(* Title: HOL/Transcendental.thy
Author: Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
Author: Lawrence C Paulson
Author: Jeremy Avigad
*)
section \<open>Power Series, Transcendental Functions etc.\<close>
theory Transcendental
imports Series Deriv NthRoot
begin
text \<open>A theorem about the factcorial function on the reals.\<close>
lemma square_fact_le_2_fact: "fact n * fact n \ (fact (2 * n) :: real)"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
have "(fact (Suc n)) * (fact (Suc n)) = of_nat (Suc n) * of_nat (Suc n) * (fact n * fact n :: real)"
by (simp add: field_simps)
also have "\ \ of_nat (Suc n) * of_nat (Suc n) * fact (2 * n)"
by (rule mult_left_mono [OF Suc]) simp
also have "\ \ of_nat (Suc (Suc (2 * n))) * of_nat (Suc (2 * n)) * fact (2 * n)"
by (rule mult_right_mono)+ (auto simp: field_simps)
also have "\ = fact (2 * Suc n)" by (simp add: field_simps)
finally show ?case .
qed
lemma fact_in_Reals: "fact n \ \"
by (induction n) auto
lemma of_real_fact [simp]: "of_real (fact n) = fact n"
by (metis of_nat_fact of_real_of_nat_eq)
lemma pochhammer_of_real: "pochhammer (of_real x) n = of_real (pochhammer x n)"
by (simp add: pochhammer_prod)
lemma norm_fact [simp]: "norm (fact n :: 'a::real_normed_algebra_1) = fact n"
proof -
have "(fact n :: 'a) = of_real (fact n)"
by simp
also have "norm \ = fact n"
by (subst norm_of_real) simp
finally show ?thesis .
qed
lemma root_test_convergence:
fixes f :: "nat \ 'a::banach"
assumes f: "(\n. root n (norm (f n))) \ x" \ \could be weakened to lim sup\
and "x < 1"
shows "summable f"
proof -
have "0 \ x"
by (rule LIMSEQ_le[OF tendsto_const f]) (auto intro!: exI[of _ 1])
from \<open>x < 1\<close> obtain z where z: "x < z" "z < 1"
by (metis dense)
from f \<open>x < z\<close> have "eventually (\<lambda>n. root n (norm (f n)) < z) sequentially"
by (rule order_tendstoD)
then have "eventually (\n. norm (f n) \ z^n) sequentially"
using eventually_ge_at_top
proof eventually_elim
fix n
assume less: "root n (norm (f n)) < z" and n: "1 \ n"
from power_strict_mono[OF less, of n] n show "norm (f n) \ z ^ n"
by simp
qed
then show "summable f"
unfolding eventually_sequentially
using z \<open>0 \<le> x\<close> by (auto intro!: summable_comparison_test[OF _ summable_geometric])
qed
subsection \<open>More facts about binomial coefficients\<close>
text \<open>
These facts could have been proven before, but having real numbers
makes the proofs a lot easier.
\<close>
lemma central_binomial_odd:
"odd n \ n choose (Suc (n div 2)) = n choose (n div 2)"
proof -
assume "odd n"
hence "Suc (n div 2) \ n" by presburger
hence "n choose (Suc (n div 2)) = n choose (n - Suc (n div 2))"
by (rule binomial_symmetric)
also from \<open>odd n\<close> have "n - Suc (n div 2) = n div 2" by presburger
finally show ?thesis .
qed
lemma binomial_less_binomial_Suc:
assumes k: "k < n div 2"
shows "n choose k < n choose (Suc k)"
proof -
from k have k': "k \ n" "Suc k \ n" by simp_all
from k' have "real (n choose k) = fact n / (fact k * fact (n - k))"
by (simp add: binomial_fact)
also from k' have "n - k = Suc (n - Suc k)" by simp
also from k' have "fact \ = (real n - real k) * fact (n - Suc k)"
by (subst fact_Suc) (simp_all add: of_nat_diff)
also from k have "fact k = fact (Suc k) / (real k + 1)" by (simp add: field_simps)
also have "fact n / (fact (Suc k) / (real k + 1) * ((real n - real k) * fact (n - Suc k))) =
(n choose (Suc k)) * ((real k + 1) / (real n - real k))"
using k by (simp add: field_split_simps binomial_fact)
also from assms have "(real k + 1) / (real n - real k) < 1" by simp
finally show ?thesis using k by (simp add: mult_less_cancel_left)
qed
lemma binomial_strict_mono:
assumes "k < k'" "2*k' \ n"
shows "n choose k < n choose k'"
proof -
from assms have "k \ k' - 1" by simp
thus ?thesis
proof (induction rule: inc_induct)
case base
with assms binomial_less_binomial_Suc[of "k' - 1" n]
show ?case by simp
next
case (step k)
from step.prems step.hyps assms have "n choose k < n choose (Suc k)"
by (intro binomial_less_binomial_Suc) simp_all
also have "\ < n choose k'" by (rule step.IH)
finally show ?case .
qed
qed
lemma binomial_mono:
assumes "k \ k'" "2*k' \ n"
shows "n choose k \ n choose k'"
using assms binomial_strict_mono[of k k' n] by (cases "k = k'") simp_all
lemma binomial_strict_antimono:
assumes "k < k'" "2 * k \ n" "k' \ n"
shows "n choose k > n choose k'"
proof -
from assms have "n choose (n - k) > n choose (n - k')"
by (intro binomial_strict_mono) (simp_all add: algebra_simps)
with assms show ?thesis by (simp add: binomial_symmetric [symmetric])
qed
lemma binomial_antimono:
assumes "k \ k'" "k \ n div 2" "k' \ n"
shows "n choose k \ n choose k'"
proof (cases "k = k'")
case False
note not_eq = False
show ?thesis
proof (cases "k = n div 2 \ odd n")
case False
with assms(2) have "2*k \ n" by presburger
with not_eq assms binomial_strict_antimono[of k k' n]
show ?thesis by simp
next
case True
have "n choose k' \ n choose (Suc (n div 2))"
proof (cases "k' = Suc (n div 2)")
case False
with assms True not_eq have "Suc (n div 2) < k'" by simp
with assms binomial_strict_antimono[of "Suc (n div 2)" k' n] True
show ?thesis by auto
qed simp_all
also from True have "\ = n choose k" by (simp add: central_binomial_odd)
finally show ?thesis .
