author  huffman 
Fri, 19 Aug 2011 18:08:05 0700  
changeset 44318  425c1f8f9487 
parent 44317  b7e9fa025f15 
child 44319  806e0390de53 
permissions  rwrr 
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(* Title: HOL/Transcendental.thy 
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Author: Jacques D. Fleuriot, University of Cambridge, University of Edinburgh 
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Author: Lawrence C Paulson 
12196  4 
*) 
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header{*Power Series, Transcendental Functions etc.*} 
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15131  8 
theory Transcendental 
25600  9 
imports Fact Series Deriv NthRoot 
15131  10 
begin 
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29164  12 
subsection {* Properties of Power Series *} 
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lemma lemma_realpow_diff: 
31017  15 
fixes y :: "'a::monoid_mult" 
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shows "p \<le> n \<Longrightarrow> y ^ (Suc n  p) = (y ^ (n  p)) * y" 
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proof  
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assume "p \<le> n" 
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hence "Suc n  p = Suc (n  p)" by (rule Suc_diff_le) 
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thus ?thesis by (simp add: power_commutes) 
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qed 
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lemma lemma_realpow_diff_sumr: 
31017  24 
fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows 
41970  25 
"(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n  p)) = 
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y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n  p))" 
29163  27 
by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac 
33549  28 
del: setsum_op_ivl_Suc) 
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15229  30 
lemma lemma_realpow_diff_sumr2: 
31017  31 
fixes y :: "'a::{comm_ring,monoid_mult}" shows 
41970  32 
"x ^ (Suc n)  y ^ (Suc n) = 
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(x  y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n  p))" 
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apply (induct n, simp) 
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apply (simp del: setsum_op_ivl_Suc) 
15561  36 
apply (subst setsum_op_ivl_Suc) 
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apply (subst lemma_realpow_diff_sumr) 
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apply (simp add: right_distrib del: setsum_op_ivl_Suc) 
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apply (subst mult_left_commute [of "x  y"]) 
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apply (erule subst) 
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apply (simp add: algebra_simps) 
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done 
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15229  44 
lemma lemma_realpow_rev_sumr: 
41970  45 
"(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n  p))) = 
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(\<Sum>p=0..<Suc n. (x ^ (n  p)) * (y ^ p))" 
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apply (rule setsum_reindex_cong [where f="\<lambda>i. n  i"]) 
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apply (rule inj_onI, simp) 
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apply auto 
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apply (rule_tac x="n  x" in image_eqI, simp, simp) 
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done 
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text{*Power series has a `circle` of convergence, i.e. if it sums for @{term 
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x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*} 
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lemma powser_insidea: 
31017  57 
fixes x z :: "'a::{real_normed_field,banach}" 
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assumes 1: "summable (\<lambda>n. f n * x ^ n)" 
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assumes 2: "norm z < norm x" 
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shows "summable (\<lambda>n. norm (f n * z ^ n))" 
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proof  
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from 2 have x_neq_0: "x \<noteq> 0" by clarsimp 
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from 1 have "(\<lambda>n. f n * x ^ n) > 0" 
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by (rule summable_LIMSEQ_zero) 
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hence "convergent (\<lambda>n. f n * x ^ n)" 
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by (rule convergentI) 
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hence "Cauchy (\<lambda>n. f n * x ^ n)" 
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by (simp add: Cauchy_convergent_iff) 
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hence "Bseq (\<lambda>n. f n * x ^ n)" 
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by (rule Cauchy_Bseq) 
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then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K" 
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by (simp add: Bseq_def, safe) 
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have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le> 
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K * norm (z ^ n) * inverse (norm (x ^ n))" 
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proof (intro exI allI impI) 
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fix n::nat assume "0 \<le> n" 
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have "norm (norm (f n * z ^ n)) * norm (x ^ n) = 
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norm (f n * x ^ n) * norm (z ^ n)" 
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by (simp add: norm_mult abs_mult) 
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also have "\<dots> \<le> K * norm (z ^ n)" 
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by (simp only: mult_right_mono 4 norm_ge_zero) 
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also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))" 
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by (simp add: x_neq_0) 
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also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)" 
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by (simp only: mult_assoc) 
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finally show "norm (norm (f n * z ^ n)) \<le> 
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K * norm (z ^ n) * inverse (norm (x ^ n))" 
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by (simp add: mult_le_cancel_right x_neq_0) 
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qed 
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moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))" 
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proof  
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from 2 have "norm (norm (z * inverse x)) < 1" 
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using x_neq_0 
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by (simp add: nonzero_norm_divide divide_inverse [symmetric]) 
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hence "summable (\<lambda>n. norm (z * inverse x) ^ n)" 
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by (rule summable_geometric) 
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hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)" 
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by (rule summable_mult) 
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thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))" 
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using x_neq_0 
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by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib 
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power_inverse norm_power mult_assoc) 
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qed 
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ultimately show "summable (\<lambda>n. norm (f n * z ^ n))" 
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by (rule summable_comparison_test) 
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qed 
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15229  108 
lemma powser_inside: 
31017  109 
fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" shows 
41970  110 
"[ summable (%n. f(n) * (x ^ n)); norm z < norm x ] 
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==> summable (%n. f(n) * (z ^ n))" 
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by (rule powser_insidea [THEN summable_norm_cancel]) 
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lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows 
41970  115 
"(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) = 
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(\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))" 
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proof (induct n) 
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case (Suc n) 
41970  119 
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(\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))" 
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using Suc.hyps unfolding One_nat_def by auto 
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also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto 
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finally show ?case . 
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qed auto 
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lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x" 
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shows "(\<lambda> n. if even n then 0 else g ((n  1) div 2)) sums x" 
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unfolding sums_def 
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proof (rule LIMSEQ_I) 
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fix r :: real assume "0 < r" 
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from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this] 
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obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n }  x) < r)" by blast 
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let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i  1) div 2)" 
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{ fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto 
41970  136 
have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }" 
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137 
using sum_split_even_odd by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

138 
hence "(norm (?SUM (2 * (m div 2))  x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

139 
moreover 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

140 
have "?SUM (2 * (m div 2)) = ?SUM m" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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141 
proof (cases "even m") 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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142 
case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] .. 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

143 
next 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

144 
case False hence "even (Suc m)" by auto 
c56a5571f60a
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hoelzl
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changeset

145 
from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

146 
have eq: "Suc (2 * (m div 2)) = m" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

147 
hence "even (2 * (m div 2))" using `odd m` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

148 
have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq .. 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
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149 
also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto 
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hoelzl
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150 
finally show ?thesis by auto 
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hoelzl
parents:
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151 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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152 
ultimately have "(norm (?SUM m  x) < r)" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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153 
} 
c56a5571f60a
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hoelzl
parents:
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154 
thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m  x) < r" by blast 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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155 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

156 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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157 
lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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158 
shows "(\<lambda> n. if even n then f (n div 2) else g ((n  1) div 2)) sums (x + y)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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159 
proof  
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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160 
let ?s = "\<lambda> n. if even n then 0 else f ((n  1) div 2)" 
c56a5571f60a
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hoelzl
parents:
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changeset

161 
{ fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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changeset

162 
by (cases B) auto } note if_sum = this 
c56a5571f60a
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hoelzl
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163 
have g_sums: "(\<lambda> n. if even n then 0 else g ((n  1) div 2)) sums x" using sums_if'[OF `g sums x`] . 
41970  164 
{ 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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165 
have "?s 0 = 0" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

