author  huffman 
Sun, 28 Aug 2011 09:20:12 0700  
changeset 44568  e6f291cb5810 
parent 44289  d81d09cdab9c 
child 44710  9caf6883f1f4 
permissions  rwrr 
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(* Title : Series.thy 
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Author : Jacques D. Fleuriot 

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Copyright : 1998 University of Cambridge 

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Converted to Isar and polished by lcp 

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Converted to setsum and polished yet more by TNN 
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Additional contributions by Jeremy Avigad 
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*) 
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header{*Finite Summation and Infinite Series*} 
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theory Series 
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New theory Probability, which contains a development of measure theory
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imports SEQ Deriv 
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begin 
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definition 
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sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool" 
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(infixr "sums" 80) where 
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"f sums s = (%n. setsum f {0..<n}) > s" 
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definition 
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summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where 
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"summable f = (\<exists>s. f sums s)" 
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definition 
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suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a" where 
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"suminf f = (THE s. f sums s)" 
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notation suminf (binder "\<Sum>" 10) 
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lemma [trans]: "f=g ==> g sums z ==> f sums z" 
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by simp 
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lemma sumr_diff_mult_const: 
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"setsum f {0..<n}  (real n*r) = setsum (%i. f i  r) {0..<n::nat}" 

15536  37 
by (simp add: diff_minus setsum_addf real_of_nat_def) 
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15542  39 
lemma real_setsum_nat_ivl_bounded: 
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"(!!p. p < n \<Longrightarrow> f(p) \<le> K) 

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\<Longrightarrow> setsum f {0..<n::nat} \<le> real n * K" 

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using setsum_bounded[where A = "{0..<n}"] 

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by (auto simp:real_of_nat_def) 

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15539  45 
(* Generalize from real to some algebraic structure? *) 
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lemma sumr_minus_one_realpow_zero [simp]: 

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"(\<Sum>i=0..<2*n. (1) ^ Suc i) = (0::real)" 
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by (induct "n", auto) 
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15539  50 
(* FIXME this is an awful lemma! *) 
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lemma sumr_one_lb_realpow_zero [simp]: 

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"(\<Sum>n=Suc 0..<n. f(n) * (0::real) ^ n) = 0" 

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by (rule setsum_0', simp) 
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lemma sumr_group: 
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"(\<Sum>m=0..<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {0 ..< n * k}" 
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apply (subgoal_tac "k = 0  0 < k", auto) 
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apply (induct "n") 
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apply (simp_all add: setsum_add_nat_ivl add_commute) 
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done 
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lemma sumr_offset3: 
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"setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)) + setsum f {0..<k}" 

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apply (subst setsum_shift_bounds_nat_ivl [symmetric]) 

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apply (simp add: setsum_add_nat_ivl add_commute) 

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done 

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lemma sumr_offset: 
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fixes f :: "nat \<Rightarrow> 'a::ab_group_add" 
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shows "(\<Sum>m=0..<n. f(m+k)) = setsum f {0..<n+k}  setsum f {0..<k}" 

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by (simp add: sumr_offset3) 

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lemma sumr_offset2: 

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"\<forall>f. (\<Sum>m=0..<n::nat. f(m+k)::real) = setsum f {0..<n+k}  setsum f {0..<k}" 

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by (simp add: sumr_offset) 
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lemma sumr_offset4: 

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"\<forall>n f. setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)::real) + setsum f {0..<k}" 
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by (clarify, rule sumr_offset3) 

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subsection{* Infinite Sums, by the Properties of Limits*} 
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(* 

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suminf is the sum 
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*) 
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lemma sums_summable: "f sums l ==> summable f" 

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by (simp add: sums_def summable_def, blast) 
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lemma summable_sums: 
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fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}" assumes "summable f" shows "f sums (suminf f)" 

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proof  

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from assms guess s unfolding summable_def sums_def_raw .. note s = this 

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then show ?thesis unfolding sums_def_raw suminf_def 

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by (rule theI, auto intro!: tendsto_unique[OF trivial_limit_sequentially]) 