qed
qed simp_all
lemma binomial_maximum: "n choose k \ n choose (n div 2)"
proof -
have "k \ n div 2 \ 2*k \ n" by linarith
consider "2*k \ n" | "2*k \ n" "k \ n" | "k > n" by linarith
thus ?thesis
proof cases
case 1
thus ?thesis by (intro binomial_mono) linarith+
next
case 2
thus ?thesis by (intro binomial_antimono) simp_all
qed (simp_all add: binomial_eq_0)
qed
lemma binomial_maximum': "(2*n) choose k \ (2*n) choose n"
using binomial_maximum[of "2*n"] by simp
lemma central_binomial_lower_bound:
assumes "n > 0"
shows "4^n / (2*real n) \ real ((2*n) choose n)"
proof -
from binomial[of 1 1 "2*n"]
have "4 ^ n = (\k\2*n. (2*n) choose k)"
by (simp add: power_mult power2_eq_square One_nat_def [symmetric] del: One_nat_def)
also have "{..2*n} = {0<..<2*n} \ {0,2*n}" by auto
also have "(\k\\. (2*n) choose k) =
(\<Sum>k\<in>{0<..<2*n}. (2*n) choose k) + (\<Sum>k\<in>{0,2*n}. (2*n) choose k)"
by (subst sum.union_disjoint) auto
also have "(\k\{0,2*n}. (2*n) choose k) \ (\k\1. (n choose k)\<^sup>2)"
by (cases n) simp_all
also from assms have "\ \ (\k\n. (n choose k)\<^sup>2)"
by (intro sum_mono2) auto
also have "\ = (2*n) choose n" by (rule choose_square_sum)
also have "(\k\{0<..<2*n}. (2*n) choose k) \ (\k\{0<..<2*n}. (2*n) choose n)"
by (intro sum_mono binomial_maximum')
also have "\ = card {0<..<2*n} * ((2*n) choose n)" by simp
also have "card {0<..<2*n} \ 2*n - 1" by (cases n) simp_all
also have "(2 * n - 1) * (2 * n choose n) + (2 * n choose n) = ((2*n) choose n) * (2*n)"
using assms by (simp add: algebra_simps)
finally have "4 ^ n \ (2 * n choose n) * (2 * n)" by simp_all
hence "real (4 ^ n) \ real ((2 * n choose n) * (2 * n))"
by (subst of_nat_le_iff)
with assms show ?thesis by (simp add: field_simps)
qed
subsection \<open>Properties of Power Series\<close>
lemma powser_zero [simp]: "(\n. f n * 0 ^ n) = f 0"
for f :: "nat \ 'a::real_normed_algebra_1"
proof -
have "(\n<1. f n * 0 ^ n) = (\n. f n * 0 ^ n)"
by (subst suminf_finite[where N="{0}"]) (auto simp: power_0_left)
then show ?thesis by simp
qed
lemma powser_sums_zero: "(\n. a n * 0^n) sums a 0"
for a :: "nat \ 'a::real_normed_div_algebra"
using sums_finite [of "{0}" "\n. a n * 0 ^ n"]
by simp
lemma powser_sums_zero_iff [simp]: "(\n. a n * 0^n) sums x \ a 0 = x"
for a :: "nat \ 'a::real_normed_div_algebra"
using powser_sums_zero sums_unique2 by blast
text \<open>
Power series has a circle or radius of convergence: if it sums for \<open>x\<close>,
then it sums absolutely for \<open>z\<close> with \<^term>\<open>\<bar>z\<bar> < \<bar>x\<bar>\<close>.\<close>
lemma powser_insidea:
fixes x z :: "'a::real_normed_div_algebra"
assumes 1: "summable (\n. f n * x^n)"
and 2: "norm z < norm x"
shows "summable (\n. norm (f n * z ^ n))"
proof -
from 2 have x_neq_0: "x \ 0" by clarsimp
from 1 have "(\n. f n * x^n) \ 0"
by (rule summable_LIMSEQ_zero)
then have "convergent (\n. f n * x^n)"
by (rule convergentI)
then have "Cauchy (\n. f n * x^n)"
by (rule convergent_Cauchy)
then have "Bseq (\n. f n * x^n)"
by (rule Cauchy_Bseq)
then obtain K where 3: "0 < K" and 4: "\n. norm (f n * x^n) \ K"
by (auto simp: Bseq_def)
have "\N. \n\N. norm (norm (f n * z ^ n)) \ K * norm (z ^ n) * inverse (norm (x^n))"
proof (intro exI allI impI)
fix n :: nat
assume "0 \ n"
have "norm (norm (f n * z ^ n)) * norm (x^n) =
norm (f n * x^n) * norm (z ^ n)"
by (simp add: norm_mult abs_mult)
also have "\ \ K * norm (z ^ n)"
by (simp only: mult_right_mono 4 norm_ge_zero)
also have "\ = K * norm (z ^ n) * (inverse (norm (x^n)) * norm (x^n))"
by (simp add: x_neq_0)
also have "\ = K * norm (z ^ n) * inverse (norm (x^n)) * norm (x^n)"
by (simp only: mult.assoc)
finally show "norm (norm (f n * z ^ n)) \ K * norm (z ^ n) * inverse (norm (x^n))"
by (simp add: mult_le_cancel_right x_neq_0)
qed
moreover have "summable (\n. K * norm (z ^ n) * inverse (norm (x^n)))"
proof -
from 2 have "norm (norm (z * inverse x)) < 1"
using x_neq_0
by (simp add: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, symmetric])
then have "summable (\n. norm (z * inverse x) ^ n)"
by (rule summable_geometric)
then have "summable (\n. K * norm (z * inverse x) ^ n)"
by (rule summable_mult)
then show "summable (\n. K * norm (z ^ n) * inverse (norm (x^n)))"
using x_neq_0
by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
power_inverse norm_power mult.assoc)
qed
ultimately show "summable (\n. norm (f n * z ^ n))"
by (rule summable_comparison_test)
qed
lemma powser_inside:
fixes f :: "nat \ 'a::{real_normed_div_algebra,banach}"
shows
"summable (\n. f n * (x^n)) \ norm z < norm x \
summable (\<lambda>n. f n * (z ^ n))"
by (rule powser_insidea [THEN summable_norm_cancel])
lemma powser_times_n_limit_0:
fixes x :: "'a::{real_normed_div_algebra,banach}"
assumes "norm x < 1"
shows "(\n. of_nat n * x ^ n) \ 0"
proof -
have "norm x / (1 - norm x) \ 0"
using assms by (auto simp: field_split_simps)
moreover obtain N where N: "norm x / (1 - norm x) < of_int N"
using ex_le_of_int by (meson ex_less_of_int)
ultimately have N0: "N>0"
by auto
then have *: "real_of_int (N + 1) * norm x / real_of_int N < 1"
using N assms by (auto simp: field_simps)
have **: "real_of_int N * (norm x * (real_of_nat (Suc n) * norm (x ^ n))) \
real_of_nat n * (norm x * ((1 + N) * norm (x ^ n)))" if "N \<le> int n" for n :: nat
proof -
from that have "real_of_int N * real_of_nat (Suc n) \ real_of_nat n * real_of_int (1 + N)"
by (simp add: algebra_simps)
then have "(real_of_int N * real_of_nat (Suc n)) * (norm x * norm (x ^ n)) \
(real_of_nat n * (1 + N)) * (norm x * norm (x ^ n))"
using N0 mult_mono by fastforce
then show ?thesis
by (simp add: algebra_simps)
qed
show ?thesis using *
by (rule summable_LIMSEQ_zero [OF summable_ratio_test, where N1="nat N"])
(simp add: N0 norm_mult field_simps ** del: of_nat_Suc of_int_add)
qed
corollary lim_n_over_pown:
fixes x :: "'a::{real_normed_field,banach}"
shows "1 < norm x \ ((\n. of_nat n / x^n) \ 0) sequentially"
using powser_times_n_limit_0 [of "inverse x"]
by (simp add: norm_divide field_split_simps)
lemma sum_split_even_odd:
fixes f :: "nat \ real"
shows "(\i<2 * n. if even i then f i else g i) = (\ii
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
have "(\i<2 * Suc n. if even i then f i else g i) =
(\<Sum>i<n. f (2 * i)) + (\<Sum>i<n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
using Suc.hyps unfolding One_nat_def by auto
also have "\ = (\ii
by auto
finally show ?case .
qed
lemma sums_if':
fixes g :: "nat \ real"
assumes "g sums x"
shows "(\ n. if even n then 0 else g ((n - 1) div 2)) sums x"
unfolding sums_def
proof (rule LIMSEQ_I)
fix r :: real
assume "0 < r"
from \<open>g sums x\<close>[unfolded sums_def, THEN LIMSEQ_D, OF this]
obtain no where no_eq: "\n. n \ no \ (norm (sum g {..
by blast
let ?SUM = "\ m. \i
have "(norm (?SUM m - x) < r)" if "m \ 2 * no" for m
proof -
from that have "m div 2 \ no" by auto
have sum_eq: "?SUM (2 * (m div 2)) = sum g {..< m div 2}"
using sum_split_even_odd by auto
then have "(norm (?SUM (2 * (m div 2)) - x) < r)"
using no_eq unfolding sum_eq using \<open>m div 2 \<ge> no\<close> by auto
moreover
have "?SUM (2 * (m div 2)) = ?SUM m"
proof (cases "even m")
case True
then show ?thesis
by (auto simp: even_two_times_div_two)
next
case False
then have eq: "Suc (2 * (m div 2)) = m" by simp
then have "even (2 * (m div 2))" using \<open>odd m\<close> by auto
have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
also have "\ = ?SUM (2 * (m div 2))" using \even (2 * (m div 2))\ by auto
finally show ?thesis by auto
qed
ultimately show ?thesis by auto
qed
then show "\no. \ m \ no. norm (?SUM m - x) < r"
by blast
qed
lemma sums_if:
fixes g :: "nat \ real"
assumes "g sums x" and "f sums y"
shows "(\ n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
proof -
let ?s = "\ n. if even n then 0 else f ((n - 1) div 2)"
have if_sum: "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
for B T E
by (cases B) auto
have g_sums: "(\ n. if even n then 0 else g ((n - 1) div 2)) sums x"
using sums_if'[OF \g sums x\] .
have if_eq: "\B T E. (if \ B then T else E) = (if B then E else T)"
by auto
have "?s sums y" using sums_if'[OF \f sums y\] .
