src/HOL/Analysis/Harmonic_Numbers.thy
 author Manuel Eberl Thu Aug 25 15:50:43 2016 +0200 (2016-08-25) changeset 63721 492bb53c3420 parent 63627 6ddb43c6b711 child 64267 b9a1486e79be permissions -rw-r--r--
More analysis lemmas
1 (*  Title:    HOL/Analysis/Harmonic_Numbers.thy
2     Author:   Manuel Eberl, TU München
3 *)
5 section \<open>Harmonic Numbers\<close>
7 theory Harmonic_Numbers
8 imports
9   Complex_Transcendental
10   Summation_Tests
11   Integral_Test
12 begin
14 text \<open>
15   The definition of the Harmonic Numbers and the Euler-Mascheroni constant.
16   Also provides a reasonably accurate approximation of @{term "ln 2 :: real"}
17   and the Euler-Mascheroni constant.
18 \<close>
20 lemma ln_2_less_1: "ln 2 < (1::real)"
21 proof -
22   have "2 < 5/(2::real)" by simp
23   also have "5/2 \<le> exp (1::real)" using exp_lower_taylor_quadratic[of 1, simplified] by simp
24   finally have "exp (ln 2) < exp (1::real)" by simp
25   thus "ln 2 < (1::real)" by (subst (asm) exp_less_cancel_iff) simp
26 qed
28 lemma setsum_Suc_diff':
29   fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
30   assumes "m \<le> n"
31   shows "(\<Sum>i = m..<n. f (Suc i) - f i) = f n - f m"
32 using assms by (induct n) (auto simp: le_Suc_eq)
35 subsection \<open>The Harmonic numbers\<close>
37 definition harm :: "nat \<Rightarrow> 'a :: real_normed_field" where
38   "harm n = (\<Sum>k=1..n. inverse (of_nat k))"
40 lemma harm_altdef: "harm n = (\<Sum>k<n. inverse (of_nat (Suc k)))"
41   unfolding harm_def by (induction n) simp_all
43 lemma harm_Suc: "harm (Suc n) = harm n + inverse (of_nat (Suc n))"
44   by (simp add: harm_def)
46 lemma harm_nonneg: "harm n \<ge> (0 :: 'a :: {real_normed_field,linordered_field})"
47   unfolding harm_def by (intro setsum_nonneg) simp_all
49 lemma harm_pos: "n > 0 \<Longrightarrow> harm n > (0 :: 'a :: {real_normed_field,linordered_field})"
50   unfolding harm_def by (intro setsum_pos) simp_all
52 lemma of_real_harm: "of_real (harm n) = harm n"
53   unfolding harm_def by simp
55 lemma norm_harm: "norm (harm n) = harm n"
56   by (subst of_real_harm [symmetric]) (simp add: harm_nonneg)
58 lemma harm_expand:
59   "harm 0 = 0"
60   "harm (Suc 0) = 1"
61   "harm (numeral n) = harm (pred_numeral n) + inverse (numeral n)"
62 proof -
63   have "numeral n = Suc (pred_numeral n)" by simp
64   also have "harm \<dots> = harm (pred_numeral n) + inverse (numeral n)"
65     by (subst harm_Suc, subst numeral_eq_Suc[symmetric]) simp
66   finally show "harm (numeral n) = harm (pred_numeral n) + inverse (numeral n)" .
67 qed (simp_all add: harm_def)
69 lemma not_convergent_harm: "\<not>convergent (harm :: nat \<Rightarrow> 'a :: real_normed_field)"
70 proof -
71   have "convergent (\<lambda>n. norm (harm n :: 'a)) \<longleftrightarrow>
72             convergent (harm :: nat \<Rightarrow> real)" by (simp add: norm_harm)
73   also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. \<Sum>k=Suc 0..Suc n. inverse (of_nat k) :: real)"
74     unfolding harm_def[abs_def] by (subst convergent_Suc_iff) simp_all
75   also have "... \<longleftrightarrow> convergent (\<lambda>n. \<Sum>k\<le>n. inverse (of_nat (Suc k)) :: real)"
76     by (subst setsum_shift_bounds_cl_Suc_ivl) (simp add: atLeast0AtMost)
77   also have "... \<longleftrightarrow> summable (\<lambda>n. inverse (of_nat n) :: real)"
78     by (subst summable_Suc_iff [symmetric]) (simp add: summable_iff_convergent')
79   also have "\<not>..." by (rule not_summable_harmonic)
80   finally show ?thesis by (blast dest: convergent_norm)
81 qed
83 lemma harm_pos_iff [simp]: "harm n > (0 :: 'a :: {real_normed_field,linordered_field}) \<longleftrightarrow> n > 0"
84   by (rule iffI, cases n, simp add: harm_expand, simp, rule harm_pos)
86 lemma ln_diff_le_inverse:
87   assumes "x \<ge> (1::real)"
88   shows   "ln (x + 1) - ln x < 1 / x"
89 proof -
90   from assms have "\<exists>z>x. z < x + 1 \<and> ln (x + 1) - ln x = (x + 1 - x) * inverse z"
91     by (intro MVT2) (auto intro!: derivative_eq_intros simp: field_simps)
92   then obtain z where z: "z > x" "z < x + 1" "ln (x + 1) - ln x = inverse z" by auto
93   have "ln (x + 1) - ln x = inverse z" by fact
94   also from z(1,2) assms have "\<dots> < 1 / x" by (simp add: field_simps)
95   finally show ?thesis .