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qed 

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lemma summable_sumr_LIMSEQ_suminf: 
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fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}" 

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shows "summable f \<Longrightarrow> (\<lambda>n. setsum f {0..<n}) > suminf f" 

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by (rule summable_sums [unfolded sums_def]) 
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lemma suminf_eq_lim: "suminf f = lim (%n. setsum f {0..<n})" 
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by (simp add: suminf_def sums_def lim_def) 
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(* 
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sum is unique 
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*) 
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lemma sums_unique: 
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fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}" 

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shows "f sums s \<Longrightarrow> (s = suminf f)" 

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apply (frule sums_summable[THEN summable_sums]) 

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apply (auto intro!: tendsto_unique[OF trivial_limit_sequentially] simp add: sums_def) 

14416  113 
done 
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lemma sums_iff: 
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fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}" 

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shows "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)" 

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by (metis summable_sums sums_summable sums_unique) 
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lemma sums_split_initial_segment: 
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fixes f :: "nat \<Rightarrow> 'a::real_normed_vector" 

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shows "f sums s ==> (\<lambda>n. f(n + k)) sums (s  (SUM i = 0..< k. f i))" 

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apply (unfold sums_def) 

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apply (simp add: sumr_offset) 

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apply (rule LIMSEQ_diff_const) 
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apply (rule LIMSEQ_ignore_initial_segment) 

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apply assumption 

128 
done 

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41970  130 
lemma summable_ignore_initial_segment: 
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fixes f :: "nat \<Rightarrow> 'a::real_normed_vector" 

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shows "summable f ==> summable (%n. f(n + k))" 

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apply (unfold summable_def) 
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apply (auto intro: sums_split_initial_segment) 

135 
done 

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lemma suminf_minus_initial_segment: 
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fixes f :: "nat \<Rightarrow> 'a::real_normed_vector" 

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shows "summable f ==> 

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suminf f = s ==> suminf (%n. f(n + k)) = s  (SUM i = 0..< k. f i)" 
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apply (frule summable_ignore_initial_segment) 

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apply (rule sums_unique [THEN sym]) 

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apply (frule summable_sums) 

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apply (rule sums_split_initial_segment) 

145 
apply auto 

146 
done 

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lemma suminf_split_initial_segment: 
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fixes f :: "nat \<Rightarrow> 'a::real_normed_vector" 

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shows "summable f ==> 

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suminf f = (SUM i = 0..< k. f i) + (\<Sum>n. f(n + k))" 

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by (auto simp add: suminf_minus_initial_segment) 
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lemma suminf_exist_split: fixes r :: real assumes "0 < r" and "summable a" 
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shows "\<exists> N. \<forall> n \<ge> N. \<bar> \<Sum> i. a (i + n) \<bar> < r" 
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proof  
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from LIMSEQ_D[OF summable_sumr_LIMSEQ_suminf[OF `summable a`] `0 < r`] 
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obtain N :: nat where "\<forall> n \<ge> N. norm (setsum a {0..<n}  suminf a) < r" by auto 
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thus ?thesis unfolding suminf_minus_initial_segment[OF `summable a` refl] abs_minus_commute real_norm_def 
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by auto 
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qed 
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41970  163 
lemma sums_Suc: 
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fixes f :: "nat \<Rightarrow> 'a::real_normed_vector" 

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assumes sumSuc: "(\<lambda> n. f (Suc n)) sums l" shows "f sums (l + f 0)" 

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proof  
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from sumSuc[unfolded sums_def] 
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have "(\<lambda>i. \<Sum>n = Suc 0..<Suc i. f n) > l" unfolding setsum_reindex[OF inj_Suc] image_Suc_atLeastLessThan[symmetric] comp_def . 
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from LIMSEQ_add_const[OF this, where b="f 0"] 
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have "(\<lambda>i. \<Sum>n = 0..<Suc i. f n) > l + f 0" unfolding add_commute setsum_head_upt_Suc[OF zero_less_Suc] . 
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thus ?thesis unfolding sums_def by (rule LIMSEQ_imp_Suc) 
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qed 
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41970  174 
lemma series_zero: 
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fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}" 