from this[unfolded sums_def, THEN LIMSEQ_Suc]
have "(\n. if even n then f (n div 2) else 0) sums y"
by (simp add: lessThan_Suc_eq_insert_0 sum.atLeast1_atMost_eq image_Suc_lessThan
if_eq sums_def cong del: if_weak_cong)
from sums_add[OF g_sums this] show ?thesis
by (simp only: if_sum)
qed
subsection \<open>Alternating series test / Leibniz formula\<close>
(* FIXME: generalise these results from the reals via type classes? *)
lemma sums_alternating_upper_lower:
fixes a :: "nat \ real"
assumes mono: "\n. a (Suc n) \ a n"
and a_pos: "\n. 0 \ a n"
and "a \ 0"
shows "\l. ((\n. (\i<2*n. (- 1)^i*a i) \ l) \ (\ n. \i<2*n. (- 1)^i*a i) \ l) \
((\<forall>n. l \<le> (\<Sum>i<2*n + 1. (- 1)^i*a i)) \<and> (\<lambda> n. \<Sum>i<2*n + 1. (- 1)^i*a i) \<longlonglongrightarrow> l)"
(is "\l. ((\n. ?f n \ l) \ _) \ ((\n. l \ ?g n) \ _)")
proof (rule nested_sequence_unique)
have fg_diff: "\n. ?f n - ?g n = - a (2 * n)" by auto
show "\n. ?f n \ ?f (Suc n)"
proof
show "?f n \ ?f (Suc n)" for n
using mono[of "2*n"] by auto
qed
show "\n. ?g (Suc n) \ ?g n"
proof
show "?g (Suc n) \ ?g n" for n
using mono[of "Suc (2*n)"] by auto
qed
show "\n. ?f n \ ?g n"
proof
show "?f n \ ?g n" for n
using fg_diff a_pos by auto
qed
show "(\n. ?f n - ?g n) \ 0"
unfolding fg_diff
proof (rule LIMSEQ_I)
fix r :: real
assume "0 < r"
with \<open>a \<longlonglongrightarrow> 0\<close>[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
by auto
then have "\n \ N. norm (- a (2 * n) - 0) < r"
by auto
then show "\N. \n \ N. norm (- a (2 * n) - 0) < r"
by auto
qed
qed
lemma summable_Leibniz':
fixes a :: "nat \ real"
assumes a_zero: "a \ 0"
and a_pos: "\n. 0 \ a n"
and a_monotone: "\n. a (Suc n) \ a n"
shows summable: "summable (\ n. (-1)^n * a n)"
and "\n. (\i<2*n. (-1)^i*a i) \ (\i. (-1)^i*a i)"
and "(\n. \i<2*n. (-1)^i*a i) \ (\i. (-1)^i*a i)"
and "\n. (\i. (-1)^i*a i) \ (\i<2*n+1. (-1)^i*a i)"
and "(\n. \i<2*n+1. (-1)^i*a i) \ (\i. (-1)^i*a i)"
proof -
let ?S = "\n. (-1)^n * a n"
let ?P = "\n. \i
let ?f = "\n. ?P (2 * n)"
let ?g = "\n. ?P (2 * n + 1)"
obtain l :: real
where below_l: "\ n. ?f n \ l"
and "?f \ l"
and above_l: "\ n. l \ ?g n"
and "?g \ l"
using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
let ?Sa = "\m. \n
have "?Sa \ l"
proof (rule LIMSEQ_I)
fix r :: real
assume "0 < r"
with \<open>?f \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
obtain f_no where f: "\n. n \ f_no \ norm (?f n - l) < r"
by auto
from \<open>0 < r\<close> \<open>?g \<longlonglongrightarrow> l\<close>[THEN LIMSEQ_D]
obtain g_no where g: "\n. n \ g_no \ norm (?g n - l) < r"
by auto
have "norm (?Sa n - l) < r" if "n \ (max (2 * f_no) (2 * g_no))" for n
proof -
from that have "n \ 2 * f_no" and "n \ 2 * g_no" by auto
show ?thesis
proof (cases "even n")
case True
then have n_eq: "2 * (n div 2) = n"
by (simp add: even_two_times_div_two)
with \<open>n \<ge> 2 * f_no\<close> have "n div 2 \<ge> f_no"
by auto
from f[OF this] show ?thesis
unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
next
case False
then have "even (n - 1)" by simp
then have n_eq: "2 * ((n - 1) div 2) = n - 1"
by (simp add: even_two_times_div_two)
then have range_eq: "n - 1 + 1 = n"
using odd_pos[OF False] by auto
from n_eq \<open>n \<ge> 2 * g_no\<close> have "(n - 1) div 2 \<ge> g_no"
by auto
from g[OF this] show ?thesis
by (simp only: n_eq range_eq)
qed
qed
then show "\no. \n \ no. norm (?Sa n - l) < r" by blast
qed
then have sums_l: "(\i. (-1)^i * a i) sums l"
by (simp only: sums_def)
then show "summable ?S"
by (auto simp: summable_def)
have "l = suminf ?S" by (rule sums_unique[OF sums_l])
fix n
show "suminf ?S \ ?g n"
unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
show "?f n \ suminf ?S"
unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
show "?g \ suminf ?S"
using \<open>?g \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
show "?f \ suminf ?S"
using \<open>?f \<longlonglongrightarrow> l\<close> \<open>l = suminf ?S\<close> by auto
qed
theorem summable_Leibniz:
fixes a :: "nat \ real"
assumes a_zero: "a \ 0"
and "monoseq a"
shows "summable (\ n. (-1)^n * a n)" (is "?summable")
and "0 < a 0 \
(\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n. (- 1)^i * a i .. \<Sum>i<2*n+1. (- 1)^i * a i})" (is "?pos")
and "a 0 < 0 \
(\<forall>n. (\<Sum>i. (- 1)^i*a i) \<in> { \<Sum>i<2*n+1. (- 1)^i * a i .. \<Sum>i<2*n. (- 1)^i * a i})" (is "?neg")
and "(\n. \i<2*n. (- 1)^i*a i) \ (\i. (- 1)^i*a i)" (is "?f")
and "(\n. \i<2*n+1. (- 1)^i*a i) \ (\i. (- 1)^i*a i)" (is "?g")
proof -
have "?summable \ ?pos \ ?neg \ ?f \ ?g"
proof (cases "(\n. 0 \ a n) \ (\m. \n\m. a n \ a m)")
case True
then have ord: "\n m. m \ n \ a n \ a m"
and ge0: "\n. 0 \ a n"
by auto
have mono: "a (Suc n) \ a n" for n
using ord[where n="Suc n" and m=n] by auto
note leibniz = summable_Leibniz'[OF \a \ 0\ ge0]
from leibniz[OF mono]
show ?thesis using \<open>0 \<le> a 0\<close> by auto
next
let ?a = "\n. - a n"
case False
with monoseq_le[OF \<open>monoseq a\<close> \<open>a \<longlonglongrightarrow> 0\<close>]
have "(\ n. a n \ 0) \ (\m. \n\m. a m \ a n)" by auto
then have ord: "\n m. m \ n \ ?a n \ ?a m" and ge0: "\ n. 0 \ ?a n"
by auto
have monotone: "?a (Suc n) \ ?a n" for n
using ord[where n="Suc n" and m=n] by auto
note leibniz =
summable_Leibniz'[OF _ ge0, of "\x. x",
OF tendsto_minus[OF \<open>a \<longlonglongrightarrow> 0\<close>, unfolded minus_zero] monotone]
have "summable (\ n. (-1)^n * ?a n)"
using leibniz(1) by auto
then obtain l where "(\ n. (-1)^n * ?a n) sums l"
unfolding summable_def by auto
from this[THEN sums_minus] have "(\ n. (-1)^n * a n) sums -l"
by auto
then have ?summable by (auto simp: summable_def)
moreover
have "\- a - - b\ = \a - b\" for a b :: real
unfolding minus_diff_minus by auto
from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
