numbers, whose
+ numerators and demoninators get very large. Using dyadic floating point numbers would be
+ much more economical.
+\<close>
+lemma pi_approx_54_decimals:
+ "abs (pi - 3.141592653589793238462643383279502884197169399375105821 :: real) \<le> inverse (10^54)"
+ (is "abs (pi - ?pi') \<le> _")
+proof -
+ def a \<equiv> "2829469759662002867886529831139137601191652261996513014734415222704732791803 /
+ 1062141879292765061960538947347721564047051545995266466660439319087625011200 :: real"
+ def b \<equiv> "13355545553549848714922837267299490903143206628621657811747118592 /
+ 23792006023392488526789546722992491355941103837356113731091180925 :: real"
+ def c \<equiv> "28274063397213534906669125255762067746830085389618481175335056 /
+ 337877029279505250241149903214554249587517250716358486542628059 :: real"
+ let ?pi'' = "3882327391761098513316067116522233897127356523627918964967729040413954225768920394233198626889767468122598417405434625348404038165437924058179155035564590497837027530349 /
+ 1235783190199688165469648572769847552336447197542738425378629633275352407743112409829873464564018488572820294102599160968781449606552922108667790799771278860366957772800"
+ def eq \<equiv> "op = :: real \<Rightarrow> real \<Rightarrow> bool"
+
+ have "abs (pi - pi_approx2 4) \<le> inverse (2^183)" by (rule pi_approx2') simp_all
+ also have "pi_approx2 4 = 48 * arctan_approx 24 (1 / 18) +
+ 32 * arctan_approx 16 (1 / 57) -
+ 20 * arctan_approx 12 (1 / 239)"
+ unfolding pi_approx2_def by simp
+ also have [unfolded eq_def]: "eq (48 * arctan_approx 24 (1 / 18)) a"
+ by (simp add: arctan_approx_def' power2_eq_square,
+ simp add: expand_arctan_approx, unfold a_def eq_def, rule refl)
+ also have [unfolded eq_def]: "eq (32 * arctan_approx 16 (1 / 57::real)) b"
+ by (simp add: arctan_approx_def' power2_eq_square,
+ simp add: expand_arctan_approx, unfold b_def eq_def, rule refl)
+ also have [unfolded eq_def]: "eq (20 * arctan_approx 12 (1 / 239::real)) c"
+ by (simp add: arctan_approx_def' power2_eq_square,
+ simp add: expand_arctan_approx, unfold c_def eq_def, rule refl)
+ also have [unfolded eq_def]:
+ "eq (a + b) (34326487387865555303797183505809267914709125998469664969258315922216638779011304447624792548723974104030355722677 /
+ 10642967245546718617684989689985787964158885991018703366677373121531695267093031090059801733340658960857196134400)"
+ by (unfold a_def b_def c_def, simp, unfold eq_def, rule refl)
+ also have [unfolded eq_def]: "eq (\<dots> - c) ?pi''"
+ by (unfold a_def b_def c_def, simp, unfold eq_def, rule refl)
+ -- \<open>This is incredibly slow because the numerators and denominators are huge.\<close>
+ finally show ?thesis by (rule approx_coarsen) simp
+qed
+
+text \<open>A 128 bit approximation of pi:\<close>
+lemma pi_approx_128:
+ "abs (pi - 1069028584064966747859680373161870783301 / 2^128) \<le> inverse (2^128)"
+ by (rule approx_coarsen[OF pi_approx_54_decimals]) simp
+
+text \<open>A 64 bit approximation of pi:\<close>
+lemma pi_approx_64:
+ "abs (pi - 57952155664616982739 / 2^64 :: real) \<le> inverse (2^64)"
+ by (rule approx_coarsen[OF pi_approx_54_decimals]) simp
end