166 
have Suc_m1: "\<And> n. Suc n  1 = n" by auto 
41550  167 
have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

168 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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169 
have "?s sums y" using sums_if'[OF `f sums y`] . 
41970  170 
from this[unfolded sums_def, THEN LIMSEQ_Suc] 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

171 
have "(\<lambda> n. if even n then f (n div 2) else 0) sums y" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

172 
unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

173 
image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def] 
31148  174 
even_Suc Suc_m1 if_eq . 
29803
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Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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175 
} from sums_add[OF g_sums this] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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176 
show ?thesis unfolding if_sum . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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177 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
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178 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

179 
subsection {* Alternating series test / Leibniz formula *} 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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180 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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181 
lemma sums_alternating_upper_lower: 
c56a5571f60a
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hoelzl
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182 
fixes a :: "nat \<Rightarrow> real" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
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changeset

183 
assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a > 0" 
41970  184 
shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. 1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. 1^i*a i) > l) \<and> 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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185 
((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. 1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. 1^i*a i) > l)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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changeset

186 
(is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)") 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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187 
proof  
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset

188 
have fg_diff: "\<And>n. ?f n  ?g n =  a (2 * n)" unfolding One_nat_def by auto 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

189 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

190 
have "\<forall> n. ?f n \<le> ?f (Suc n)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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191 
proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

192 
moreover 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

193 
have "\<forall> n. ?g (Suc n) \<le> ?g n" 
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset

194 
proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"] 
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset

195 
unfolding One_nat_def by auto qed 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

196 
moreover 
41970  197 
have "\<forall> n. ?f n \<le> ?g n" 
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset

198 
proof fix n show "?f n \<le> ?g n" using fg_diff a_pos 
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset

199 
unfolding One_nat_def by auto qed 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

200 
moreover 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

201 
have "(\<lambda> n. ?f n  ?g n) > 0" unfolding fg_diff 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

202 
proof (rule LIMSEQ_I) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

203 
fix r :: real assume "0 < r" 
41970  204 
with `a > 0`[THEN LIMSEQ_D] 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

205 
obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n  0) < r" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

206 
hence "\<forall> n \<ge> N. norm ( a (2 * n)  0) < r" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

207 
thus "\<exists> N. \<forall> n \<ge> N. norm ( a (2 * n)  0) < r" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

208 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

209 
ultimately 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

210 
show ?thesis by (rule lemma_nest_unique) 
41970  211 
qed 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

212 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

213 
lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

214 
assumes a_zero: "a > 0" and a_pos: "\<And> n. 0 \<le> a n" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

215 
and a_monotone: "\<And> n. a (Suc n) \<le> a n" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

216 
shows summable: "summable (\<lambda> n. (1)^n * a n)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

217 
and "\<And>n. (\<Sum>i=0..<2*n. (1)^i*a i) \<le> (\<Sum>i. (1)^i*a i)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

218 
and "(\<lambda>n. \<Sum>i=0..<2*n. (1)^i*a i) > (\<Sum>i. (1)^i*a i)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

219 
and "\<And>n. (\<Sum>i. (1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (1)^i*a i)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

220 
and "(\<lambda>n. \<Sum>i=0..<2*n+1. (1)^i*a i) > (\<Sum>i. (1)^i*a i)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

221 
proof  
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

222 
let "?S n" = "(1)^n * a n" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

223 
let "?P n" = "\<Sum>i=0..<n. ?S i" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

224 
let "?f n" = "?P (2 * n)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

225 
let "?g n" = "?P (2 * n + 1)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

226 
obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f > l" and above_l: "\<forall> n. l \<le> ?g n" and "?g > l" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

227 
using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast 
41970  228 

29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

229 
let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

230 
have "?Sa > l" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

231 
proof (rule LIMSEQ_I) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

232 
fix r :: real assume "0 < r" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

233 

41970  234 
with `?f > l`[THEN LIMSEQ_D] 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

235 
obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n  l) < r" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

236 

41970  237 
from `0 < r` `?g > l`[THEN LIMSEQ_D] 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

238 
obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n  l) < r" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

239 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

240 
{ fix n :: nat 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

241 
assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

242 
have "norm (?Sa n  l) < r" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

243 
proof (cases "even n") 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

244 
case True from even_nat_div_two_times_two[OF this] 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

245 
have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

246 
with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

247 
from f[OF this] 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

248 
show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost . 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
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diff
changeset

249 
next 
35213  250 
case False hence "even (n  1)" by simp 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

251 
from even_nat_div_two_times_two[OF this] 
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252 
have n_eq: "2 * ((n  1) div 2) = n  1" unfolding numeral_2_eq_2[symmetric] by auto 
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253 
hence range_eq: "n  1 + 1 = n" using odd_pos[OF False] by auto 
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254 

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255 
from n_eq `n \<ge> 2 * g_no` have "(n  1) div 2 \<ge> g_no" by auto 
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256 
from g[OF this] 
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257 
show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq . 
29803
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258 
qed 
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259 
} 
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260 
thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n  l) < r" by blast 
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261 
qed 
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262 
hence sums_l: "(\<lambda>i. (1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] . 
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263 
thus "summable ?S" using summable_def by auto 
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264 

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265 
have "l = suminf ?S" using sums_unique[OF sums_l] . 
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266 

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267 
{ fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto } 
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268 
{ fix n show "?f n \<le> suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto } 
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269 
show "?g > suminf ?S" using `?g > l` `l = suminf ?S` by auto 
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270 
show "?f > suminf ?S" using `?f > l` `l = suminf ?S` by auto 
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271 
qed 
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272 

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273 
theorem summable_Leibniz: fixes a :: "nat \<Rightarrow> real" 
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274 
assumes a_zero: "a > 0" and "monoseq a" 
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275 
shows "summable (\<lambda> n. (1)^n * a n)" (is "?summable") 
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276 
and "0 < a 0 \<longrightarrow> (\<forall>n. (\<Sum>i. 1^i*a i) \<in> { \<Sum>i=0..<2*n. 1^i * a i .. \<Sum>i=0..<2*n+1. 1^i * a i})" (is "?pos") 
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277 
and "a 0 < 0 \<longrightarrow> (\<forall>n. (\<Sum>i. 1^i*a i) \<in> { \<Sum>i=0..<2*n+1. 1^i * a i .. \<Sum>i=0..<2*n. 1^i * a i})" (is "?neg") 
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278 
and "(\<lambda>n. \<Sum>i=0..<2*n. 1^i*a i) > (\<Sum>i. 1^i*a i)" (is "?f") 
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279 
and "(\<lambda>n. \<Sum>i=0..<2*n+1. 1^i*a i) > (\<Sum>i. 1^i*a i)" (is "?g") 
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280 
proof  
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281 
have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g" 
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282 
proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)") 
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283 
case True 
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284 
hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto 
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285 
{ fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=n] by auto } 
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286 
note leibniz = summable_Leibniz'[OF `a > 0` ge0] and mono = this 
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287 
from leibniz[OF mono] 
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288 
show ?thesis using `0 \<le> a 0` by auto 
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289 
next 
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290 
let ?a = "\<lambda> n.  a n" 
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291 
case False 
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292 
with monoseq_le[OF `monoseq a` `a > 0`] 
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293 
have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto 
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294 
hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto 
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295 
{ fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto } 
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296 
note monotone = this 
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297 
note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF LIMSEQ_minus[OF `a > 0`, unfolded minus_zero] monotone] 
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298 
have "summable (\<lambda> n. (1)^n * ?a n)" using leibniz(1) by auto 
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299 
then obtain l where "(\<lambda> n. (1)^n * ?a n) sums l" unfolding summable_def by auto 
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300 
from this[THEN sums_minus] 
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301 
have "(\<lambda> n. (1)^n * a n) sums l" by auto 
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302 
hence ?summable unfolding summable_def by auto 
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303 
moreover 
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304 
have "\<And> a b :: real. \<bar>  a   b \<bar> = \<bar>a  b\<bar>" unfolding minus_diff_minus by auto 
41970  305 