96 qed
98 lemma ln_le_harm: "ln (real n + 1) \<le> (harm n :: real)"
99 proof (induction n)
100   fix n assume IH: "ln (real n + 1) \<le> harm n"
101   have "ln (real (Suc n) + 1) = ln (real n + 1) + (ln (real n + 2) - ln (real n + 1))" by simp
102   also have "(ln (real n + 2) - ln (real n + 1)) \<le> 1 / real (Suc n)"
103     using ln_diff_le_inverse[of "real n + 1"] by (simp add: add_ac)
104   also note IH
105   also have "harm n + 1 / real (Suc n) = harm (Suc n)" by (simp add: harm_Suc field_simps)
106   finally show "ln (real (Suc n) + 1) \<le> harm (Suc n)" by - simp
107 qed (simp_all add: harm_def)
110 subsection \<open>The Euler--Mascheroni constant\<close>
112 text \<open>
113   The limit of the difference between the partial harmonic sum and the natural logarithm
114   (approximately 0.577216). This value occurs e.g. in the definition of the Gamma function.
115  \<close>
116 definition euler_mascheroni :: "'a :: real_normed_algebra_1" where
117   "euler_mascheroni = of_real (lim (\<lambda>n. harm n - ln (of_nat n)))"
119 lemma of_real_euler_mascheroni [simp]: "of_real euler_mascheroni = euler_mascheroni"
120   by (simp add: euler_mascheroni_def)
122 interpretation euler_mascheroni: antimono_fun_sum_integral_diff "\<lambda>x. inverse (x + 1)"
123   by unfold_locales (auto intro!: continuous_intros)
125 lemma euler_mascheroni_sum_integral_diff_series:
126   "euler_mascheroni.sum_integral_diff_series n = harm (Suc n) - ln (of_nat (Suc n))"
127 proof -
128   have "harm (Suc n) = (\<Sum>k=0..n. inverse (of_nat k + 1) :: real)" unfolding harm_def
129     unfolding One_nat_def by (subst setsum_shift_bounds_cl_Suc_ivl) (simp add: add_ac)
130   moreover have "((\<lambda>x. inverse (x + 1) :: real) has_integral ln (of_nat n + 1) - ln (0 + 1))
131                    {0..of_nat n}"
132     by (intro fundamental_theorem_of_calculus)
133        (auto intro!: derivative_eq_intros simp: divide_inverse
134            has_field_derivative_iff_has_vector_derivative[symmetric])
135   hence "integral {0..of_nat n} (\<lambda>x. inverse (x + 1) :: real) = ln (of_nat (Suc n))"
136     by (auto dest!: integral_unique)
137   ultimately show ?thesis
138     by (simp add: euler_mascheroni.sum_integral_diff_series_def atLeast0AtMost)
139 qed
141 lemma euler_mascheroni_sequence_decreasing:
142   "m > 0 \<Longrightarrow> m \<le> n \<Longrightarrow> harm n - ln (of_nat n) \<le> harm m - ln (of_nat m :: real)"
143   by (cases m, simp, cases n, simp, hypsubst,
144       subst (1 2) euler_mascheroni_sum_integral_diff_series [symmetric],
145       rule euler_mascheroni.sum_integral_diff_series_antimono, simp)
147 lemma euler_mascheroni_sequence_nonneg:
148   "n > 0 \<Longrightarrow> harm n - ln (of_nat n) \<ge> (0::real)"
149   by (cases n, simp, hypsubst, subst euler_mascheroni_sum_integral_diff_series [symmetric],
150       rule euler_mascheroni.sum_integral_diff_series_nonneg)
152 lemma euler_mascheroni_convergent: "convergent (\<lambda>n. harm n - ln (of_nat n) :: real)"
153 proof -
154   have A: "(\<lambda>n. harm (Suc n) - ln (of_nat (Suc n))) =
155              euler_mascheroni.sum_integral_diff_series"
156     by (subst euler_mascheroni_sum_integral_diff_series [symmetric]) (rule refl)
157   have "convergent (\<lambda>n. harm (Suc n) - ln (of_nat (Suc n) :: real))"