176 
assumes "\<forall>m. n \<le> m \<longrightarrow> f m = 0" 

177 
shows "f sums (setsum f {0..<n})" 

178 
proof  

179 
{ fix k :: nat have "setsum f {0..<k + n} = setsum f {0..<n}" 

180 
using assms by (induct k) auto } 

181 
note setsum_const = this 

182 
show ?thesis 

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unfolding sums_def 

184 
apply (rule LIMSEQ_offset[of _ n]) 

185 
unfolding setsum_const 

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apply (rule tendsto_const) 
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done 
188 
qed 

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41970  190 
lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0" 
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unfolding sums_def by (simp add: tendsto_const) 
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41970  193 
lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)" 
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by (rule sums_zero [THEN sums_summable]) 
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41970  196 
lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0" 
23121  197 
by (rule sums_zero [THEN sums_unique, symmetric]) 
41970  198 

23119  199 
lemma (in bounded_linear) sums: 
200 
"(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)" 

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unfolding sums_def by (drule tendsto, simp only: setsum) 
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203 
lemma (in bounded_linear) summable: 

204 
"summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))" 

205 
unfolding summable_def by (auto intro: sums) 

206 

207 
lemma (in bounded_linear) suminf: 

208 
"summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))" 

23121  209 
by (intro sums_unique sums summable_sums) 
23119  210 

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lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real] 
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lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real] 
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lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real] 
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20692  215 
lemma sums_mult: 
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fixes c :: "'a::real_normed_algebra" 

217 
shows "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)" 

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by (rule bounded_linear.sums [OF bounded_linear_mult_right]) 
14416  219 

20692  220 
lemma summable_mult: 
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fixes c :: "'a::real_normed_algebra" 

23121  222 
shows "summable f \<Longrightarrow> summable (%n. c * f n)" 
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by (rule bounded_linear.summable [OF bounded_linear_mult_right]) 
16819  224 

20692  225 
lemma suminf_mult: 
226 
fixes c :: "'a::real_normed_algebra" 

41970  227 
shows "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f" 
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by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric]) 
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20692  230 
lemma sums_mult2: 
231 
fixes c :: "'a::real_normed_algebra" 

232 
shows "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)" 

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by (rule bounded_linear.sums [OF bounded_linear_mult_left]) 
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20692  235 
lemma summable_mult2: 
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fixes c :: "'a::real_normed_algebra" 

237 
shows "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)" 

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by (rule bounded_linear.summable [OF bounded_linear_mult_left]) 
16819  239 

20692  240 
lemma suminf_mult2: 
241 
fixes c :: "'a::real_normed_algebra" 

242 
shows "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)" 

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by (rule bounded_linear.suminf [OF bounded_linear_mult_left]) 
16819  244 

20692  245 
lemma sums_divide: 
246 
fixes c :: "'a::real_normed_field" 

247 
shows "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)" 

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by (rule bounded_linear.sums [OF bounded_linear_divide]) 
14416  249 

20692  250 
lemma summable_divide: 
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fixes c :: "'a::real_normed_field" 

252 
shows "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)" 

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by (rule bounded_linear.summable [OF bounded_linear_divide]) 
16819  254 

20692  255 
lemma suminf_divide: 
256 
fixes c :: "'a::real_normed_field" 

257 
shows "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c" 

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by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric]) 
16819  259 

41970  260 
lemma sums_add: 
261 
fixes a b :: "'a::real_normed_field" 

262 
shows "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) sums (a + b)" 

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unfolding sums_def by (simp add: setsum_addf tendsto_add) 
16819  264 

41970  265 
lemma summable_add: 
266 
fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field" 

267 
shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n + Y n)" 

23121  268 
unfolding summable_def by (auto intro: sums_add) 
16819  269 

270 
lemma suminf_add: 

41970  271 
fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field" 
272 
shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X + suminf Y = (\<Sum>n. X n + Y n)" 