have move_minus: "(\n. - ((- 1) ^ n * a n)) = - (\n. (- 1) ^ n * a n)"
by auto
have ?pos using \<open>0 \<le> ?a 0\<close> by auto
moreover have ?neg
using leibniz(2,4)
unfolding mult_minus_right sum_negf move_minus neg_le_iff_le
by auto
moreover have ?f and ?g
using leibniz(3,5)[unfolded mult_minus_right sum_negf move_minus, THEN tendsto_minus_cancel]
by auto
ultimately show ?thesis by auto
qed
then show ?summable and ?pos and ?neg and ?f and ?g
by safe
qed
subsection \<open>Term-by-Term Differentiability of Power Series\<close>
definition diffs :: "(nat \ 'a::ring_1) \ nat \ 'a"
where "diffs c = (\n. of_nat (Suc n) * c (Suc n))"
text \<open>Lemma about distributing negation over it.\<close>
lemma diffs_minus: "diffs (\n. - c n) = (\n. - diffs c n)"
by (simp add: diffs_def)
lemma diffs_equiv:
fixes x :: "'a::{real_normed_vector,ring_1}"
shows "summable (\n. diffs c n * x^n) \
(\<lambda>n. of_nat n * c n * x^(n - Suc 0)) sums (\<Sum>n. diffs c n * x^n)"
unfolding diffs_def
by (simp add: summable_sums sums_Suc_imp)
lemma lemma_termdiff1:
fixes z :: "'a :: {monoid_mult,comm_ring}"
shows "(\p
(\<Sum>p<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
by (auto simp: algebra_simps power_add [symmetric])
lemma sumr_diff_mult_const2: "sum f {..i
for r :: "'a::ring_1"
by (simp add: sum_subtractf)
lemma lemma_termdiff2:
fixes h :: "'a::field"
assumes h: "h \ 0"
shows "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
h * (\<Sum>p< n - Suc 0. \<Sum>q< n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))"
(is "?lhs = ?rhs")
proof (cases n)
case (Suc m)
have 0: "\x k. (\n
(\<Sum>j<Suc k. h * ((h + z) ^ j * z ^ (x + k - j)))"
by (auto simp add: power_add [symmetric] mult.commute intro: sum.cong)
have *: "(\i
(\<Sum>i<m. \<Sum>j<m - i. h * ((z + h) ^ j * z ^ (m - Suc j)))"
by (force simp add: less_iff_Suc_add sum_distrib_left diff_power_eq_sum ac_simps 0
simp del: sum.lessThan_Suc power_Suc intro: sum.cong)
have "h * ?lhs = (z + h) ^ n - z ^ n - h * of_nat n * z ^ (n - Suc 0)"
by (simp add: right_diff_distrib diff_divide_distrib h mult.assoc [symmetric])
also have "... = h * ((\p
by (simp add: Suc diff_power_eq_sum h right_diff_distrib [symmetric] mult.assoc
del: power_Suc sum.lessThan_Suc of_nat_Suc)
also have "... = h * ((\p
by (subst sum.nat_diff_reindex[symmetric]) simp
also have "... = h * (\i
by (simp add: sum_subtractf)
also have "... = h * ?rhs"
by (simp add: lemma_termdiff1 sum_distrib_left Suc *)
finally have "h * ?lhs = h * ?rhs" .
then show ?thesis
by (simp add: h)
qed auto
lemma real_sum_nat_ivl_bounded2:
fixes K :: "'a::linordered_semidom"
assumes f: "\p::nat. p < n \ f p \ K" and K: "0 \ K"
shows "sum f {.. of_nat n * K"
proof -
have "sum f {.. (\i
by (rule sum_mono [OF f]) auto
also have "... \ of_nat n * K"
by (auto simp: mult_right_mono K)
finally show ?thesis .
qed
lemma lemma_termdiff3:
fixes h z :: "'a::real_normed_field"
assumes 1: "h \ 0"
and 2: "norm z \ K"
and 3: "norm (z + h) \ K"
shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) \
of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
proof -
have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
norm (\<Sum>p<n - Suc 0. \<Sum>q<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult.commute norm_mult)
also have "\ \ of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
proof (rule mult_right_mono [OF _ norm_ge_zero])
from norm_ge_zero 2 have K: "0 \ K"
by (rule order_trans)
have le_Kn: "norm ((z + h) ^ i * z ^ j) \ K ^ n" if "i + j = n" for i j n
proof -
have "norm (z + h) ^ i * norm z ^ j \ K ^ i * K ^ j"
by (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
also have "... = K^n"
by (metis power_add that)
finally show ?thesis
by (simp add: norm_mult norm_power)
qed
then have "\p q.
\<lbrakk>p < n; q < n - Suc 0\<rbrakk> \<Longrightarrow> norm ((z + h) ^ q * z ^ (n - 2 - q)) \<le> K ^ (n - 2)"
by (simp del: subst_all)
then
show "norm (\pq
of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
by (intro order_trans [OF norm_sum]
real_sum_nat_ivl_bounded2 mult_nonneg_nonneg of_nat_0_le_iff zero_le_power K)
qed
also have "\ = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
by (simp only: mult.assoc)
finally show ?thesis .
qed
lemma lemma_termdiff4:
fixes f :: "'a::real_normed_vector \ 'b::real_normed_vector"
and k :: real
assumes k: "0 < k"
and le: "\h. h \ 0 \ norm h < k \ norm (f h) \ K * norm h"
shows "f \0\ 0"
proof (rule tendsto_norm_zero_cancel)
show "(\h. norm (f h)) \0\ 0"
proof (rule real_tendsto_sandwich)
show "eventually (\h. 0 \ norm (f h)) (at 0)"
by simp
show "eventually (\h. norm (f h) \ K * norm h) (at 0)"
using k by (auto simp: eventually_at dist_norm le)
show "(\h. 0) \(0::'a)\ (0::real)"
by (rule tendsto_const)
have "(\h. K * norm h) \(0::'a)\ K * norm (0::'a)"
by (intro tendsto_intros)
then show "(\h. K * norm h) \(0::'a)\ 0"
by simp
qed
qed
lemma lemma_termdiff5:
fixes g :: "'a::real_normed_vector \ nat \ 'b::banach"
and k :: real
assumes k: "0 < k"
and f: "summable f"
and le: "\h n. h \ 0 \ norm h < k \ norm (g h n) \ f n * norm h"
shows "(\h. suminf (g h)) \0\ 0"
proof (rule lemma_termdiff4 [OF k])
fix h :: 'a
assume "h \ 0" and "norm h < k"
then have 1: "\n. norm (g h n) \ f n * norm h"
by (simp add: le)
then have "\N. \n\N. norm (norm (g h n)) \ f n * norm h"
by simp
moreover from f have 2: "summable (\n. f n * norm h)"
by (rule summable_mult2)
ultimately have 3: "summable (\n. norm (g h n))"
by (rule summable_comparison_test)
then have "norm (suminf (g h)) \ (\n. norm (g h n))"
by (rule summable_norm)
also from 1 3 2 have "(\n. norm (g h n)) \ (\n. f n * norm h)"
by (simp add: suminf_le)
also from f have "(\n. f n * norm h) = suminf f * norm h"
by (rule suminf_mult2 [symmetric])
finally show "norm (suminf (g h)) \ suminf f * norm h" .