158     by (subst A) (fact euler_mascheroni.sum_integral_diff_series_convergent)
159   thus ?thesis by (subst (asm) convergent_Suc_iff)
160 qed
162 lemma euler_mascheroni_LIMSEQ:
163   "(\<lambda>n. harm n - ln (of_nat n) :: real) \<longlonglongrightarrow> euler_mascheroni"
164   unfolding euler_mascheroni_def
165   by (simp add: convergent_LIMSEQ_iff [symmetric] euler_mascheroni_convergent)
167 lemma euler_mascheroni_LIMSEQ_of_real:
168   "(\<lambda>n. of_real (harm n - ln (of_nat n))) \<longlonglongrightarrow>
169       (euler_mascheroni :: 'a :: {real_normed_algebra_1, topological_space})"
170 proof -
171   have "(\<lambda>n. of_real (harm n - ln (of_nat n))) \<longlonglongrightarrow> (of_real (euler_mascheroni) :: 'a)"
172     by (intro tendsto_of_real euler_mascheroni_LIMSEQ)
173   thus ?thesis by simp
174 qed
176 lemma euler_mascheroni_sum_real:
177   "(\<lambda>n. inverse (of_nat (n+1)) + ln (of_nat (n+1)) - ln (of_nat (n+2)) :: real)
178        sums euler_mascheroni"
179  using sums_add[OF telescope_sums[OF LIMSEQ_Suc[OF euler_mascheroni_LIMSEQ]]
180                    telescope_sums'[OF LIMSEQ_inverse_real_of_nat]]
181   by (simp_all add: harm_def algebra_simps)
183 lemma euler_mascheroni_sum:
184   "(\<lambda>n. inverse (of_nat (n+1)) + of_real (ln (of_nat (n+1))) - of_real (ln (of_nat (n+2))))
185        sums (euler_mascheroni :: 'a :: {banach, real_normed_field})"
186 proof -
187   have "(\<lambda>n. of_real (inverse (of_nat (n+1)) + ln (of_nat (n+1)) - ln (of_nat (n+2))))
188        sums (of_real euler_mascheroni :: 'a :: {banach, real_normed_field})"
189     by (subst sums_of_real_iff) (rule euler_mascheroni_sum_real)
190   thus ?thesis by simp
191 qed
193 lemma alternating_harmonic_series_sums: "(\<lambda>k. (-1)^k / real_of_nat (Suc k)) sums ln 2"
194 proof -
195   let ?f = "\<lambda>n. harm n - ln (real_of_nat n)"
196   let ?g = "\<lambda>n. if even n then 0 else (2::real)"
197   let ?em = "\<lambda>n. harm n - ln (real_of_nat n)"
198   have "eventually (\<lambda>n. ?em (2*n) - ?em n + ln 2 = (\<Sum>k<2*n. (-1)^k / real_of_nat (Suc k))) at_top"
199     using eventually_gt_at_top[of "0::nat"]
200   proof eventually_elim
201     fix n :: nat assume n: "n > 0"
202     have "(\<Sum>k<2*n. (-1)^k / real_of_nat (Suc k)) =
203               (\<Sum>k<2*n. ((-1)^k + ?g k) / of_nat (Suc k)) - (\<Sum>k<2*n. ?g k / of_nat (Suc k))"
204       by (simp add: setsum.distrib algebra_simps divide_inverse)
205     also have "(\<Sum>k<2*n. ((-1)^k + ?g k) / real_of_nat (Suc k)) = harm (2*n)"
206       unfolding harm_altdef by (intro setsum.cong) (auto simp: field_simps)
207     also have "(\<Sum>k<2*n. ?g k / real_of_nat (Suc k)) = (\<Sum>k|k<2*n \<and> odd k. ?g k / of_nat (Suc k))"
208       by (intro setsum.mono_neutral_right) auto
209     also have "\<dots> = (\<Sum>k|k<2*n \<and> odd k. 2 / (real_of_nat (Suc k)))"
210       by (intro setsum.cong) auto
211     also have "(\<Sum>k|k<2*n \<and> odd k. 2 / (real_of_nat (Suc k))) = harm n"
212       unfolding harm_altdef
213       by (intro setsum.reindex_cong[of "\<lambda>n. 2*n+1"]) (auto simp: inj_on_def field_simps elim!: oddE)
214     also have "harm (2*n) - harm n = ?em (2*n) - ?em n + ln 2" using n
215       by (simp_all add: algebra_simps ln_mult)
216     finally show "?em (2*n) - ?em n + ln 2 = (\<Sum>k<2*n. (-1)^k / real_of_nat (Suc k))" ..