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306 
from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus] 
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307 
have move_minus: "(\<Sum>n.  (1 ^ n * a n)) =  (\<Sum>n. 1 ^ n * a n)" by auto 
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308 

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309 
have ?pos using `0 \<le> ?a 0` by auto 
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310 
moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto 
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311 
moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN LIMSEQ_minus_cancel] by auto 
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312 
ultimately show ?thesis by auto 
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313 
qed 
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314 
from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1] 
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315 
this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2] 
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316 
show ?summable and ?pos and ?neg and ?f and ?g . 
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317 
qed 
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318 

29164  319 
subsection {* TermbyTerm Differentiability of Power Series *} 
23043  320 

321 
definition 

23082
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322 
diffs :: "(nat => 'a::ring_1) => nat => 'a" where 
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323 
"diffs c = (%n. of_nat (Suc n) * c(Suc n))" 
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324 

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325 
text{*Lemma about distributing negation over it*} 
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326 
lemma diffs_minus: "diffs (%n.  c n) = (%n.  diffs c n)" 
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327 
by (simp add: diffs_def) 
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328 

29163  329 
lemma sums_Suc_imp: 
330 
assumes f: "f 0 = 0" 

331 
shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s" 

332 
unfolding sums_def 

333 
apply (rule LIMSEQ_imp_Suc) 

334 
apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric]) 

335 
apply (simp only: setsum_shift_bounds_Suc_ivl) 

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336 
done 
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337 

15229  338 
lemma diffs_equiv: 
41970  339 
fixes x :: "'a::{real_normed_vector, ring_1}" 
340 
shows "summable (%n. (diffs c)(n) * (x ^ n)) ==> 

341 
(%n. of_nat n * c(n) * (x ^ (n  Suc 0))) sums 

15546  342 
(\<Sum>n. (diffs c)(n) * (x ^ n))" 
29163  343 
unfolding diffs_def 
344 
apply (drule summable_sums) 

345 
apply (rule sums_Suc_imp, simp_all) 

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346 
done 
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347 

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348 
lemma lemma_termdiff1: 
31017  349 
fixes z :: "'a :: {monoid_mult,comm_ring}" shows 
41970  350 
"(\<Sum>p=0..<m. (((z + h) ^ (m  p)) * (z ^ p))  (z ^ m)) = 
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351 
(\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m  p))  (z ^ (m  p))))" 
41550  352 
by(auto simp add: algebra_simps power_add [symmetric]) 
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353 

23082
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354 
lemma sumr_diff_mult_const2: 
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355 
"setsum f {0..<n}  of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i  r)" 
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356 
by (simp add: setsum_subtractf) 
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357 

15229  358 
lemma lemma_termdiff2: 
31017  359 
fixes h :: "'a :: {field}" 
20860  360 
assumes h: "h \<noteq> 0" shows 
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361 
"((z + h) ^ n  z ^ n) / h  of_nat n * z ^ (n  Suc 0) = 
20860  362 
h * (\<Sum>p=0..< n  Suc 0. \<Sum>q=0..< n  Suc 0  p. 
23082
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363 
(z + h) ^ q * z ^ (n  2  q))" (is "?lhs = ?rhs") 
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364 
apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h) 
20860  365 
apply (simp add: right_diff_distrib diff_divide_distrib h) 
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366 
apply (simp add: mult_assoc [symmetric]) 
20860  367 
apply (cases "n", simp) 
368 
apply (simp add: lemma_realpow_diff_sumr2 h 

369 
right_diff_distrib [symmetric] mult_assoc 

30273
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370 
del: power_Suc setsum_op_ivl_Suc of_nat_Suc) 
20860  371 
apply (subst lemma_realpow_rev_sumr) 
23082
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372 
apply (subst sumr_diff_mult_const2) 
20860  373 
apply simp 
374 
apply (simp only: lemma_termdiff1 setsum_right_distrib) 

375 
apply (rule setsum_cong [OF refl]) 

15539  376 
apply (simp add: diff_minus [symmetric] less_iff_Suc_add) 
20860  377 
apply (clarify) 
378 
apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac 

30273
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379 
del: setsum_op_ivl_Suc power_Suc) 
20860  380 
apply (subst mult_assoc [symmetric], subst power_add [symmetric]) 
381 
apply (simp add: mult_ac) 

382 
done 

383 

384 
lemma real_setsum_nat_ivl_bounded2: 

35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34974
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changeset

385 
fixes K :: "'a::linordered_semidom" 
23082
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386 
assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K" 
ffef77eed382
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387 
assumes K: "0 \<le> K" 
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388 
shows "setsum f {0..<nk} \<le> of_nat n * K" 
ffef77eed382
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389 
apply (rule order_trans [OF setsum_mono]) 
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390 
apply (rule f, simp) 
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391 
apply (simp add: mult_right_mono K) 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

392 
done 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

393 

15229  394 
lemma lemma_termdiff3: 
31017  395 
fixes h z :: "'a::{real_normed_field}" 
20860  396 
assumes 1: "h \<noteq> 0" 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

397 
assumes 2: "norm z \<le> K" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

398 
assumes 3: "norm (z + h) \<le> K" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

399 
shows "norm (((z + h) ^ n  z ^ n) / h  of_nat n * z ^ (n  Suc 0)) 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

400 
\<le> of_nat n * of_nat (n  Suc 0) * K ^ (n  2) * norm h" 
20860  401 
proof  
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

402 
have "norm (((z + h) ^ n  z ^ n) / h  of_nat n * z ^ (n  Suc 0)) = 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

403 
norm (\<Sum>p = 0..<n  Suc 0. \<Sum>q = 0..<n  Suc 0  p. 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

404 
(z + h) ^ q * z ^ (n  2  q)) * norm h" 
20860  405 
apply (subst lemma_termdiff2 [OF 1]) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

406 
apply (subst norm_mult) 
20860  407 
apply (rule mult_commute) 
408 
done 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

409 
also have "\<dots> \<le> of_nat n * (of_nat (n  Suc 0) * K ^ (n  2)) * norm h" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

410 
proof (rule mult_right_mono [OF _ norm_ge_zero]) 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

411 
from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans) 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

412 
have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n" 
20860  413 
apply (erule subst) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

414 
apply (simp only: norm_mult norm_power power_add) 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

415 
apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K) 
20860  416 
done 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

417 
show "norm (\<Sum>p = 0..<n  Suc 0. \<Sum>q = 0..<n  Suc 0  p. 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

418 
(z + h) ^ q * z ^ (n  2  q)) 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

419 
\<le> of_nat n * (of_nat (n  Suc 0) * K ^ (n  2))" 
20860  420 
apply (intro 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

421 
order_trans [OF norm_setsum] 
20860  422 
real_setsum_nat_ivl_bounded2 
423 
mult_nonneg_nonneg 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

424 
zero_le_imp_of_nat 
20860  425 
zero_le_power K) 
426 
apply (rule le_Kn, simp) 

427 
done 

428 
qed 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

429 
also have "\<dots> = of_nat n * of_nat (n  Suc 0) * K ^ (n  2) * norm h" 
20860  430 
by (simp only: mult_assoc) 
431 
finally show ?thesis . 