23121  273 
by (intro sums_unique sums_add summable_sums) 
14416  274 

41970  275 
lemma sums_diff: 
276 
fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field" 

277 
shows "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n  Y n) sums (a  b)" 

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unfolding sums_def by (simp add: setsum_subtractf tendsto_diff) 
23121  279 

41970  280 
lemma summable_diff: 
281 
fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field" 

282 
shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n  Y n)" 

23121  283 
unfolding summable_def by (auto intro: sums_diff) 
14416  284 

285 
lemma suminf_diff: 

41970  286 
fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field" 
287 
shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X  suminf Y = (\<Sum>n. X n  Y n)" 

23121  288 
by (intro sums_unique sums_diff summable_sums) 
14416  289 

41970  290 
lemma sums_minus: 
291 
fixes X :: "nat \<Rightarrow> 'a::real_normed_field" 

292 
shows "X sums a ==> (\<lambda>n.  X n) sums ( a)" 

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unfolding sums_def by (simp add: setsum_negf tendsto_minus) 
16819  294 

41970  295 
lemma summable_minus: 
296 
fixes X :: "nat \<Rightarrow> 'a::real_normed_field" 

297 
shows "summable X \<Longrightarrow> summable (\<lambda>n.  X n)" 

23121  298 
unfolding summable_def by (auto intro: sums_minus) 
16819  299 

41970  300 
lemma suminf_minus: 
301 
fixes X :: "nat \<Rightarrow> 'a::real_normed_field" 

302 
shows "summable X \<Longrightarrow> (\<Sum>n.  X n) =  (\<Sum>n. X n)" 

23121  303 
by (intro sums_unique [symmetric] sums_minus summable_sums) 
14416  304 

305 
lemma sums_group: 

41970  306 
fixes f :: "nat \<Rightarrow> 'a::real_normed_field" 
307 
shows "[summable f; 0 < k ] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)" 

14416  308 
apply (drule summable_sums) 
20692  309 
apply (simp only: sums_def sumr_group) 
31336
e17f13cd1280
generalize constants in SEQ.thy to class metric_space
huffman
parents:
31017
diff
changeset

310 
apply (unfold LIMSEQ_iff, safe) 
20692  311 
apply (drule_tac x="r" in spec, safe) 
312 
apply (rule_tac x="no" in exI, safe) 

313 
apply (drule_tac x="n*k" in spec) 

314 
apply (erule mp) 

315 
apply (erule order_trans) 

316 
apply simp 

14416  317 
done 
318 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

319 
text{*A summable series of positive terms has limit that is at least as 
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

320 
great as any partial sum.*} 
14416  321 

33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

322 
lemma pos_summable: 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

323 
fixes f:: "nat \<Rightarrow> real" 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

324 
assumes pos: "!!n. 0 \<le> f n" and le: "!!n. setsum f {0..<n} \<le> x" 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

325 
shows "summable f" 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

326 
proof  
41970  327 
have "convergent (\<lambda>n. setsum f {0..<n})" 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

328 
proof (rule Bseq_mono_convergent) 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

329 
show "Bseq (\<lambda>n. setsum f {0..<n})" 
33536  330 
by (rule f_inc_g_dec_Beq_f [of "(\<lambda>n. setsum f {0..<n})" "\<lambda>n. x"]) 
41970  331 
(auto simp add: le pos) 
332 
next 

33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

333 
show "\<forall>m n. m \<le> n \<longrightarrow> setsum f {0..<m} \<le> setsum f {0..<n}" 
41970  334 
by (auto intro: setsum_mono2 pos) 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

335 
qed 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

336 
then obtain L where "(%n. setsum f {0..<n}) > L" 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

337 
by (blast dest: convergentD) 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

338 
thus ?thesis 
41970  339 
by (force simp add: summable_def sums_def) 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

340 
qed 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

341 

20692  342 
lemma series_pos_le: 
343 
fixes f :: "nat \<Rightarrow> real" 

344 
shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 \<le> f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} \<le> suminf f" 

14416  345 
apply (drule summable_sums) 
346 
apply (simp add: sums_def) 