217   qed
218   moreover have "(\<lambda>n. ?em (2*n) - ?em n + ln (2::real))
219                      \<longlonglongrightarrow> euler_mascheroni - euler_mascheroni + ln 2"
220     by (intro tendsto_intros euler_mascheroni_LIMSEQ filterlim_compose[OF euler_mascheroni_LIMSEQ]
221               filterlim_subseq) (auto simp: subseq_def)
222   hence "(\<lambda>n. ?em (2*n) - ?em n + ln (2::real)) \<longlonglongrightarrow> ln 2" by simp
223   ultimately have "(\<lambda>n. (\<Sum>k<2*n. (-1)^k / real_of_nat (Suc k))) \<longlonglongrightarrow> ln 2"
224     by (rule Lim_transform_eventually)
226   moreover have "summable (\<lambda>k. (-1)^k * inverse (real_of_nat (Suc k)))"
227     using LIMSEQ_inverse_real_of_nat
228     by (intro summable_Leibniz(1) decseq_imp_monoseq decseq_SucI) simp_all
229   hence A: "(\<lambda>n. \<Sum>k<n. (-1)^k / real_of_nat (Suc k)) \<longlonglongrightarrow> (\<Sum>k. (-1)^k / real_of_nat (Suc k))"
230     by (simp add: summable_sums_iff divide_inverse sums_def)
231   from filterlim_compose[OF this filterlim_subseq[of "op * (2::nat)"]]
232     have "(\<lambda>n. \<Sum>k<2*n. (-1)^k / real_of_nat (Suc k)) \<longlonglongrightarrow> (\<Sum>k. (-1)^k / real_of_nat (Suc k))"
233     by (simp add: subseq_def)
234   ultimately have "(\<Sum>k. (- 1) ^ k / real_of_nat (Suc k)) = ln 2" by (intro LIMSEQ_unique)
235   with A show ?thesis by (simp add: sums_def)
236 qed
238 lemma alternating_harmonic_series_sums':
239   "(\<lambda>k. inverse (real_of_nat (2*k+1)) - inverse (real_of_nat (2*k+2))) sums ln 2"
240 unfolding sums_def
241 proof (rule Lim_transform_eventually)
242   show "(\<lambda>n. \<Sum>k<2*n. (-1)^k / (real_of_nat (Suc k))) \<longlonglongrightarrow> ln 2"
243     using alternating_harmonic_series_sums unfolding sums_def
244     by (rule filterlim_compose) (rule mult_nat_left_at_top, simp)
245   show "eventually (\<lambda>n. (\<Sum>k<2*n. (-1)^k / (real_of_nat (Suc k))) =
246             (\<Sum>k<n. inverse (real_of_nat (2*k+1)) - inverse (real_of_nat (2*k+2)))) sequentially"
247   proof (intro always_eventually allI)
248     fix n :: nat
249     show "(\<Sum>k<2*n. (-1)^k / (real_of_nat (Suc k))) =
250               (\<Sum>k<n. inverse (real_of_nat (2*k+1)) - inverse (real_of_nat (2*k+2)))"
251       by (induction n) (simp_all add: inverse_eq_divide)
252   qed
253 qed
256 subsection \<open>Bounds on the Euler--Mascheroni constant\<close>
258 (* TODO: Move? *)
259 lemma ln_inverse_approx_le:
260   assumes "(x::real) > 0" "a > 0"
261   shows   "ln (x + a) - ln x \<le> a * (inverse x + inverse (x + a))/2" (is "_ \<le> ?A")
262 proof -
263   define f' where "f' = (inverse (x + a) - inverse x)/a"
264   have f'_nonpos: "f' \<le> 0" using assms by (simp add: f'_def divide_simps)
265   let ?f = "\<lambda>t. (t - x) * f' + inverse x"
266   let ?F = "\<lambda>t. (t - x)^2 * f' / 2 + t * inverse x"
267   have diff: "\<forall>t\<in>{x..x+a}. (?F has_vector_derivative ?f t)
268                                (at t within {x..x+a})" using assms
269     by (auto intro!: derivative_eq_intros
270              simp: has_field_derivative_iff_has_vector_derivative[symmetric])
271   from assms have "(?f has_integral (?F (x+a) - ?F x)) {x..x+a}"
272     by (intro fundamental_theorem_of_calculus[OF _ diff])
273        (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] field_simps
274              intro!: derivative_eq_intros)
275   also have "?F (x+a) - ?F x = (a*2 + f'*a\<^sup>2*x) / (2*x)" using assms by (simp add: field_simps)
276   also have "f'*a^2 = - (a^2) / (x*(x + a))" using assms
277     by (simp add: divide_simps f'_def power2_eq_square)
278   also have "(a*2 + - a\<^sup>2/(x*(x+a))*x) / (2*x) = ?A" using assms
279     by (simp add: divide_simps power2_eq_square) (simp add: algebra_simps)
280   finally have int1: "((\<lambda>t. (t - x) * f' + inverse x) has_integral ?A) {x..x + a}" .