432 
qed 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

433 

20860  434 
lemma lemma_termdiff4: 
31017  435 
fixes f :: "'a::{real_normed_field} \<Rightarrow> 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

436 
'b::real_normed_vector" 
20860  437 
assumes k: "0 < (k::real)" 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

438 
assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h" 
20860  439 
shows "f  0 > 0" 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31271
diff
changeset

440 
unfolding LIM_eq diff_0_right 
29163  441 
proof (safe) 
442 
let ?h = "of_real (k / 2)::'a" 

443 
have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all 

444 
hence "norm (f ?h) \<le> K * norm ?h" by (rule le) 

445 
hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero]) 

446 
hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff) 

447 

20860  448 
fix r::real assume r: "0 < r" 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

449 
show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" 
20860  450 
proof (cases) 
451 
assume "K = 0" 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

452 
with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)" 
20860  453 
by simp 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

454 
thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" .. 
20860  455 
next 
456 
assume K_neq_zero: "K \<noteq> 0" 

457 
with zero_le_K have K: "0 < K" by simp 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

458 
show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" 
20860  459 
proof (rule exI, safe) 
460 
from k r K show "0 < min k (r * inverse K / 2)" 

461 
by (simp add: mult_pos_pos positive_imp_inverse_positive) 

462 
next 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

463 
fix x::'a 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

464 
assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

465 
from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2" 
20860  466 
by simp_all 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

467 
from x1 x3 le have "norm (f x) \<le> K * norm x" by simp 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

468 
also from x4 K have "K * norm x < K * (r * inverse K / 2)" 
20860  469 
by (rule mult_strict_left_mono) 
470 
also have "\<dots> = r / 2" 

471 
using K_neq_zero by simp 

472 
also have "r / 2 < r" 

473 
using r by simp 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

474 
finally show "norm (f x) < r" . 
20860  475 
qed 
476 
qed 

477 
qed 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

478 

15229  479 
lemma lemma_termdiff5: 
31017  480 
fixes g :: "'a::{real_normed_field} \<Rightarrow> 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

481 
nat \<Rightarrow> 'b::banach" 
20860  482 
assumes k: "0 < (k::real)" 
483 
assumes f: "summable f" 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

484 
assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h" 
20860  485 
shows "(\<lambda>h. suminf (g h))  0 > 0" 
486 
proof (rule lemma_termdiff4 [OF k]) 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

487 
fix h::'a assume "h \<noteq> 0" and "norm h < k" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

488 
hence A: "\<forall>n. norm (g h n) \<le> f n * norm h" 
20860  489 
by (simp add: le) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

490 
hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h" 
20860  491 
by simp 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

492 
moreover from f have B: "summable (\<lambda>n. f n * norm h)" 
20860  493 
by (rule summable_mult2) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

494 
ultimately have C: "summable (\<lambda>n. norm (g h n))" 
20860  495 
by (rule summable_comparison_test) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

496 
hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

497 
by (rule summable_norm) 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

498 
also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)" 
20860  499 
by (rule summable_le) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

500 
also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h" 
20860  501 
by (rule suminf_mult2 [symmetric]) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

502 
finally show "norm (suminf (g h)) \<le> suminf f * norm h" . 
20860  503 
qed 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

504 

89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

505 

89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

506 
text{* FIXME: Long proofs*} 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

507 

89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

508 
lemma termdiffs_aux: 
31017  509 
fixes x :: "'a::{real_normed_field,banach}" 
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

510 
assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)" 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

511 
assumes 2: "norm x < norm K" 
20860  512 
shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n  x ^ n) / h 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

513 
 of_nat n * x ^ (n  Suc 0)))  0 > 0" 
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

514 
proof  
20860  515 
from dense [OF 2] 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

516 
obtain r where r1: "norm x < r" and r2: "r < norm K" by fast 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

517 
from norm_ge_zero r1 have r: "0 < r" 
20860  518 
by (rule order_le_less_trans) 
519 
hence r_neq_0: "r \<noteq> 0" by simp 

520 
show ?thesis 

20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

521 
proof (rule lemma_termdiff5) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

522 
show "0 < r  norm x" using r1 by simp 
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

523 
next 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

524 
from r r2 have "norm (of_real r::'a) < norm K" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

525 
by simp 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

526 
with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))" 
20860  527 
by (rule powser_insidea) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

528 
hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

529 
using r 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

530 
by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc) 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

531 
hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n  Suc 0))" 
20860  532 
by (rule diffs_equiv [THEN sums_summable]) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

533 
also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n  Suc 0)) 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

534 
= (\<lambda>n. diffs (%m. of_nat (m  Suc 0) * norm (c m) * inverse r) n * (r ^ n))" 
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

535 
apply (rule ext) 
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

536 
apply (simp add: diffs_def) 
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

537 
apply (case_tac n, simp_all add: r_neq_0) 
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

538 
done 
41970  539 
finally have "summable 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

540 
(\<lambda>n. of_nat n * (of_nat (n  Suc 0) * norm (c n) * inverse r) * r ^ (n  Suc 0))" 
20860  541 
by (rule diffs_equiv [THEN sums_summable]) 
542 
also have 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

543 
"(\<lambda>n. of_nat n * (of_nat (n  Suc 0) * norm (c n) * inverse r) * 
20860  544 
r ^ (n  Suc 0)) = 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

545 
(\<lambda>n. norm (c n) * of_nat n * of_nat (n  Suc 0) * r ^ (n  2))" 
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

546 
apply (rule ext) 
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

547 
apply (case_tac "n", simp) 
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

548 
apply (case_tac "nat", simp) 
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

549 
apply (simp add: r_neq_0) 
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

550 
done 
20860  551 
finally show 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

552 
"summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n  Suc 0) * r ^ (n  2))" . 
20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

553 
next 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

554 
fix h::'a and n::nat 
20860  555 
assume h: "h \<noteq> 0" 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

556 
assume "norm h < r  norm x" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

557 
hence "norm x + norm h < r" by simp 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

558 
with norm_triangle_ineq have xh: "norm (x + h) < r" 
20860  559 
by (rule order_le_less_trans) 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

560 
show "norm (c n * (((x + h) ^ n  x ^ n) / h  of_nat n * x ^ (n  Suc 0))) 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

561 
\<le> norm (c n) * of_nat n * of_nat (n  Suc 0) * r ^ (n  2) * norm h" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

562 
apply (simp only: norm_mult mult_assoc) 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

563 
apply (rule mult_left_mono [OF _ norm_ge_zero]) 
20860  564 
apply (simp (no_asm) add: mult_assoc [symmetric]) 
565 
apply (rule lemma_termdiff3) 

566 
apply (rule h) 

567 
apply (rule r1 [THEN order_less_imp_le]) 

568 
apply (rule xh [THEN order_less_imp_le]) 

569 
done 

20849
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

570 
qed 
389cd9c8cfe1
rewrite proofs of powser_insidea and termdiffs_aux
huffman
parents:
20692
diff
changeset

571 
qed 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
19765
diff
changeset

572 

20860  573 
lemma termdiffs: 
31017  574 
fixes K x :: "'a::{real_normed_field,banach}" 
20860  575 
assumes 1: "summable (\<lambda>n. c n * K ^ n)" 
576 
assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)" 