44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44289
diff
changeset

347 
apply (cut_tac k = "setsum f {0..<n}" in tendsto_const) 
15539  348 
apply (erule LIMSEQ_le, blast) 
20692  349 
apply (rule_tac x="n" in exI, clarify) 
15539  350 
apply (rule setsum_mono2) 
351 
apply auto 

14416  352 
done 
353 

354 
lemma series_pos_less: 

20692  355 
fixes f :: "nat \<Rightarrow> real" 
356 
shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} < suminf f" 

357 
apply (rule_tac y="setsum f {0..<Suc n}" in order_less_le_trans) 

358 
apply simp 

359 
apply (erule series_pos_le) 

360 
apply (simp add: order_less_imp_le) 

361 
done 

362 

363 
lemma suminf_gt_zero: 

364 
fixes f :: "nat \<Rightarrow> real" 

365 
shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f" 

366 
by (drule_tac n="0" in series_pos_less, simp_all) 

367 

368 
lemma suminf_ge_zero: 

369 
fixes f :: "nat \<Rightarrow> real" 

370 
shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 \<le> suminf f" 

371 
by (drule_tac n="0" in series_pos_le, simp_all) 

372 

373 
lemma sumr_pos_lt_pair: 

374 
fixes f :: "nat \<Rightarrow> real" 

375 
shows "\<lbrakk>summable f; 

376 
\<forall>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk> 

377 
\<Longrightarrow> setsum f {0..<k} < suminf f" 

30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset

378 
unfolding One_nat_def 
20692  379 
apply (subst suminf_split_initial_segment [where k="k"]) 
380 
apply assumption 

381 
apply simp 

382 
apply (drule_tac k="k" in summable_ignore_initial_segment) 

383 
apply (drule_tac k="Suc (Suc 0)" in sums_group, simp) 

384 
apply simp 

385 
apply (frule sums_unique) 

386 
apply (drule sums_summable) 

387 
apply simp 

388 
apply (erule suminf_gt_zero) 

389 
apply (simp add: add_ac) 

14416  390 
done 
391 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

392 
text{*Sum of a geometric progression.*} 
14416  393 

17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16819
diff
changeset

394 
lemmas sumr_geometric = geometric_sum [where 'a = real] 
14416  395 

20692  396 
lemma geometric_sums: 
31017  397 
fixes x :: "'a::{real_normed_field}" 
20692  398 
shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) sums (1 / (1  x))" 
399 
proof  

400 
assume less_1: "norm x < 1" 

401 
hence neq_1: "x \<noteq> 1" by auto 

402 
hence neq_0: "x  1 \<noteq> 0" by simp 

403 
from less_1 have lim_0: "(\<lambda>n. x ^ n) > 0" 

404 
by (rule LIMSEQ_power_zero) 

22719
c51667189bd3
lemma geometric_sum no longer needs class division_by_zero
huffman
parents:
21404
diff
changeset

405 
hence "(\<lambda>n. x ^ n / (x  1)  1 / (x  1)) > 0 / (x  1)  1 / (x  1)" 
44568
e6f291cb5810
discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44289
diff
changeset

406 
using neq_0 by (intro tendsto_intros) 
20692  407 
hence "(\<lambda>n. (x ^ n  1) / (x  1)) > 1 / (1  x)" 
408 
by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib) 

409 
thus "(\<lambda>n. x ^ n) sums (1 / (1  x))" 

410 
by (simp add: sums_def geometric_sum neq_1) 

411 
qed 

412 

413 
lemma summable_geometric: 

31017  414 
fixes x :: "'a::{real_normed_field}" 
20692  415 
shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)" 
416 
by (rule geometric_sums [THEN sums_summable]) 

14416  417 

36409  418 
lemma half: "0 < 1 / (2::'a::{number_ring,linordered_field_inverse_zero})" 
41970  419 
by arith 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

420 

7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

421 
lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1" 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

422 
proof  
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

423 
have 2: "(\<lambda>n. (1/2::real)^n) sums 2" using geometric_sums [of "1/2::real"] 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