282   from assms have int2: "(inverse has_integral (ln (x + a) - ln x)) {x..x+a}"
283     by (intro fundamental_theorem_of_calculus)
284        (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] divide_simps
285              intro!: derivative_eq_intros)
286   hence "ln (x + a) - ln x = integral {x..x+a} inverse" by (simp add: integral_unique)
287   also have ineq: "\<forall>xa\<in>{x..x + a}. inverse xa \<le> (xa - x) * f' + inverse x"
288   proof
289     fix t assume t': "t \<in> {x..x+a}"
290     with assms have t: "0 \<le> (t - x) / a" "(t - x) / a \<le> 1" by simp_all
291     have "inverse t = inverse ((1 - (t - x) / a) *\<^sub>R x + ((t - x) / a) *\<^sub>R (x + a))" (is "_ = ?A")
292       using assms t' by (simp add: field_simps)
293     also from assms have "convex_on {x..x+a} inverse" by (intro convex_on_inverse) auto
294     from convex_onD_Icc[OF this _ t] assms
295       have "?A \<le> (1 - (t - x) / a) * inverse x + (t - x) / a * inverse (x + a)" by simp
296     also have "\<dots> = (t - x) * f' + inverse x" using assms
297       by (simp add: f'_def divide_simps) (simp add: f'_def field_simps)
298     finally show "inverse t \<le> (t - x) * f' + inverse x" .
299   qed
300   hence "integral {x..x+a} inverse \<le> integral {x..x+a} ?f" using f'_nonpos assms
301     by (intro integral_le has_integral_integrable[OF int1] has_integral_integrable[OF int2] ineq)
302   also have "\<dots> = ?A" using int1 by (rule integral_unique)
303   finally show ?thesis .
304 qed
306 lemma ln_inverse_approx_ge:
307   assumes "(x::real) > 0" "x < y"
308   shows   "ln y - ln x \<ge> 2 * (y - x) / (x + y)" (is "_ \<ge> ?A")
309 proof -
310   define m where "m = (x+y)/2"
311   define f' where "f' = -inverse (m^2)"
312   from assms have m: "m > 0" by (simp add: m_def)
313   let ?F = "\<lambda>t. (t - m)^2 * f' / 2 + t / m"
314   from assms have "((\<lambda>t. (t - m) * f' + inverse m) has_integral (?F y - ?F x)) {x..y}"
315     by (intro fundamental_theorem_of_calculus)
316        (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] divide_simps
317              intro!: derivative_eq_intros)
318   also from m have "?F y - ?F x = ((y - m)^2 - (x - m)^2) * f' / 2 + (y - x) / m"
319     by (simp add: field_simps)
320   also have "((y - m)^2 - (x - m)^2) = 0" by (simp add: m_def power2_eq_square field_simps)
321   also have "0 * f' / 2 + (y - x) / m = ?A" by (simp add: m_def)
322   finally have int1: "((\<lambda>t. (t - m) * f' + inverse m) has_integral ?A) {x..y}" .
324   from assms have int2: "(inverse has_integral (ln y - ln x)) {x..y}"
325     by (intro fundamental_theorem_of_calculus)
326        (auto simp: has_field_derivative_iff_has_vector_derivative[symmetric] divide_simps
327              intro!: derivative_eq_intros)
328   hence "ln y - ln x = integral {x..y} inverse" by (simp add: integral_unique)
329   also have ineq: "\<forall>xa\<in>{x..y}. inverse xa \<ge> (xa - m) * f' + inverse m"
330   proof
331     fix t assume t: "t \<in> {x..y}"
332     from t assms have "inverse t - inverse m \<ge> f' * (t - m)"
333       by (intro convex_on_imp_above_tangent[of "{0<..}"] convex_on_inverse)
334          (auto simp: m_def interior_open f'_def power2_eq_square intro!: derivative_eq_intros)
335     thus "(t - m) * f' + inverse m \<le> inverse t" by (simp add: algebra_simps)
336   qed
337   hence "integral {x..y} inverse \<ge> integral {x..y} (\<lambda>t. (t - m) * f' + inverse m)"
338     using int1 int2 by (intro integral_le has_integral_integrable)
339   also have "integral {x..y} (\<lambda>t. (t - m) * f' + inverse m) = ?A"
340     using integral_unique[OF int1] by simp
341   finally show ?thesis .