577 
assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)" 

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

578 
assumes 4: "norm x < norm K" 
20860  579 
shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)" 
29163  580 
unfolding deriv_def 
581 
proof (rule LIM_zero_cancel) 

20860  582 
show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n)  suminf (\<lambda>n. c n * x ^ n)) / h 
583 
 suminf (\<lambda>n. diffs c n * x ^ n))  0 > 0" 

584 
proof (rule LIM_equal2) 

29163  585 
show "0 < norm K  norm x" using 4 by (simp add: less_diff_eq) 
20860  586 
next 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

587 
fix h :: 'a 
20860  588 
assume "h \<noteq> 0" 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

589 
assume "norm (h  0) < norm K  norm x" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

590 
hence "norm x + norm h < norm K" by simp 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

591 
hence 5: "norm (x + h) < norm K" 
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

592 
by (rule norm_triangle_ineq [THEN order_le_less_trans]) 
20860  593 
have A: "summable (\<lambda>n. c n * x ^ n)" 
594 
by (rule powser_inside [OF 1 4]) 

595 
have B: "summable (\<lambda>n. c n * (x + h) ^ n)" 

596 
by (rule powser_inside [OF 1 5]) 

597 
have C: "summable (\<lambda>n. diffs c n * x ^ n)" 

598 
by (rule powser_inside [OF 2 4]) 

599 
show "((\<Sum>n. c n * (x + h) ^ n)  (\<Sum>n. c n * x ^ n)) / h 

41970  600 
 (\<Sum>n. diffs c n * x ^ n) = 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

601 
(\<Sum>n. c n * (((x + h) ^ n  x ^ n) / h  of_nat n * x ^ (n  Suc 0)))" 
20860  602 
apply (subst sums_unique [OF diffs_equiv [OF C]]) 
603 
apply (subst suminf_diff [OF B A]) 

604 
apply (subst suminf_divide [symmetric]) 

605 
apply (rule summable_diff [OF B A]) 

606 
apply (subst suminf_diff) 

607 
apply (rule summable_divide) 

608 
apply (rule summable_diff [OF B A]) 

609 
apply (rule sums_summable [OF diffs_equiv [OF C]]) 

29163  610 
apply (rule arg_cong [where f="suminf"], rule ext) 
29667  611 
apply (simp add: algebra_simps) 
20860  612 
done 
613 
next 

614 
show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n  x ^ n) / h  

23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

615 
of_nat n * x ^ (n  Suc 0)))  0 > 0" 
20860  616 
by (rule termdiffs_aux [OF 3 4]) 
617 
qed 

618 
qed 

619 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

620 

29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

621 
subsection {* Derivability of power series *} 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

622 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

623 
lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

624 
assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

625 
and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

626 
and "summable (f' x0)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

627 
and "summable L" and L_def: "\<And> n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar> f x n  f y n \<bar> \<le> L n * \<bar> x  y \<bar>" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

628 
shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

629 
unfolding deriv_def 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

630 
proof (rule LIM_I) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

631 
fix r :: real assume "0 < r" hence "0 < r/3" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

632 

41970  633 
obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3" 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

634 
using suminf_exist_split[OF `0 < r/3` `summable L`] by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

635 

41970  636 
obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3" 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

637 
using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

638 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

639 
let ?N = "Suc (max N_L N_f')" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

640 
have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

641 
L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

642 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

643 
let "?diff i x" = "(f (x0 + x) i  f x0 i) / x" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

644 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

645 
let ?r = "r / (3 * real ?N)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

646 
have "0 < 3 * real ?N" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

647 
from divide_pos_pos[OF `0 < r` this] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

648 
have "0 < ?r" . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

649 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

650 
let "?s n" = "SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x  f' x0 n \<bar> < ?r)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

651 
def S' \<equiv> "Min (?s ` { 0 ..< ?N })" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

652 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

653 
have "0 < S'" unfolding S'_def 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

654 
proof (rule iffD2[OF Min_gr_iff]) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

655 
show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

656 
proof (rule ballI) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

657 
fix x assume "x \<in> ?s ` {0..<?N}" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

658 
then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast 
41970  659 
from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def] 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

660 
obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x  f' x0 n\<bar> < ?r)" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

661 
have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

662 
thus "0 < x" unfolding `x = ?s n` . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

663 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

664 
qed auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

665 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

666 
def S \<equiv> "min (min (x0  a) (b  x0)) S'" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

667 
hence "0 < S" and S_a: "S \<le> x0  a" and S_b: "S \<le> b  x0" and "S \<le> S'" using x0_in_I and `0 < S'` 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

668 
by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

669 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

670 
{ fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

671 
hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto 
41970  672 

29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

673 
note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

674 
note div_smbl = summable_divide[OF diff_smbl] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

675 
note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

676 
note ign = summable_ignore_initial_segment[where k="?N"] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

677 
note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

678 
note div_shft_smbl = summable_divide[OF diff_shft_smbl] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

679 
note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

680 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

681 
{ fix n 
41970  682 
have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x)  x0 \<bar> / \<bar> x \<bar>" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

683 
using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide . 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

684 
hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

685 
} note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

686 
from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

687 
have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

688 
hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

689 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

690 
have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x  f' x0 n \<bar> \<le> (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x  f' x0 n \<bar>)" .. 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

691 
also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

692 
proof (rule setsum_strict_mono) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

693 
fix n assume "n \<in> { 0 ..< ?N}" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

694 
have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

695 
also have "S \<le> S'" using `S \<le> S'` . 
41970  696 
also have "S' \<le> ?s n" unfolding S'_def 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

697 
proof (rule Min_le_iff[THEN iffD2]) 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

698 
have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

699 
thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

700 
qed auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

701 
finally have "\<bar> x \<bar> < ?s n" . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

702 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

703 
from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

704 
have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x  f' x0 n\<bar> < ?r" . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

705 
with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n` 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

706 
show "\<bar>?diff n x  f' x0 n\<bar> < ?r" by blast 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

707 
qed auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

708 
also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

709 
also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

710 
also have "\<dots> = r/3" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

711 
finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x  f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

712 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

713 
from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]] 
41970  714 
have "\<bar> (suminf (f (x0 + x))  (suminf (f x0))) / x  suminf (f' x0) \<bar> = 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

715 
\<bar> \<Sum>n. ?diff n x  f' x0 n \<bar>" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

716 
also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x)  (\<Sum> n. f' x0 (n + ?N)) \<bar>" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

717 
also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto 
41970  718 
also have "\<dots> < r /3 + r/3 + r/3" 
36842  719 
using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3` 
720 
by (rule add_strict_mono [OF add_less_le_mono]) 

29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

721 
finally have "\<bar> (suminf (f (x0 + x))  (suminf (f x0))) / x  suminf (f' x0) \<bar> < r" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

722 
by auto 
41970  723 
} thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x  0) < s \<longrightarrow> 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

724 
norm (((\<Sum>n. f (x0 + x) n)  (\<Sum>n. f x0 n)) / x  (\<Sum>n. f' x0 n)) < r" using `0 < S` 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

725 
unfolding real_norm_def diff_0_right by blast 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

726 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

727 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

728 
lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

729 
assumes converges: "\<And> x. x \<in> {R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

730 
and x0_in_I: "x0 \<in> {R <..< R}" and "0 < R" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

731 
shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

732 
(is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))") 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