qed
(* FIXME: Long proofs *)
lemma termdiffs_aux:
fixes x :: "'a::{real_normed_field,banach}"
assumes 1: "summable (\n. diffs (diffs c) n * K ^ n)"
and 2: "norm x < norm K"
shows "(\h. \n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \0\ 0"
proof -
from dense [OF 2] obtain r where r1: "norm x < r" and r2: "r < norm K"
by fast
from norm_ge_zero r1 have r: "0 < r"
by (rule order_le_less_trans)
then have r_neq_0: "r \ 0" by simp
show ?thesis
proof (rule lemma_termdiff5)
show "0 < r - norm x"
using r1 by simp
from r r2 have "norm (of_real r::'a) < norm K"
by simp
with 1 have "summable (\n. norm (diffs (diffs c) n * (of_real r ^ n)))"
by (rule powser_insidea)
then have "summable (\n. diffs (diffs (\n. norm (c n))) n * r ^ n)"
using r by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
then have "summable (\n. of_nat n * diffs (\n. norm (c n)) n * r ^ (n - Suc 0))"
by (rule diffs_equiv [THEN sums_summable])
also have "(\n. of_nat n * diffs (\n. norm (c n)) n * r ^ (n - Suc 0)) =
(\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
by (simp add: diffs_def r_neq_0 fun_eq_iff split: nat_diff_split)
finally have "summable
(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
by (rule diffs_equiv [THEN sums_summable])
also have
"(\n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0)) =
(\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
by (rule ext) (simp add: r_neq_0 split: nat_diff_split)
finally show "summable (\n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
next
fix h :: 'a and n
assume h: "h \ 0"
assume "norm h < r - norm x"
then have "norm x + norm h < r" by simp
with norm_triangle_ineq
have xh: "norm (x + h) < r"
by (rule order_le_less_trans)
have "norm (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))
\<le> real n * (real (n - Suc 0) * (r ^ (n - 2) * norm h))"
by (metis (mono_tags, lifting) h mult.assoc lemma_termdiff3 less_eq_real_def r1 xh)
then show "norm (c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \
norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
by (simp only: norm_mult mult.assoc mult_left_mono [OF _ norm_ge_zero])
qed
qed
lemma termdiffs:
fixes K x :: "'a::{real_normed_field,banach}"
assumes 1: "summable (\n. c n * K ^ n)"
and 2: "summable (\n. (diffs c) n * K ^ n)"
and 3: "summable (\n. (diffs (diffs c)) n * K ^ n)"
and 4: "norm x < norm K"
shows "DERIV (\x. \n. c n * x^n) x :> (\n. (diffs c) n * x^n)"
unfolding DERIV_def
proof (rule LIM_zero_cancel)
show "(\h. (suminf (\n. c n * (x + h) ^ n) - suminf (\n. c n * x^n)) / h
- suminf (\<lambda>n. diffs c n * x^n)) \<midarrow>0\<rightarrow> 0"
proof (rule LIM_equal2)
show "0 < norm K - norm x"
using 4 by (simp add: less_diff_eq)
next
fix h :: 'a
assume "norm (h - 0) < norm K - norm x"
then have "norm x + norm h < norm K" by simp
then have 5: "norm (x + h) < norm K"
by (rule norm_triangle_ineq [THEN order_le_less_trans])
have "summable (\n. c n * x^n)"
and "summable (\n. c n * (x + h) ^ n)"
and "summable (\n. diffs c n * x^n)"
using 1 2 4 5 by (auto elim: powser_inside)
then have "((\n. c n * (x + h) ^ n) - (\n. c n * x^n)) / h - (\n. diffs c n * x^n) =
(\<Sum>n. (c n * (x + h) ^ n - c n * x^n) / h - of_nat n * c n * x ^ (n - Suc 0))"
by (intro sums_unique sums_diff sums_divide diffs_equiv summable_sums)
then show "((\n. c n * (x + h) ^ n) - (\n. c n * x^n)) / h - (\n. diffs c n * x^n) =
(\<Sum>n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0)))"
by (simp add: algebra_simps)
next
show "(\h. \n. c n * (((x + h) ^ n - x^n) / h - of_nat n * x ^ (n - Suc 0))) \0\ 0"
by (rule termdiffs_aux [OF 3 4])
qed
qed
subsection \<open>The Derivative of a Power Series Has the Same Radius of Convergence\<close>
lemma termdiff_converges:
fixes x :: "'a::{real_normed_field,banach}"
assumes K: "norm x < K"
and sm: "\x. norm x < K \ summable(\n. c n * x ^ n)"
shows "summable (\n. diffs c n * x ^ n)"
proof (cases "x = 0")
case True
then show ?thesis
using powser_sums_zero sums_summable by auto
next
case False
then have "K > 0"
using K less_trans zero_less_norm_iff by blast
then obtain r :: real where r: "norm x < norm r" "norm r < K" "r > 0"
using K False
by (auto simp: field_simps abs_less_iff add_pos_pos intro: that [of "(norm x + K) / 2"])
have to0: "(\n. of_nat n * (x / of_real r) ^ n) \ 0"
using r by (simp add: norm_divide powser_times_n_limit_0 [of "x / of_real r"])
obtain N where N: "\n. n\N \ real_of_nat n * norm x ^ n < r ^ n"
using r LIMSEQ_D [OF to0, of 1]
by (auto simp: norm_divide norm_mult norm_power field_simps)
have "summable (\n. (of_nat n * c n) * x ^ n)"
proof (rule summable_comparison_test')
show "summable (\n. norm (c n * of_real r ^ n))"
apply (rule powser_insidea [OF sm [of "of_real ((r+K)/2)"]])
using N r norm_of_real [of "r + K", where 'a = 'a] by auto
show "\n. N \ n \ norm (of_nat n * c n * x ^ n) \ norm (c n * of_real r ^ n)"
using N r by (fastforce simp add: norm_mult norm_power less_eq_real_def)
qed
then have "summable (\n. (of_nat (Suc n) * c(Suc n)) * x ^ Suc n)"
using summable_iff_shift [of "\n. of_nat n * c n * x ^ n" 1]
by simp
then have "summable (\n. (of_nat (Suc n) * c(Suc n)) * x ^ n)"
using False summable_mult2 [of "\n. (of_nat (Suc n) * c(Suc n) * x ^ n) * x" "inverse x"]
by (simp add: mult.assoc) (auto simp: ac_simps)
then show ?thesis
by (simp add: diffs_def)
qed
lemma termdiff_converges_all:
fixes x :: "'a::{real_normed_field,banach}"
assumes "\x. summable (\n. c n * x^n)"
shows "summable (\n. diffs c n * x^n)"
by (rule termdiff_converges [where K = "1 + norm x"]) (use assms in auto)
lemma termdiffs_strong:
fixes K x :: "'a::{real_normed_field,banach}"
assumes sm: "summable (\n. c n * K ^ n)"
and K: "norm x < norm K"
shows "DERIV (\x. \n. c n * x^n) x :> (\n. diffs c n * x^n)"
proof -
have "norm K + norm x < norm K + norm K"
using K by force
then have K2: "norm ((of_real (norm K) + of_real (norm x)) / 2 :: 'a) < norm K"
by (auto simp: norm_triangle_lt norm_divide field_simps)
then have [simp]: "norm ((of_real (norm K) + of_real (norm x)) :: 'a) < norm K * 2"
by simp
have "summable (\n. c n * (of_real (norm x + norm K) / 2) ^ n)"
by (metis K2 summable_norm_cancel [OF powser_insidea [OF sm]] add.commute of_real_add)
moreover have "\x. norm x < norm K \ summable (\n. diffs c n * x ^ n)"
by (blast intro: sm termdiff_converges powser_inside)
moreover have "\x. norm x < norm K \ summable (\n. diffs(diffs c) n * x ^ n)"
by (blast intro: sm termdiff_converges powser_inside)
ultimately show ?thesis
by (rule termdiffs [where K = "of_real (norm x + norm K) / 2"])
(use K in \<open>auto simp: field_simps simp flip: of_real_add\<close>)
qed
lemma termdiffs_strong_converges_everywhere:
fixes K x :: "'a::{real_normed_field,banach}"
assumes "\y. summable (\n. c n * y ^ n)"
shows "((\x. \n. c n * x^n) has_field_derivative (\n. diffs c n * x^n)) (at x)"
using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
by (force simp del: of_real_add)
lemma termdiffs_strong':
fixes z :: "'a :: {real_normed_field,banach}"
assumes "\z. norm z < K \ summable (\n. c n * z ^ n)"
assumes "norm z < K"
shows "((\z. \n. c n * z^n) has_field_derivative (\n. diffs c n * z^n)) (at z)"
proof (rule termdiffs_strong)
define L :: real where "L = (norm z + K) / 2"
have "0 \ norm z" by simp
also note \<open>norm z < K\<close>
finally have K: "K \ 0" by simp
from assms K have L: "L \ 0" "norm z < L" "L < K" by (simp_all add: L_def)
from L show "norm z < norm (of_real L :: 'a)" by simp
from L show "summable (\n. c n * of_real L ^ n)" by (intro assms(1)) simp_all
qed
lemma termdiffs_sums_strong:
fixes z :: "'a :: {banach,real_normed_field}"
assumes sums: "\z. norm z < K \ (\n. c n * z ^ n) sums f z"
assumes deriv: "(f has_field_derivative f') (at z)"
assumes norm: "norm z < K"
shows "(\n. diffs c n * z ^ n) sums f'"
proof -
have summable: "summable (\n. diffs c n * z^n)"
by (intro termdiff_converges[OF norm] sums_summable[OF sums])
from norm have "eventually (\z. z \ norm -` {..