342 qed
345 lemma euler_mascheroni_lower:
346         "euler_mascheroni \<ge> harm (Suc n) - ln (real_of_nat (n + 2)) + 1/real_of_nat (2 * (n + 2))"
347   and euler_mascheroni_upper:
348         "euler_mascheroni \<le> harm (Suc n) - ln (real_of_nat (n + 2)) + 1/real_of_nat (2 * (n + 1))"
349 proof -
350   define D :: "_ \<Rightarrow> real"
351     where "D n = inverse (of_nat (n+1)) + ln (of_nat (n+1)) - ln (of_nat (n+2))" for n
352   let ?g = "\<lambda>n. ln (of_nat (n+2)) - ln (of_nat (n+1)) - inverse (of_nat (n+1)) :: real"
353   define inv where [abs_def]: "inv n = inverse (real_of_nat n)" for n
354   fix n :: nat
355   note summable = sums_summable[OF euler_mascheroni_sum_real, folded D_def]
356   have sums: "(\<lambda>k. (inv (Suc (k + (n+1))) - inv (Suc (Suc k + (n+1))))/2) sums ((inv (Suc (0 + (n+1))) - 0)/2)"
357     unfolding inv_def
358     by (intro sums_divide telescope_sums' LIMSEQ_ignore_initial_segment LIMSEQ_inverse_real_of_nat)
359   have sums': "(\<lambda>k. (inv (Suc (k + n)) - inv (Suc (Suc k + n)))/2) sums ((inv (Suc (0 + n)) - 0)/2)"
360     unfolding inv_def
361     by (intro sums_divide telescope_sums' LIMSEQ_ignore_initial_segment LIMSEQ_inverse_real_of_nat)
362   from euler_mascheroni_sum_real have "euler_mascheroni = (\<Sum>k. D k)"
363     by (simp add: sums_iff D_def)
364   also have "\<dots> = (\<Sum>k. D (k + Suc n)) + (\<Sum>k\<le>n. D k)"
365     by (subst suminf_split_initial_segment[OF summable, of "Suc n"],
366         subst lessThan_Suc_atMost) simp
367   finally have sum: "(\<Sum>k\<le>n. D k) - euler_mascheroni = -(\<Sum>k. D (k + Suc n))" by simp
369   note sum
370   also have "\<dots> \<le> -(\<Sum>k. (inv (k + Suc n + 1) - inv (k + Suc n + 2)) / 2)"
371   proof (intro le_imp_neg_le suminf_le allI summable_ignore_initial_segment[OF summable])
372     fix k' :: nat
373     define k where "k = k' + Suc n"
374     hence k: "k > 0" by (simp add: k_def)
375     have "real_of_nat (k+1) > 0" by (simp add: k_def)
376     with ln_inverse_approx_le[OF this zero_less_one]
377       have "ln (of_nat k + 2) - ln (of_nat k + 1) \<le> (inv (k+1) + inv (k+2))/2"
379     hence "(inv (k+1) - inv (k+2))/2 \<le> inv (k+1) + ln (of_nat (k+1)) - ln (of_nat (k+2))"
380       by (simp add: field_simps)
381     also have "\<dots> = D k" unfolding D_def inv_def ..
382     finally show "D (k' + Suc n) \<ge> (inv (k' + Suc n + 1) - inv (k' + Suc n + 2)) / 2"
383       by (simp add: k_def)
384     from sums_summable[OF sums]
385       show "summable (\<lambda>k. (inv (k + Suc n + 1) - inv (k + Suc n + 2))/2)" by simp
386   qed
387   also from sums have "\<dots> = -inv (n+2) / 2" by (simp add: sums_iff)
388   finally have "euler_mascheroni \<ge> (\<Sum>k\<le>n. D k) + 1 / (of_nat (2 * (n+2)))"
389     by (simp add: inv_def field_simps)
390   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)))"
391     unfolding harm_altdef D_def by (subst lessThan_Suc_atMost) (simp add:  setsum.distrib setsum_subtractf)
392   also have "(\<Sum>k\<le>n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1))) = ln (of_nat (n+2))"
393     by (subst atLeast0AtMost [symmetric], subst setsum_Suc_diff) simp_all
394   finally show "euler_mascheroni \<ge> harm (Suc n) - ln (real_of_nat (n + 2)) + 1/real_of_nat (2 * (n + 2))"
395     by simp
397   note sum
398   also have "-(\<Sum>k. D (k + Suc n)) \<ge> -(\<Sum>k. (inv (Suc (k + n)) - inv (Suc (Suc k + n)))/2)"
399   proof (intro le_imp_neg_le suminf_le allI summable_ignore_initial_segment[OF summable])
400     fix k' :: nat
401     define k where "k = k' + Suc n"
402     hence k: "k > 0" by (simp add: k_def)
403     have "real_of_nat (k+1) > 0" by (simp add: k_def)
404     from ln_inverse_approx_ge[of "of_nat k + 1" "of_nat k + 2"]