733 
proof  
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

734 
{ fix R' assume "0 < R'" and "R' < R" and "R' < x0" and "x0 < R'" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

735 
hence "x0 \<in> {R' <..< R'}" and "R' \<in> {R <..< R}" and "x0 \<in> {R <..< R}" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

736 
have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

737 
proof (rule DERIV_series') 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

738 
show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

739 
proof  
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

740 
have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

741 
hence in_Rball: "(R' + R) / 2 \<in> {R <..< R}" using `R' < R` by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

742 
have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

743 
from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

744 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

745 
{ fix n x y assume "x \<in> {R' <..< R'}" and "y \<in> {R' <..< R'}" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

746 
show "\<bar>?f x n  ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>xy\<bar>" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

747 
proof  
41970  748 
have "\<bar>f n * x ^ (Suc n)  f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>xy\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n  p)\<bar>" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

749 
unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto 
41970  750 
also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>xy\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

751 
proof (rule mult_left_mono) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

752 
have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n  p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n  p)\<bar>)" by (rule setsum_abs) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

753 
also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

754 
proof (rule setsum_mono) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

755 
fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

756 
{ fix n fix x :: real assume "x \<in> {R'<..<R'}" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

757 
hence "\<bar>x\<bar> \<le> R'" by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

758 
hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

759 
} from mult_mono[OF this[OF `x \<in> {R'<..<R'}`, of p] this[OF `y \<in> {R'<..<R'}`, of "np"]] `0 < R'` 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

760 
have "\<bar>x^p * y^(np)\<bar> \<le> R'^p * R'^(np)" unfolding abs_mult by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

761 
thus "\<bar>x^p * y^(np)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

762 
qed 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

763 
also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

764 
finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n  p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] . 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

765 
show "0 \<le> \<bar>f n\<bar> * \<bar>x  y\<bar>" unfolding abs_mult[symmetric] by auto 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

766 
qed 
36777
be5461582d0f
avoid using realspecific versions of generic lemmas
huffman
parents:
36776
diff
changeset

767 
also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x  y\<bar>" unfolding abs_mult mult_assoc[symmetric] by algebra 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

768 
finally show ?thesis . 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

769 
qed } 
31881  770 
{ fix n show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

771 
by (auto intro!: DERIV_intros simp del: power_Suc) } 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

772 
{ fix x assume "x \<in> {R' <..< R'}" hence "R' \<in> {R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

773 
have "summable (\<lambda> n. f n * x^n)" 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

774 
proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {R <..< R}`] `norm x < norm R'`]], rule allI) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

775 
fix n 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

776 
have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

777 
show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

778 
by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right]) 
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

779 
qed 
36777
be5461582d0f
avoid using realspecific versions of generic lemmas
huffman
parents:
36776
diff
changeset

780 
from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute] 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

781 
show "summable (?f x)" by auto } 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

782 
show "summable (?f' x0)" using converges[OF `x0 \<in> {R <..< R}`] . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

783 
show "x0 \<in> {R' <..< R'}" using `x0 \<in> {R' <..< R'}` . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

784 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

785 
} note for_subinterval = this 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

786 
let ?R = "(R + \<bar>x0\<bar>) / 2" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

787 
have "\<bar>x0\<bar> < ?R" using assms by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

788 
hence " ?R < x0" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

789 
proof (cases "x0 < 0") 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

790 
case True 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

791 
hence " x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

792 
thus ?thesis unfolding neg_less_iff_less[symmetric, of " x0"] by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

793 
next 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

794 
case False 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

795 
have " ?R < 0" using assms by auto 
41970  796 
also have "\<dots> \<le> x0" using False by auto 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

797 
finally show ?thesis . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

798 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

799 
hence "0 < ?R" "?R < R" " ?R < x0" and "x0 < ?R" using assms by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

800 
from for_subinterval[OF this] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

801 
show ?thesis . 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

802 
qed 
29695  803 

29164  804 
subsection {* Exponential Function *} 
23043  805 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

806 
definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

807 
"exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))" 
23043  808 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

809 
lemma summable_exp_generic: 
31017  810 
fixes x :: "'a::{real_normed_algebra_1,banach}" 
25062  811 
defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)" 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

812 
shows "summable S" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

813 
proof  
25062  814 
have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30082
diff
changeset

815 
unfolding S_def by (simp del: mult_Suc) 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

816 
obtain r :: real where r0: "0 < r" and r1: "r < 1" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

817 
using dense [OF zero_less_one] by fast 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

818 
obtain N :: nat where N: "norm x < real N * r" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

819 
using reals_Archimedean3 [OF r0] by fast 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

820 
from r1 show ?thesis 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

821 
proof (rule ratio_test [rule_format]) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

822 
fix n :: nat 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

823 
assume n: "N \<le> n" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

824 
have "norm x \<le> real N * r" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

825 
using N by (rule order_less_imp_le) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

826 
also have "real N * r \<le> real (Suc n) * r" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

827 
using r0 n by (simp add: mult_right_mono) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

828 
finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

829 
using norm_ge_zero by (rule mult_right_mono) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

830 
hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

831 
by (rule order_trans [OF norm_mult_ineq]) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

832 
hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

833 
by (simp add: pos_divide_le_eq mult_ac) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

834 
thus "norm (S (Suc n)) \<le> r * norm (S n)" 
35216  835 
by (simp add: S_Suc inverse_eq_divide) 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

836 
qed 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

837 
qed 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

838 

4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

839 
lemma summable_norm_exp: 
31017  840 
fixes x :: "'a::{real_normed_algebra_1,banach}" 
25062  841 
shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))" 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

842 
proof (rule summable_norm_comparison_test [OF exI, rule_format]) 
25062  843 
show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))" 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

844 
by (rule summable_exp_generic) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

845 
next 
25062  846 
fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)" 
35216  847 
by (simp add: norm_power_ineq) 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

848 
qed 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

849 

23043  850 
lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)" 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

851 
by (insert summable_exp_generic [where x=x], simp) 
23043  852 

25062  853 
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x" 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

854 
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums]) 
23043  855 

856 

41970  857 
lemma exp_fdiffs: 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

858 
"diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))" 
23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23413
diff
changeset

859 
by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult 
23082
ffef77eed382
generalize powerseries and termdiffs lemmas using axclasses
huffman
parents:
23069
diff
changeset

860 
del: mult_Suc of_nat_Suc) 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

861 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

862 
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

863 
by (simp add: diffs_def) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

864 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

865 
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)" 
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

866 
unfolding exp_def scaleR_conv_of_real 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

867 
apply (rule DERIV_cong) 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

868 
apply (rule termdiffs [where K="of_real (1 + norm x)"]) 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

869 
apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

870 
apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+ 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

871 
apply (simp del: of_real_add) 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

872 
done 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

873 

44311  874 
lemma isCont_exp: "isCont exp x" 
875 
by (rule DERIV_exp [THEN DERIV_isCont]) 

876 

877 
lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a" 

878 
by (rule isCont_o2 [OF _ isCont_exp]) 

879 

880 
lemma tendsto_exp [tendsto_intros]: 

881 
"(f > a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) > exp a) F" 

882 
by (rule isCont_tendsto_compose [OF isCont_exp]) 

23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset

883 

95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset

884 

29167  885 
subsubsection {* Properties of the Exponential Function *} 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

886 

23278  887 
lemma powser_zero: 
31017  888 
fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}" 
23278  889 
shows "(\<Sum>n. f n * 0 ^ n) = f 0" 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

890 
proof  
23278  891 
have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)" 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