424 
by auto 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

425 
have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)" 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

426 
by simp 
44282
f0de18b62d63
remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents:
41970
diff
changeset

427 
thus ?thesis using sums_divide [OF 2, of 2] 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

428 
by simp 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

429 
qed 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

430 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

431 
text{*Cauchytype criterion for convergence of series (c.f. Harrison)*} 
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

432 

15539  433 
lemma summable_convergent_sumr_iff: 
434 
"summable f = convergent (%n. setsum f {0..<n})" 

14416  435 
by (simp add: summable_def sums_def convergent_def) 
436 

41970  437 
lemma summable_LIMSEQ_zero: 
438 
fixes f :: "nat \<Rightarrow> 'a::real_normed_field" 

439 
shows "summable f \<Longrightarrow> f > 0" 

20689  440 
apply (drule summable_convergent_sumr_iff [THEN iffD1]) 
20692  441 
apply (drule convergent_Cauchy) 
31336
e17f13cd1280
generalize constants in SEQ.thy to class metric_space
huffman
parents:
31017
diff
changeset

442 
apply (simp only: Cauchy_iff LIMSEQ_iff, safe) 
20689  443 
apply (drule_tac x="r" in spec, safe) 
444 
apply (rule_tac x="M" in exI, safe) 

445 
apply (drule_tac x="Suc n" in spec, simp) 

446 
apply (drule_tac x="n" in spec, simp) 

447 
done 

448 

32707
836ec9d0a0c8
New lemmas involving the real numbers, especially limits and series
paulson
parents:
31336
diff
changeset

449 
lemma suminf_le: 
836ec9d0a0c8
New lemmas involving the real numbers, especially limits and series
paulson
parents:
31336
diff
changeset

450 
fixes x :: real 
836ec9d0a0c8
New lemmas involving the real numbers, especially limits and series
paulson
parents:
31336
diff
changeset

451 
shows "summable f \<Longrightarrow> (!!n. setsum f {0..<n} \<le> x) \<Longrightarrow> suminf f \<le> x" 
41970  452 
by (simp add: summable_convergent_sumr_iff suminf_eq_lim lim_le) 
32707
836ec9d0a0c8
New lemmas involving the real numbers, especially limits and series
paulson
parents:
31336
diff
changeset

453 

14416  454 
lemma summable_Cauchy: 
41970  455 
"summable (f::nat \<Rightarrow> 'a::banach) = 
20848  456 
(\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. norm (setsum f {m..<n}) < e)" 
31336
e17f13cd1280
generalize constants in SEQ.thy to class metric_space
huffman
parents:
31017
diff
changeset

457 
apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_iff, safe) 
20410  458 
apply (drule spec, drule (1) mp) 
459 
apply (erule exE, rule_tac x="M" in exI, clarify) 

460 
apply (rule_tac x="m" and y="n" in linorder_le_cases) 

461 
apply (frule (1) order_trans) 

462 
apply (drule_tac x="n" in spec, drule (1) mp) 

463 
apply (drule_tac x="m" in spec, drule (1) mp) 

464 
apply (simp add: setsum_diff [symmetric]) 

465 
apply simp 

466 
apply (drule spec, drule (1) mp) 

467 
apply (erule exE, rule_tac x="N" in exI, clarify) 

468 
apply (rule_tac x="m" and y="n" in linorder_le_cases) 

20552
2c31dd358c21
generalized types of many constants to work over arbitrary vector spaces;
huffman
parents:
20432
diff
changeset

469 
apply (subst norm_minus_commute) 
20410  470 
apply (simp add: setsum_diff [symmetric]) 
471 
apply (simp add: setsum_diff [symmetric]) 

14416  472 
done 
473 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

474 
text{*Comparison test*} 
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

475 

20692  476 
lemma norm_setsum: 
477 
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" 

478 
shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))" 

479 
apply (case_tac "finite A") 

480 
apply (erule finite_induct) 

481 
apply simp 

482 
apply simp 

483 
apply (erule order_trans [OF norm_triangle_ineq add_left_mono]) 