405       have "2 / (2 * real_of_nat k + 3) \<le> ln (of_nat (k+2)) - ln (real_of_nat (k+1))"
407     hence "D k \<le> 1 / real_of_nat (k+1) - 2 / (2 * real_of_nat k + 3)"
408       by (simp add: D_def inverse_eq_divide inv_def)
409     also have "\<dots> = inv ((k+1)*(2*k+3))" unfolding inv_def by (simp add: field_simps)
410     also have "\<dots> \<le> inv (2*k*(k+1))" unfolding inv_def using k
411       by (intro le_imp_inverse_le)
412          (simp add: algebra_simps, simp del: of_nat_add)
413     also have "\<dots> = (inv k - inv (k+1))/2" unfolding inv_def using k
414       by (simp add: divide_simps del: of_nat_mult) (simp add: algebra_simps)
415     finally show "D k \<le> (inv (Suc (k' + n)) - inv (Suc (Suc k' + n)))/2" unfolding k_def by simp
416   next
417     from sums_summable[OF sums']
418       show "summable (\<lambda>k. (inv (Suc (k + n)) - inv (Suc (Suc k + n)))/2)" by simp
419   qed
420   also from sums' have "(\<Sum>k. (inv (Suc (k + n)) - inv (Suc (Suc k + n)))/2) = inv (n+1)/2"
421     by (simp add: sums_iff)
422   finally have "euler_mascheroni \<le> (\<Sum>k\<le>n. D k) + 1 / of_nat (2 * (n+1))"
423     by (simp add: inv_def field_simps)
424   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)))"
425     unfolding harm_altdef D_def by (subst lessThan_Suc_atMost) (simp add:  setsum.distrib setsum_subtractf)
426   also have "(\<Sum>k\<le>n. ln (real_of_nat (Suc k+1)) - ln (of_nat (k+1))) = ln (of_nat (n+2))"
427     by (subst atLeast0AtMost [symmetric], subst setsum_Suc_diff) simp_all
428   finally show "euler_mascheroni \<le> harm (Suc n) - ln (real_of_nat (n + 2)) + 1/real_of_nat (2 * (n + 1))"
429     by simp
430 qed
432 lemma euler_mascheroni_pos: "euler_mascheroni > (0::real)"
433   using euler_mascheroni_lower[of 0] ln_2_less_1 by (simp add: harm_def)
435 context
436 begin
438 private lemma ln_approx_aux:
439   fixes n :: nat and x :: real
440   defines "y \<equiv> (x-1)/(x+1)"
441   assumes x: "x > 0" "x \<noteq> 1"
442   shows "inverse (2*y^(2*n+1)) * (ln x - (\<Sum>k<n. 2*y^(2*k+1) / of_nat (2*k+1))) \<in>
443             {0..(1 / (1 - y^2) / of_nat (2*n+1))}"
444 proof -
445   from x have norm_y: "norm y < 1" unfolding y_def by simp
446   from power_strict_mono[OF this, of 2] have norm_y': "norm y^2 < 1" by simp
448   let ?f = "\<lambda>k. 2 * y ^ (2*k+1) / of_nat (2*k+1)"
449   note sums = ln_series_quadratic[OF x(1)]
450   define c where "c = inverse (2*y^(2*n+1))"
451   let ?d = "c * (ln x - (\<Sum>k<n. ?f k))"
452   have "\<forall>k. y\<^sup>2^k / of_nat (2*(k+n)+1) \<le> y\<^sup>2 ^ k / of_nat (2*n+1)"
453     by (intro allI divide_left_mono mult_right_mono mult_pos_pos zero_le_power[of "y^2"]) simp_all
454   moreover {
455     have "(\<lambda>k. ?f (k + n)) sums (ln x - (\<Sum>k<n. ?f k))"
456       using sums_split_initial_segment[OF sums] by (simp add: y_def)
457     hence "(\<lambda>k. c * ?f (k + n)) sums ?d" by (rule sums_mult)
458     also have "(\<lambda>k. c * (2*y^(2*(k+n)+1) / of_nat (2*(k+n)+1))) =
459                    (\<lambda>k. (c * (2*y^(2*n+1))) * ((y^2)^k / of_nat (2*(k+n)+1)))"
460       by (simp only: ring_distribs power_add power_mult) (simp add: mult_ac)
461     also from x have "c * (2*y^(2*n+1)) = 1" by (simp add: c_def y_def)
462     finally have "(\<lambda>k. (y^2)^k / of_nat (2*(k+n)+1)) sums ?d" by simp
463   } note sums' = this
464   moreover from norm_y' have "(\<lambda>k. (y^2)^k / of_nat (2*n+1)) sums (1 / (1 - y^2) / of_nat (2*n+1))"
465     by (intro sums_divide geometric_sums) (simp_all add: norm_power)
466   ultimately have "?d \<le> (1 / (1 - y^2) / of_nat (2*n+1))" by (rule sums_le)
467   moreover have "c * (ln x - (\<Sum>k<n. 2 * y ^ (2 * k + 1) / real_of_nat (2 * k + 1))) \<ge> 0"
468     by (intro sums_le[OF _ sums_zero sums']) simp_all