by (intro eventually_nhds_in_open open_vimage)
(simp_all add: continuous_on_norm)
hence eq: "eventually (\z. (\n. c n * z^n) = f z) (nhds z)"
by eventually_elim (insert sums, simp add: sums_iff)
have "((\z. \n. c n * z^n) has_field_derivative (\n. diffs c n * z^n)) (at z)"
by (intro termdiffs_strong'[OF _ norm] sums_summable[OF sums])
hence "(f has_field_derivative (\n. diffs c n * z^n)) (at z)"
by (subst (asm) DERIV_cong_ev[OF refl eq refl])
from this and deriv have "(\n. diffs c n * z^n) = f'" by (rule DERIV_unique)
with summable show ?thesis by (simp add: sums_iff)
qed
lemma isCont_powser:
fixes K x :: "'a::{real_normed_field,banach}"
assumes "summable (\n. c n * K ^ n)"
assumes "norm x < norm K"
shows "isCont (\x. \n. c n * x^n) x"
using termdiffs_strong[OF assms] by (blast intro!: DERIV_isCont)
lemmas isCont_powser' = isCont_o2[OF _ isCont_powser]
lemma isCont_powser_converges_everywhere:
fixes K x :: "'a::{real_normed_field,banach}"
assumes "\y. summable (\n. c n * y ^ n)"
shows "isCont (\x. \n. c n * x^n) x"
using termdiffs_strong[OF assms[of "of_real (norm x + 1)"], of x]
by (force intro!: DERIV_isCont simp del: of_real_add)
lemma powser_limit_0:
fixes a :: "nat \ 'a::{real_normed_field,banach}"
assumes s: "0 < s"
and sm: "\x. norm x < s \ (\n. a n * x ^ n) sums (f x)"
shows "(f \ a 0) (at 0)"
proof -
have "norm (of_real s / 2 :: 'a) < s"
using s by (auto simp: norm_divide)
then have "summable (\n. a n * (of_real s / 2) ^ n)"
by (rule sums_summable [OF sm])
then have "((\x. \n. a n * x ^ n) has_field_derivative (\n. diffs a n * 0 ^ n)) (at 0)"
by (rule termdiffs_strong) (use s in \<open>auto simp: norm_divide\<close>)
then have "isCont (\x. \n. a n * x ^ n) 0"
by (blast intro: DERIV_continuous)
then have "((\x. \n. a n * x ^ n) \ a 0) (at 0)"
by (simp add: continuous_within)
moreover have "(\x. f x - (\n. a n * x ^ n)) \0\ 0"
apply (clarsimp simp: LIM_eq)
apply (rule_tac x=s in exI)
using s sm sums_unique by fastforce
ultimately show ?thesis
by (rule Lim_transform)
qed
lemma powser_limit_0_strong:
fixes a :: "nat \ 'a::{real_normed_field,banach}"
assumes s: "0 < s"
and sm: "\x. x \ 0 \ norm x < s \ (\n. a n * x ^ n) sums (f x)"
shows "(f \ a 0) (at 0)"
proof -
have *: "((\x. if x = 0 then a 0 else f x) \ a 0) (at 0)"
by (rule powser_limit_0 [OF s]) (auto simp: powser_sums_zero sm)
show ?thesis
using "*" by (auto cong: Lim_cong_within)
qed
subsection \<open>Derivability of power series\<close>
lemma DERIV_series':
fixes f :: "real \ nat \ real"
assumes DERIV_f: "\ n. DERIV (\ x. f x n) x0 :> (f' x0 n)"
and allf_summable: "\ x. x \ {a <..< b} \ summable (f x)"
and x0_in_I: "x0 \ {a <..< b}"
and "summable (f' x0)"
and "summable L"
and L_def: "\n x y. x \ {a <..< b} \ y \ {a <..< b} \ \f x n - f y n\ \ L n * \x - y\"
shows "DERIV (\ x. suminf (f x)) x0 :> (suminf (f' x0))"
unfolding DERIV_def
proof (rule LIM_I)
fix r :: real
assume "0 < r" then have "0 < r/3" by auto
obtain N_L where N_L: "\ n. N_L \ n \ \ \ i. L (i + n) \ < r/3"
using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable L\<close>] by auto
obtain N_f' where N_f': "\ n. N_f' \ n \ \ \ i. f' x0 (i + n) \ < r/3"
using suminf_exist_split[OF \<open>0 < r/3\<close> \<open>summable (f' x0)\<close>] by auto
let ?N = "Suc (max N_L N_f')"
have "\ \ i. f' x0 (i + ?N) \ < r/3" (is "?f'_part < r/3")
and L_estimate: "\ \ i. L (i + ?N) \ < r/3"
using N_L[of "?N"] and N_f' [of "?N"] by auto
let ?diff = "\i x. (f (x0 + x) i - f x0 i) / x"
let ?r = "r / (3 * real ?N)"
from \<open>0 < r\<close> have "0 < ?r" by simp
let ?s = "\n. SOME s. 0 < s \ (\ x. x \ 0 \ \ x \ < s \ \ ?diff n x - f' x0 n \ < ?r)"
define S' where "S' = Min (?s ` {..< ?N })"
have "0 < S'"
unfolding S'_def
proof (rule iffD2[OF Min_gr_iff])
show "\x \ (?s ` {..< ?N }). 0 < x"
proof
fix x
assume "x \ ?s ` {..
then obtain n where "x = ?s n" and "n \ {..
using image_iff[THEN iffD1] by blast
from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>, unfolded real_norm_def]
obtain s where s_bound: "0 < s \ (\x. x \ 0 \ \x\ < s \ \?diff n x - f' x0 n\ < ?r)"
by auto
have "0 < ?s n"
by (rule someI2[where a=s]) (auto simp: s_bound simp del: of_nat_Suc)
then show "0 < x" by (simp only: \<open>x = ?s n\<close>)
qed
qed auto
define S where "S = min (min (x0 - a) (b - x0)) S'"
then have "0 < S" and S_a: "S \ x0 - a" and S_b: "S \ b - x0"
and "S \ S'" using x0_in_I and \0 < S'\
by auto
have "\(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\ < r"
if "x \ 0" and "\x\ < S" for x
proof -
from that have x_in_I: "x0 + x \ {a <..< b}"
using S_a S_b by auto
note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
note div_smbl = summable_divide[OF diff_smbl]
note all_smbl = summable_diff[OF div_smbl \<open>summable (f' x0)\<close>]
note ign = summable_ignore_initial_segment[where k="?N"]
note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
note div_shft_smbl = summable_divide[OF diff_shft_smbl]
note all_shft_smbl = summable_diff[OF div_smbl ign[OF \<open>summable (f' x0)\<close>]]
have 1: "\(\?diff (n + ?N) x\)\ \ L (n + ?N)" for n
proof -
have "\?diff (n + ?N) x\ \ L (n + ?N) * \(x0 + x) - x0\ / \x\"
using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
by (simp only: abs_divide)
with \<open>x \<noteq> 0\<close> show ?thesis by auto
qed
note 2 = summable_rabs_comparison_test[OF _ ign[OF \<open>summable L\<close>]]
from 1 have "\ \ i. ?diff (i + ?N) x \ \ (\ i. L (i + ?N))"
by (metis (lifting) abs_idempotent
order_trans[OF summable_rabs[OF 2] suminf_le[OF _ 2 ign[OF \<open>summable L\<close>]]])
then have "\\i. ?diff (i + ?N) x\ \ r / 3" (is "?L_part \ r/3")
using L_estimate by auto
have "\\n \ (\n?diff n x - f' x0 n\)" ..
also have "\ < (\n
proof (rule sum_strict_mono)
fix n
assume "n \ {..< ?N}"
have "\x\ < S" using \\x\ < S\ .
also have "S \ S'" using \S \ S'\ .
also have "S' \ ?s n"
unfolding S'_def
proof (rule Min_le_iff[THEN iffD2])
have "?s n \ (?s ` {.. ?s n \ ?s n"
using \<open>n \<in> {..< ?N}\<close> by auto
then show "\ a \ (?s ` {.. ?s n"
by blast
qed auto
finally have "\x\ < ?s n" .
from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF \<open>0 < ?r\<close>,
unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
have "\x. x \ 0 \ \x\ < ?s n \ \?diff n x - f' x0 n\ < ?r" .
with \<open>x \<noteq> 0\<close> and \<open>\<bar>x\<bar> < ?s n\<close> show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
by blast
qed auto
also have "\ = of_nat (card {..
by (rule sum_constant)
also have "\ = real ?N * ?r"
by simp
also have "\ = r/3"
by (auto simp del: of_nat_Suc)
finally have "\\n < r / 3" (is "?diff_part < r / 3") .