484 
apply simp 

485 
done 

486 

14416  487 
lemma summable_comparison_test: 
20848  488 
fixes f :: "nat \<Rightarrow> 'a::banach" 
489 
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f" 

20692  490 
apply (simp add: summable_Cauchy, safe) 
491 
apply (drule_tac x="e" in spec, safe) 

492 
apply (rule_tac x = "N + Na" in exI, safe) 

14416  493 
apply (rotate_tac 2) 
494 
apply (drule_tac x = m in spec) 

495 
apply (auto, rotate_tac 2, drule_tac x = n in spec) 

20848  496 
apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans) 
497 
apply (rule norm_setsum) 

15539  498 
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans) 
22998  499 
apply (auto intro: setsum_mono simp add: abs_less_iff) 
14416  500 
done 
501 

20848  502 
lemma summable_norm_comparison_test: 
503 
fixes f :: "nat \<Rightarrow> 'a::banach" 

504 
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> 

505 
\<Longrightarrow> summable (\<lambda>n. norm (f n))" 

506 
apply (rule summable_comparison_test) 

507 
apply (auto) 

508 
done 

509 

14416  510 
lemma summable_rabs_comparison_test: 
20692  511 
fixes f :: "nat \<Rightarrow> real" 
512 
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable (\<lambda>n. \<bar>f n\<bar>)" 

14416  513 
apply (rule summable_comparison_test) 
15543  514 
apply (auto) 
14416  515 
done 
516 

23084  517 
text{*Summability of geometric series for real algebras*} 
518 

519 
lemma complete_algebra_summable_geometric: 

31017  520 
fixes x :: "'a::{real_normed_algebra_1,banach}" 
23084  521 
shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)" 
522 
proof (rule summable_comparison_test) 

523 
show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n" 

524 
by (simp add: norm_power_ineq) 

525 
show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)" 

526 
by (simp add: summable_geometric) 

527 
qed 

528 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

529 
text{*Limit comparison property for series (c.f. jrh)*} 
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

530 

14416  531 
lemma summable_le: 
20692  532 
fixes f g :: "nat \<Rightarrow> real" 
533 
shows "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g" 

14416  534 
apply (drule summable_sums)+ 
20692  535 
apply (simp only: sums_def, erule (1) LIMSEQ_le) 
14416  536 
apply (rule exI) 
15539  537 
apply (auto intro!: setsum_mono) 
14416  538 
done 
539 

540 
lemma summable_le2: 

20692  541 
fixes f g :: "nat \<Rightarrow> real" 
542 
shows "\<lbrakk>\<forall>n. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f \<and> suminf f \<le> suminf g" 

20848  543 
apply (subgoal_tac "summable f") 
544 
apply (auto intro!: summable_le) 

22998  545 
apply (simp add: abs_le_iff) 
20848  546 
apply (rule_tac g="g" in summable_comparison_test, simp_all) 
14416  547 
done 
548 

19106
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

549 
(* specialisation for the common 0 case *) 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

550 
lemma suminf_0_le: 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

551 
fixes f::"nat\<Rightarrow>real" 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

552 
assumes gt0: "\<forall>n. 0 \<le> f n" and sm: "summable f" 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

553 
shows "0 \<le> suminf f" 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

554 
proof  
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

555 
let ?g = "(\<lambda>n. (0::real))" 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

556 
from gt0 have "\<forall>n. ?g n \<le> f n" by simp 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

557 
moreover have "summable ?g" by (rule summable_zero) 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

558 
moreover from sm have "summable f" . 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

559 
ultimately have "suminf ?g \<le> suminf f" by (rule summable_le) 
44289  560 
then show "0 \<le> suminf f" by simp 
41970  561 
qed 
19106
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

562 

6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

563 

15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

564 
text{*Absolute convergence imples normal convergence*} 
20848  565 
lemma summable_norm_cancel: 
566 
fixes f :: "nat \<Rightarrow> 'a::banach" 

567 
shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f" 

20692  568 
apply (simp only: summable_Cauchy, safe) 
569 
apply (drule_tac x="e" in spec, safe) 