892 
by (rule sums_unique [OF series_zero], simp add: power_0_left) 
30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset

893 
thus ?thesis unfolding One_nat_def by simp 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

894 
qed 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

895 

23278  896 
lemma exp_zero [simp]: "exp 0 = 1" 
897 
unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero) 

898 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

899 
lemma setsum_cl_ivl_Suc2: 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

900 
"(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))" 
28069  901 
by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

902 
del: setsum_cl_ivl_Suc) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

903 

4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

904 
lemma exp_series_add: 
31017  905 
fixes x y :: "'a::{real_field}" 
25062  906 
defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)" 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

907 
shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n  i))" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

908 
proof (induct n) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

909 
case 0 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

910 
show ?case 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

911 
unfolding S_def by simp 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

912 
next 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

913 
case (Suc n) 
25062  914 
have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)" 
30273
ecd6f0ca62ea
declare power_Suc [simp]; remove redundant typespecific versions of power_Suc
huffman
parents:
30082
diff
changeset

915 
unfolding S_def by (simp del: mult_Suc) 
25062  916 
hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)" 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

917 
by simp 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

918 

25062  919 
have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n" 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

920 
by (simp only: times_S) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

921 
also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (ni))" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

922 
by (simp only: Suc) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

923 
also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (ni)) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

924 
+ y * (\<Sum>i=0..n. S x i * S y (ni))" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

925 
by (rule left_distrib) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

926 
also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (ni)) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

927 
+ (\<Sum>i=0..n. S x i * (y * S y (ni)))" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

928 
by (simp only: setsum_right_distrib mult_ac) 
25062  929 
also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (ni))) 
930 
+ (\<Sum>i=0..n. real (Suc ni) *\<^sub>R (S x i * S y (Suc ni)))" 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

931 
by (simp add: times_S Suc_diff_le) 
25062  932 
also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (ni))) = 
933 
(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc ni)))" 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

934 
by (subst setsum_cl_ivl_Suc2, simp) 
25062  935 
also have "(\<Sum>i=0..n. real (Suc ni) *\<^sub>R (S x i * S y (Suc ni))) = 
936 
(\<Sum>i=0..Suc n. real (Suc ni) *\<^sub>R (S x i * S y (Suc ni)))" 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

937 
by (subst setsum_cl_ivl_Suc, simp) 
25062  938 
also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc ni))) + 
939 
(\<Sum>i=0..Suc n. real (Suc ni) *\<^sub>R (S x i * S y (Suc ni))) = 

940 
(\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc ni)))" 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

941 
by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric] 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

942 
real_of_nat_add [symmetric], simp) 
25062  943 
also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc ni))" 
23127  944 
by (simp only: scaleR_right.setsum) 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

945 
finally show 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

946 
"S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n  i))" 
35216  947 
by (simp del: setsum_cl_ivl_Suc) 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

948 
qed 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

949 

4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

950 
lemma exp_add: "exp (x + y) = exp x * exp y" 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

951 
unfolding exp_def 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

952 
by (simp only: Cauchy_product summable_norm_exp exp_series_add) 
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

953 

29170  954 
lemma mult_exp_exp: "exp x * exp y = exp (x + y)" 
955 
by (rule exp_add [symmetric]) 

956 

23241  957 
lemma exp_of_real: "exp (of_real x) = of_real (exp x)" 
958 
unfolding exp_def 

44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
43335
diff
changeset

959 
apply (subst suminf_of_real) 
23241  960 
apply (rule summable_exp_generic) 
961 
apply (simp add: scaleR_conv_of_real) 

962 
done 

963 

29170  964 
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0" 
965 
proof 

966 
have "exp x * exp ( x) = 1" by (simp add: mult_exp_exp) 

967 
also assume "exp x = 0" 

968 
finally show "False" by simp 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

969 
qed 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

970 

29170  971 
lemma exp_minus: "exp ( x) = inverse (exp x)" 
972 
by (rule inverse_unique [symmetric], simp add: mult_exp_exp) 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

973 

29170  974 
lemma exp_diff: "exp (x  y) = exp x / exp y" 
975 
unfolding diff_minus divide_inverse 

976 
by (simp add: exp_add exp_minus) 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

977 

29167  978 

979 
subsubsection {* Properties of the Exponential Function on Reals *} 

980 

29170  981 
text {* Comparisons of @{term "exp x"} with zero. *} 
29167  982 

983 
text{*Proof: because every exponential can be seen as a square.*} 

984 
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)" 

985 
proof  

986 
have "0 \<le> exp (x/2) * exp (x/2)" by simp 

987 
thus ?thesis by (simp add: exp_add [symmetric]) 

988 
qed 

989 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

990 
lemma exp_gt_zero [simp]: "0 < exp (x::real)" 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

991 
by (simp add: order_less_le) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

992 

29170  993 
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0" 
994 
by (simp add: not_less) 

995 

996 
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0" 

997 
by (simp add: not_le) 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

998 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

999 
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x" 
29165
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset

1000 
by simp 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1001 

89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1002 
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n" 
15251  1003 
apply (induct "n") 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1004 
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1005 
done 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1006 

29170  1007 
text {* Strict monotonicity of exponential. *} 
1008 

1009 
lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)" 

1010 
apply (drule order_le_imp_less_or_eq, auto) 

1011 
apply (simp add: exp_def) 

36777
be5461582d0f
avoid using realspecific versions of generic lemmas
huffman
parents:
36776
diff
changeset

1012 
apply (rule order_trans) 
29170  1013 
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le) 
1014 
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff) 

1015 
done 

1016 

1017 
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x" 

1018 
proof  

1019 
assume x: "0 < x" 

1020 
hence "1 < 1 + x" by simp 

1021 
also from x have "1 + x \<le> exp x" 

1022 
by (simp add: exp_ge_add_one_self_aux) 

1023 
finally show ?thesis . 

1024 
qed 

1025 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1026 
lemma exp_less_mono: 
23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

1027 
fixes x y :: real 
29165
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset

1028 
assumes "x < y" shows "exp x < exp y" 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1029 
proof  
29165
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset

1030 
from `x < y` have "0 < y  x" by simp 
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset

1031 
hence "1 < exp (y  x)" by (rule exp_gt_one) 
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset

1032 
hence "1 < exp y / exp x" by (simp only: exp_diff) 
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset

1033 
thus "exp x < exp y" by simp 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1034 
qed 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1035 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

1036 
lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y" 
29170  1037 
apply (simp add: linorder_not_le [symmetric]) 
1038 
apply (auto simp add: order_le_less exp_less_mono) 

15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1039 
done 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1040 

29170  1041 
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y" 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1042 
by (auto intro: exp_less_mono exp_less_cancel) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1043 

29170  1044 
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y" 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1045 
by (auto simp add: linorder_not_less [symmetric]) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1046 

29170  1047 
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y" 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1048 
by (simp add: order_eq_iff) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1049 

29170  1050 
text {* Comparisons of @{term "exp x"} with one. *} 
1051 

1052 
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x" 

1053 
using exp_less_cancel_iff [where x=0 and y=x] by simp 

1054 

1055 
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0" 

1056 
using exp_less_cancel_iff [where x=x and y=0] by simp 

1057 

1058 
lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x" 

1059 
using exp_le_cancel_iff [where x=0 and y=x] by simp 

1060 

1061 
lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0" 

1062 
using exp_le_cancel_iff [where x=x and y=0] by simp 

1063 

1064 
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0" 