469   ultimately show ?thesis unfolding c_def by simp
470 qed
472 lemma
473   fixes n :: nat and x :: real
474   defines "y \<equiv> (x-1)/(x+1)"
475   defines "approx \<equiv> (\<Sum>k<n. 2*y^(2*k+1) / of_nat (2*k+1))"
476   defines "d \<equiv> y^(2*n+1) / (1 - y^2) / of_nat (2*n+1)"
477   assumes x: "x > 1"
478   shows   ln_approx_bounds: "ln x \<in> {approx..approx + 2*d}"
479   and     ln_approx_abs:    "abs (ln x - (approx + d)) \<le> d"
480 proof -
481   define c where "c = 2*y^(2*n+1)"
482   from x have c_pos: "c > 0" unfolding c_def y_def
483     by (intro mult_pos_pos zero_less_power) simp_all
484   have A: "inverse c * (ln x - (\<Sum>k<n. 2*y^(2*k+1) / of_nat (2*k+1))) \<in>
485               {0.. (1 / (1 - y^2) / of_nat (2*n+1))}" using assms unfolding y_def c_def
486     by (intro ln_approx_aux) simp_all
487   hence "inverse c * (ln x - (\<Sum>k<n. 2*y^(2*k+1)/of_nat (2*k+1))) \<le> (1 / (1-y^2) / of_nat (2*n+1))"
488     by simp
489   hence "(ln x - (\<Sum>k<n. 2*y^(2*k+1) / of_nat (2*k+1))) / c \<le> (1 / (1 - y^2) / of_nat (2*n+1))"
490     by (auto simp add: divide_simps)
491   with c_pos have "ln x \<le> c / (1 - y^2) / of_nat (2*n+1) + approx"
492     by (subst (asm) pos_divide_le_eq) (simp_all add: mult_ac approx_def)
493   moreover {
494     from A c_pos have "0 \<le> c * (inverse c * (ln x - (\<Sum>k<n. 2*y^(2*k+1) / of_nat (2*k+1))))"
495       by (intro mult_nonneg_nonneg[of c]) simp_all
496     also have "\<dots> = (c * inverse c) * (ln x - (\<Sum>k<n. 2*y^(2*k+1) / of_nat (2*k+1)))"
497       by (simp add: mult_ac)
498     also from c_pos have "c * inverse c = 1" by simp
499     finally have "ln x \<ge> approx" by (simp add: approx_def)
500   }
501   ultimately show "ln x \<in> {approx..approx + 2*d}" by (simp add: c_def d_def)
502   thus "abs (ln x - (approx + d)) \<le> d" by auto
503 qed
505 end
507 lemma euler_mascheroni_bounds:
508   fixes n :: nat assumes "n \<ge> 1" defines "t \<equiv> harm n - ln (of_nat (Suc n)) :: real"
509   shows "euler_mascheroni \<in> {t + inverse (of_nat (2*(n+1)))..t + inverse (of_nat (2*n))}"
510   using assms euler_mascheroni_upper[of "n-1"] euler_mascheroni_lower[of "n-1"]
511   unfolding t_def by (cases n) (simp_all add: harm_Suc t_def inverse_eq_divide)
513 lemma euler_mascheroni_bounds':
514   fixes n :: nat assumes "n \<ge> 1" "ln (real_of_nat (Suc n)) \<in> {l<..<u}"
515   shows "euler_mascheroni \<in>
516            {harm n - u + inverse (of_nat (2*(n+1)))<..<harm n - l + inverse (of_nat (2*n))}"
517   using euler_mascheroni_bounds[OF assms(1)] assms(2) by auto
520 text \<open>
521   Approximation of @{term "ln 2"}. The lower bound is accurate to about 0.03; the upper
522   bound is accurate to about 0.0015.
523 \<close>
524 lemma ln2_ge_two_thirds: "2/3 \<le> ln (2::real)"
525   and ln2_le_25_over_36: "ln (2::real) \<le> 25/36"
526   using ln_approx_bounds[of 2 1, simplified, simplified eval_nat_numeral, simplified] by simp_all
529 text \<open>
530   Approximation of the Euler--Mascheroni constant. The lower bound is accurate to about 0.0015;
531   the upper bound is accurate to about 0.015.
532 \<close>
533 lemma euler_mascheroni_gt_19_over_33: "(euler_mascheroni :: real) > 19/33" (is ?th1)
534   and euler_mascheroni_less_13_over_22: "(euler_mascheroni :: real) < 13/22" (is ?th2)
535 proof -
536   have "ln (real (Suc 7)) = 3 * ln 2" by (simp add: ln_powr [symmetric] powr_numeral)
537   also from ln_approx_bounds[of 2 3] have "\<dots> \<in> {3*307/443<..<3*4615/6658}"
538     by (simp add: eval_nat_numeral)
539   finally have "ln (real (Suc 7)) \<in> \<dots>" .
540   from euler_mascheroni_bounds'[OF _ this] have "?th1 \<and> ?th2" by (simp_all add: harm_expand)
541   thus ?th1 ?th2 by blast+
542 qed
544 end