from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
have "\(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\ =
\<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
unfolding suminf_diff[OF div_smbl \<open>summable (f' x0)\<close>, symmetric]
using suminf_divide[OF diff_smbl, symmetric] by auto
also have "\ \ ?diff_part + \(\n. ?diff (n + ?N) x) - (\ n. f' x0 (n + ?N))\"
unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
unfolding suminf_diff[OF div_shft_smbl ign[OF \<open>summable (f' x0)\<close>]]
apply (simp only: add.commute)
using abs_triangle_ineq by blast
also have "\ \ ?diff_part + ?L_part + ?f'_part"
using abs_triangle_ineq4 by auto
also have "\ < r /3 + r/3 + r/3"
using \<open>?diff_part < r/3\<close> \<open>?L_part \<le> r/3\<close> and \<open>?f'_part < r/3\<close>
by (rule add_strict_mono [OF add_less_le_mono])
finally show ?thesis
by auto
qed
then show "\s > 0. \ x. x \ 0 \ norm (x - 0) < s \
norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
using \<open>0 < S\<close> by auto
qed
lemma DERIV_power_series':
fixes f :: "nat \ real"
assumes converges: "\x. x \ {-R <..< R} \ summable (\n. f n * real (Suc n) * x^n)"
and x0_in_I: "x0 \ {-R <..< R}"
and "0 < R"
shows "DERIV (\x. (\n. f n * x^(Suc n))) x0 :> (\n. f n * real (Suc n) * x0^n)"
(is "DERIV (\x. suminf (?f x)) x0 :> suminf (?f' x0)")
proof -
have for_subinterval: "DERIV (\x. suminf (?f x)) x0 :> suminf (?f' x0)"
if "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'" for R'
proof -
from that have "x0 \ {-R' <..< R'}" and "R' \ {-R <..< R}" and "x0 \ {-R <..< R}"
by auto
show ?thesis
proof (rule DERIV_series')
show "summable (\ n. \f n * real (Suc n) * R'^n\)"
proof -
have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
then have in_Rball: "(R' + R) / 2 \ {-R <..< R}"
using \<open>R' < R\<close> by auto
have "norm R' < norm ((R' + R) / 2)"
using \<open>0 < R'\<close> \<open>0 < R\<close> \<open>R' < R\<close> by (auto simp: field_simps)
from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
by auto
qed
next
fix n x y
assume "x \ {-R' <..< R'}" and "y \ {-R' <..< R'}"
show "\?f x n - ?f y n\ \ \f n * real (Suc n) * R'^n\ * \x-y\"
proof -
have "\f n * x ^ (Suc n) - f n * y ^ (Suc n)\ =
(\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p<Suc n. x ^ p * y ^ (n - p)\<bar>"
unfolding right_diff_distrib[symmetric] diff_power_eq_sum abs_mult
by auto
also have "\ \ (\f n\ * \x-y\) * (\real (Suc n)\ * \R' ^ n\)"
proof (rule mult_left_mono)
have "\\p \ (\px ^ p * y ^ (n - p)\)"
by (rule sum_abs)
also have "\ \ (\p
proof (rule sum_mono)
fix p
assume "p \ {..
then have "p \ n" by auto
have "\x^n\ \ R'^n" if "x \ {-R'<..
proof -
from that have "\x\ \ R'" by auto
then show ?thesis
unfolding power_abs by (rule power_mono) auto
qed
from mult_mono[OF this[OF \<open>x \<in> {-R'<..<R'}\<close>, of p] this[OF \<open>y \<in> {-R'<..<R'}\<close>, of "n-p"]]
and \<open>0 < R'\<close>
have "\x^p * y^(n - p)\ \ R'^p * R'^(n - p)"
unfolding abs_mult by auto
then show "\x^p * y^(n - p)\ \ R'^n"
unfolding power_add[symmetric] using \<open>p \<le> n\<close> by auto
qed
also have "\ = real (Suc n) * R' ^ n"
unfolding sum_constant card_atLeastLessThan by auto
finally show "\\p \ \real (Suc n)\ * \R' ^ n\"
unfolding abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF \<open>0 < R'\<close>]]]
by linarith
show "0 \ \f n\ * \x - y\"
unfolding abs_mult[symmetric] by auto
qed
also have "\ = \f n * real (Suc n) * R' ^ n\ * \x - y\"
unfolding abs_mult mult.assoc[symmetric] by algebra
finally show ?thesis .
qed
next
show "DERIV (\x. ?f x n) x0 :> ?f' x0 n" for n
by (auto intro!: derivative_eq_intros simp del: power_Suc)
next
fix x
assume "x \ {-R' <..< R'}"
then have "R' \ {-R <..< R}" and "norm x < norm R'"
using assms \<open>R' < R\<close> by auto
have "summable (\n. f n * x^n)"
proof (rule summable_comparison_test, intro exI allI impI)
fix n
have le: "\f n\ * 1 \ \f n\ * real (Suc n)"
by (rule mult_left_mono) auto
show "norm (f n * x^n) \ norm (f n * real (Suc n) * x^n)"
unfolding real_norm_def abs_mult
using le mult_right_mono by fastforce
qed (rule powser_insidea[OF converges[OF \<open>R' \<in> {-R <..< R}\<close>] \<open>norm x < norm R'\<close>])
from this[THEN summable_mult2[where c=x], simplified mult.assoc, simplified mult.commute]
show "summable (?f x)" by auto
next
show "summable (?f' x0)"
using converges[OF \<open>x0 \<in> {-R <..< R}\<close>] .
show "x0 \ {-R' <..< R'}"
using \<open>x0 \<in> {-R' <..< R'}\<close> .
qed
qed
let ?R = "(R + \x0\) / 2"
have "\x0\ < ?R"
using assms by (auto simp: field_simps)
then have "- ?R < x0"
proof (cases "x0 < 0")
case True
then have "- x0 < ?R"
using \<open>\<bar>x0\<bar> < ?R\<close> by auto
then show ?thesis
unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
next
case False
have "- ?R < 0" using assms by auto
also have "\ \ x0" using False by auto
finally show ?thesis .
qed
then have "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
using assms by (auto simp: field_simps)
from for_subinterval[OF this] show ?thesis .