1065 
using exp_inj_iff [where x=x and y=0] by simp 

1066 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

1067 
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y  1 & exp(x::real) = y" 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1068 
apply (rule IVT) 
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset

1069 
apply (auto intro: isCont_exp simp add: le_diff_eq) 
41970  1070 
apply (subgoal_tac "1 + (y  1) \<le> exp (y  1)") 
29165
562f95f06244
cleaned up some proofs; removed redundant simp rules
huffman
parents:
29164
diff
changeset

1071 
apply simp 
17014
ad5ceb90877d
renamed exp_ge_add_one_self to exp_ge_add_one_self_aux
avigad
parents:
16924
diff
changeset

1072 
apply (rule exp_ge_add_one_self_aux, simp) 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1073 
done 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1074 

23115
4615b2078592
generalized exp to work over any complete field; new proof of exp_add
huffman
parents:
23112
diff
changeset

1075 
lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y" 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1076 
apply (rule_tac x = 1 and y = y in linorder_cases) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1077 
apply (drule order_less_imp_le [THEN lemma_exp_total]) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1078 
apply (rule_tac [2] x = 0 in exI) 
36776
c137ae7673d3
remove a couple of redundant lemmas; simplify some proofs
huffman
parents:
35216
diff
changeset

1079 
apply (frule_tac [3] one_less_inverse) 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1080 
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1081 
apply (rule_tac x = "x" in exI) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1082 
apply (simp add: exp_minus) 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1083 
done 
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1084 

89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1085 

29164  1086 
subsection {* Natural Logarithm *} 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1087 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1088 
definition ln :: "real \<Rightarrow> real" where 
23043  1089 
"ln x = (THE u. exp u = x)" 
1090 

1091 
lemma ln_exp [simp]: "ln (exp x) = x" 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1092 
by (simp add: ln_def) 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1093 

22654
c2b6b5a9e136
new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents:
22653
diff
changeset

1094 
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x" 
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1095 
by (auto dest: exp_total) 
22654
c2b6b5a9e136
new simp rule exp_ln; new standard proof of DERIV_exp_ln_one; changed imports
huffman
parents:
22653
diff
changeset

1096 

29171  1097 
lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x" 
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1098 
by (metis exp_gt_zero exp_ln) 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1099 

29171  1100 
lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y" 
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1101 
by (erule subst, rule ln_exp) 
29171  1102 

1103 
lemma ln_one [simp]: "ln 1 = 0" 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1104 
by (rule ln_unique, simp) 
29171  1105 

1106 
lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y" 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1107 
by (rule ln_unique, simp add: exp_add) 
29171  1108 

1109 
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) =  ln x" 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1110 
by (rule ln_unique, simp add: exp_minus) 
29171  1111 

1112 
lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x  ln y" 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1113 
by (rule ln_unique, simp add: exp_diff) 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1114 

29171  1115 
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x" 
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1116 
by (rule ln_unique, simp add: exp_real_of_nat_mult) 
29171  1117 

1118 
lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y" 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1119 
by (subst exp_less_cancel_iff [symmetric], simp) 
29171  1120 

1121 
lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y" 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1122 
by (simp add: linorder_not_less [symmetric]) 
29171  1123 

1124 
lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y" 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1125 
by (simp add: order_eq_iff) 
29171  1126 

1127 
lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x" 

44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1128 
apply (rule exp_le_cancel_iff [THEN iffD1]) 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1129 
apply (simp add: exp_ge_add_one_self_aux) 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1130 
done 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1131 

29171  1132 
lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x" 
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1133 
by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1134 

d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1135 
lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x" 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1136 
using ln_le_cancel_iff [of 1 x] by simp 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1137 

d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1138 
lemma ln_ge_zero_imp_ge_one: "\<lbrakk>0 \<le> ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 \<le> x" 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1139 
using ln_le_cancel_iff [of 1 x] by simp 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1140 

d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1141 
lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> (0 \<le> ln x) = (1 \<le> x)" 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1142 
using ln_le_cancel_iff [of 1 x] by simp 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1143 

d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1144 
lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (x < 1)" 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1145 
using ln_less_cancel_iff [of x 1] by simp 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1146 

d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1147 
lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x" 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1148 
using ln_less_cancel_iff [of 1 x] by simp 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1149 

d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1150 
lemma ln_gt_zero_imp_gt_one: "\<lbrakk>0 < ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 < x" 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1151 
using ln_less_cancel_iff [of 1 x] by simp 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1152 

d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1153 
lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (1 < x)" 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1154 
using ln_less_cancel_iff [of 1 x] by simp 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1155 

d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1156 
lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x = 0) = (x = 1)" 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1157 
using ln_inj_iff [of x 1] by simp 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1158 

d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1159 
lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0" 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1160 
by simp 
15077
89840837108e
converting Hyperreal/Transcendental to Isar script
paulson
parents:
15013
diff
changeset

1161 

23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset

1162 
lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x" 
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1163 
apply (subgoal_tac "isCont ln (exp (ln x))", simp) 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1164 
apply (rule isCont_inverse_function [where f=exp], simp_all) 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1165 
done 
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset

1166 

95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset

1167 
lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x" 
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1168 
apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"]) 
44317
b7e9fa025f15
remove redundant lemma lemma_DERIV_subst in favor of DERIV_cong
huffman
parents:
44316
diff
changeset

1169 
apply (erule DERIV_cong [OF DERIV_exp exp_ln]) 
44308
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1170 
apply (simp_all add: abs_if isCont_ln) 
d2a6f9af02f4
Transcendental.thy: remove several unused lemmas and simplify some proofs
huffman
parents:
44307
diff
changeset

1171 
done 
23045
95e04f335940
add lemmas about inverse functions; cleaned up proof of polar_ex
huffman
parents:
23043
diff
changeset

1172 

33667  1173 
lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x" 
1174 
by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse) 

1175 

29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1176 
lemma ln_series: assumes "0 < x" and "x < 2" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1177 
shows "ln x = (\<Sum> n. (1)^n * (1 / real (n + 1)) * (x  1)^(Suc n))" (is "ln x = suminf (?f (x  1))") 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1178 
proof  
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1179 
let "?f' x n" = "(1)^n * (x  1)^n" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1180 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1181 
have "ln x  suminf (?f (x  1)) = ln 1  suminf (?f (1  1))" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1182 
proof (rule DERIV_isconst3[where x=x]) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1183 
fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1184 
have "norm (1  x) < 1" using `0 < x` and `x < 2` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1185 
have "1 / x = 1 / (1  (1  x))" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1186 
also have "\<dots> = (\<Sum> n. (1  x)^n)" using geometric_sums[OF `norm (1  x) < 1`] by (rule sums_unique) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1187 
also have "\<dots> = suminf (?f' x)" unfolding power_mult_distrib[symmetric] by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto) 
36777
be5461582d0f
avoid using realspecific versions of generic lemmas
huffman
parents:
36776
diff
changeset

1188 
finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1189 
moreover 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1190 
have repos: "\<And> h x :: real. h  1 + x = h + x  1" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1191 
have "DERIV (\<lambda>x. suminf (?f x)) (x  1) :> (\<Sum>n. (1)^n * (1 / real (n + 1)) * real (Suc n) * (x  1) ^ n)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1192 
proof (rule DERIV_power_series') 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1193 
show "x  1 \<in> { 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29695
diff
changeset

1194 
{ fix x :: real assume "x \<in> { 1<..<1}" hence "norm (x) < 1" by auto 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
32047
diff
changeset

1195 