qed
lemma geometric_deriv_sums:
fixes z :: "'a :: {real_normed_field,banach}"
assumes "norm z < 1"
shows "(\n. of_nat (Suc n) * z ^ n) sums (1 / (1 - z)^2)"
proof -
have "(\n. diffs (\n. 1) n * z^n) sums (1 / (1 - z)^2)"
proof (rule termdiffs_sums_strong)
fix z :: 'a assume "norm z < 1"
thus "(\n. 1 * z^n) sums (1 / (1 - z))" by (simp add: geometric_sums)
qed (insert assms, auto intro!: derivative_eq_intros simp: power2_eq_square)
thus ?thesis unfolding diffs_def by simp
qed
lemma isCont_pochhammer [continuous_intros]: "isCont (\z. pochhammer z n) z"
for z :: "'a::real_normed_field"
by (induct n) (auto simp: pochhammer_rec')
lemma continuous_on_pochhammer [continuous_intros]: "continuous_on A (\z. pochhammer z n)"
for A :: "'a::real_normed_field set"
by (intro continuous_at_imp_continuous_on ballI isCont_pochhammer)
lemmas continuous_on_pochhammer' [continuous_intros] =
continuous_on_compose2[OF continuous_on_pochhammer _ subset_UNIV]
subsection \<open>Exponential Function\<close>
definition exp :: "'a \ 'a::{real_normed_algebra_1,banach}"
where "exp = (\x. \n. x^n /\<^sub>R fact n)"
lemma summable_exp_generic:
fixes x :: "'a::{real_normed_algebra_1,banach}"
defines S_def: "S \ \n. x^n /\<^sub>R fact n"
shows "summable S"
proof -
have S_Suc: "\n. S (Suc n) = (x * S n) /\<^sub>R (Suc n)"
unfolding S_def by (simp del: mult_Suc)
obtain r :: real where r0: "0 < r" and r1: "r < 1"
using dense [OF zero_less_one] by fast
obtain N :: nat where N: "norm x < real N * r"
using ex_less_of_nat_mult r0 by auto
from r1 show ?thesis
proof (rule summable_ratio_test [rule_format])
fix n :: nat
assume n: "N \ n"
have "norm x \ real N * r"
using N by (rule order_less_imp_le)
also have "real N * r \ real (Suc n) * r"
using r0 n by (simp add: mult_right_mono)
finally have "norm x * norm (S n) \ real (Suc n) * r * norm (S n)"
using norm_ge_zero by (rule mult_right_mono)
then have "norm (x * S n) \ real (Suc n) * r * norm (S n)"
by (rule order_trans [OF norm_mult_ineq])
then have "norm (x * S n) / real (Suc n) \ r * norm (S n)"
by (simp add: pos_divide_le_eq ac_simps)
then show "norm (S (Suc n)) \ r * norm (S n)"
by (simp add: S_Suc inverse_eq_divide)
qed
qed
lemma summable_norm_exp: "summable (\n. norm (x^n /\<^sub>R fact n))"
for x :: "'a::{real_normed_algebra_1,banach}"
proof (rule summable_norm_comparison_test [OF exI, rule_format])
show "summable (\n. norm x^n /\<^sub>R fact n)"
by (rule summable_exp_generic)
show "norm (x^n /\<^sub>R fact n) \ norm x^n /\<^sub>R fact n" for n
by (simp add: norm_power_ineq)
qed
lemma summable_exp: "summable (\n. inverse (fact n) * x^n)"
for x :: "'a::{real_normed_field,banach}"
using summable_exp_generic [where x=x]
by (simp add: scaleR_conv_of_real nonzero_of_real_inverse)
lemma exp_converges: "(\n. x^n /\<^sub>R fact n) sums exp x"
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
lemma exp_fdiffs:
"diffs (\n. inverse (fact n)) = (\n. inverse (fact n :: 'a::{real_normed_field,banach}))"
by (simp add: diffs_def mult_ac nonzero_inverse_mult_distrib nonzero_of_real_inverse
del: mult_Suc of_nat_Suc)
lemma diffs_of_real: "diffs (\n. of_real (f n)) = (\n. of_real (diffs f n))"
by (simp add: diffs_def)
lemma DERIV_exp [simp]: "DERIV exp x :> exp x"
unfolding exp_def scaleR_conv_of_real
proof (rule DERIV_cong)
have sinv: "summable (\n. of_real (inverse (fact n)) * x ^ n)" for x::'a
by (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])
note xx = exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real]
show "((\x. \n. of_real (inverse (fact n)) * x ^ n) has_field_derivative
(\<Sum>n. diffs (\<lambda>n. of_real (inverse (fact n))) n * x ^ n)) (at x)"
by (rule termdiffs [where K="of_real (1 + norm x)"]) (simp_all only: diffs_of_real exp_fdiffs sinv norm_of_real)
show "(\n. diffs (\n. of_real (inverse (fact n))) n * x ^ n) = (\n. of_real (inverse (fact n)) * x ^ n)"
by (simp add: diffs_of_real exp_fdiffs)
qed
declare DERIV_exp[THEN DERIV_chain2, derivative_intros]
and DERIV_exp[THEN DERIV_chain2, unfolded has_field_derivative_def, derivative_intros]
lemmas has_derivative_exp[derivative_intros] = DERIV_exp[THEN DERIV_compose_FDERIV]
lemma norm_exp: "norm (exp x) \ exp (norm x)"
proof -
from summable_norm[OF summable_norm_exp, of x]
have "norm (exp x) \ (\n. inverse (fact n) * norm (x^n))"
by (simp add: exp_def)
also have "\ \ exp (norm x)"
using summable_exp_generic[of "norm x"] summable_norm_exp[of x]
by (auto simp: exp_def intro!: suminf_le norm_power_ineq)
finally show ?thesis .
qed
lemma isCont_exp: "isCont exp x"
for x :: "'a::{real_normed_field,banach}"
by (rule DERIV_exp [THEN DERIV_isCont])
lemma isCont_exp' [simp]: "isCont f a \ isCont (\x. exp (f x)) a"
for f :: "_ \'a::{real_normed_field,banach}"
by (rule isCont_o2 [OF _ isCont_exp])
lemma tendsto_exp [tendsto_intros]: "(f \ a) F \ ((\x. exp (f x)) \ exp a) F"
for f:: "_ \'a::{real_normed_field,banach}"
by (rule isCont_tendsto_compose [OF isCont_exp])
lemma continuous_exp [continuous_intros]: "continuous F f \ continuous F (\x. exp (f x))"
for f :: "_ \'a::{real_normed_field,banach}"
unfolding continuous_def by (rule tendsto_exp)
lemma continuous_on_exp [continuous_intros]: "continuous_on s f \ continuous_on s (\x. exp (f x))"
for f :: "_ \'a::{real_normed_field,banach}"
unfolding continuous_on_def by (auto intro: tendsto_exp)
subsubsection \<open>Properties of the Exponential Function\<close>
lemma exp_zero [simp]: "exp 0 = 1"
unfolding exp_def by (simp add: scaleR_conv_of_real)
lemma exp_series_add_commuting:
fixes x y :: "'a::{real_normed_algebra_1,banach}"
defines S_def: "S \ \x n. x^n /\<^sub>R fact n"
assumes comm: "x * y = y * x"
shows "S (x + y) n = (\i\n. S x i * S y (n - i))"
proof (induct n)
case 0
show ?case
unfolding S_def by simp
next
case (Suc n)
have S_Suc: "\x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
unfolding S_def by (simp del: mult_Suc)
then have times_S: "\x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
by simp
have S_comm: "\n. S x n * y = y * S x n"
by (simp add: power_commuting_commutes comm S_def)
have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * (\i\n. S x i * S y (n - i))"
by (metis Suc.hyps times_S)
also have "\ = x * (\i\n. S x i * S y (n - i)) + y * (\i\n. S x i * S y (n - i))"
by (rule distrib_right)
also have "\ = (\i\n. x * S x i * S y (n - i)) + (\i\n. S x i * y * S y (n - i))"
by (simp add: sum_distrib_left ac_simps S_comm)
also have "\ = (\i\n. x * S x i * S y (n - i)) + (\i\n. S x i * (y * S y (n - i)))"
by (simp add: ac_simps)
also have "\ = (\i\n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i)))
+ (\<Sum>i\<le>n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))"
by (simp add: times_S Suc_diff_le)
also have "(\i\n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i)))
= (\<Sum>i\<le>Suc n. real i *\<^sub>R (S x i * S y (Suc n - i)))"
by (subst sum.atMost_Suc_shift) simp
also have "(\i\n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))
= (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))"
by simp
also have "(\i\Suc n. real i *\<^sub>R (S x i * S y (Suc n - i)))
+ (\<Sum>i\<le>Suc n. real (Suc n - i) *\<^sub>R (S x i * S y (Suc n - i)))
= (\<Sum>i\<le>Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n - i)))"
by (simp flip: sum.distrib scaleR_add_left of_nat_add)
also have "\ = real (Suc n) *\<^sub>R (\i\Suc n. S x i * S y (Suc n - i))"
by (simp only: scaleR_right.sum)
finally show "S (x + y) (Suc n) = (\i\Suc n. S x i * S y (Suc n - i))"
by (simp del: sum.cl_ivl_Suc)
qed
lemma exp_add_commuting: "x * y = y * x \ exp (x + y) = exp x * exp y"
by (simp only: exp_def Cauchy_product summable_norm_exp exp_series_add_commuting)
lemma exp_times_arg_commute: "exp A * A = A * exp A"
by (simp add: exp_def suminf_mult[symmetric] summable_exp_generic power_commutes suminf_mult2)
lemma exp_add: "exp (x + y) = exp x * exp y"
for x y :: "'a::{real_normed_field,banach}"
by (rule exp_add_commuting) (simp add: ac_simps)
lemma exp_double: "exp(2 * z) = exp z ^ 2"
by (simp add: exp_add_commuting mult_2 power2_eq_square)
lemmas mult_exp_exp = exp_add [symmetric]
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
unfolding exp_def
apply (subst suminf_of_real [OF summable_exp_generic])
apply (simp add: scaleR_conv_of_real)
done
lemmas of_real_exp = exp_of_real[symmetric]
corollary exp_in_Reals [simp]: "z \ \ \ exp z \ \"
by (metis Reals_cases Reals_of_real exp_of_real)
lemma exp_not_eq_zero [simp]: "exp x \ 0"
proof
have "exp x * exp (- x) = 1"
by (simp add: exp_add_commuting[symmetric])
also assume "exp x = 0"
finally show False by simp
qed
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