570 
apply (rule_tac x="N" in exI, safe) 

571 
apply (drule_tac x="m" in spec, safe) 

20848  572 
apply (rule order_le_less_trans [OF norm_setsum]) 
573 
apply (rule order_le_less_trans [OF abs_ge_self]) 

20692  574 
apply simp 
14416  575 
done 
576 

20848  577 
lemma summable_rabs_cancel: 
578 
fixes f :: "nat \<Rightarrow> real" 

579 
shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f" 

580 
by (rule summable_norm_cancel, simp) 

581 

15085
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removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
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15053
diff
changeset

582 
text{*Absolute convergence of series*} 
20848  583 
lemma summable_norm: 
584 
fixes f :: "nat \<Rightarrow> 'a::banach" 

585 
shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))" 

44568
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discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44289
diff
changeset

586 
by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel 
20848  587 
summable_sumr_LIMSEQ_suminf norm_setsum) 
588 

14416  589 
lemma summable_rabs: 
20692  590 
fixes f :: "nat \<Rightarrow> real" 
591 
shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)" 

20848  592 
by (fold real_norm_def, rule summable_norm) 
14416  593 

594 
subsection{* The Ratio Test*} 

595 

20848  596 
lemma norm_ratiotest_lemma: 
22852  597 
fixes x y :: "'a::real_normed_vector" 
20848  598 
shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0" 
599 
apply (subgoal_tac "norm x \<le> 0", simp) 

600 
apply (erule order_trans) 

601 
apply (simp add: mult_le_0_iff) 

602 
done 

603 

14416  604 
lemma rabs_ratiotest_lemma: "[ c \<le> 0; abs x \<le> c * abs y ] ==> x = (0::real)" 
20848  605 
by (erule norm_ratiotest_lemma, simp) 
14416  606 

607 
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)" 

608 
apply (drule le_imp_less_or_eq) 

609 
apply (auto dest: less_imp_Suc_add) 

610 
done 

611 

612 
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)" 

613 
by (auto simp add: le_Suc_ex) 

614 

615 
(*All this trouble just to get 0<c *) 

616 
lemma ratio_test_lemma2: 

20848  617 
fixes f :: "nat \<Rightarrow> 'a::banach" 
618 
shows "\<lbrakk>\<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> 0 < c \<or> summable f" 

14416  619 
apply (simp (no_asm) add: linorder_not_le [symmetric]) 
620 
apply (simp add: summable_Cauchy) 

15543  621 
apply (safe, subgoal_tac "\<forall>n. N < n > f (n) = 0") 
622 
prefer 2 

623 
apply clarify 

30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
29803
diff
changeset

624 
apply(erule_tac x = "n  Suc 0" in allE) 
15543  625 
apply (simp add:diff_Suc split:nat.splits) 
20848  626 
apply (blast intro: norm_ratiotest_lemma) 
14416  627 
apply (rule_tac x = "Suc N" in exI, clarify) 
15543  628 
apply(simp cong:setsum_ivl_cong) 
14416  629 
done 
630 

631 
lemma ratio_test: 

20848  632 
fixes f :: "nat \<Rightarrow> 'a::banach" 
633 
shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f" 

14416  634 
apply (frule ratio_test_lemma2, auto) 
41970  635 
apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n" 
15234
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simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

636 
in summable_comparison_test) 
14416  637 
apply (rule_tac x = N in exI, safe) 
638 
apply (drule le_Suc_ex_iff [THEN iffD1]) 

22959  639 
apply (auto simp add: power_add field_power_not_zero) 
15539  640 
apply (induct_tac "na", auto) 
20848  641 
apply (rule_tac y = "c * norm (f (N + n))" in order_trans) 
14416  642 
apply (auto intro: mult_right_mono simp add: summable_def) 
20848  643 
apply (rule_tac x = "norm (f N) * (1/ (1  c)) / (c ^ N)" in exI) 
41970  644 
apply (rule sums_divide) 
27108  645 
apply (rule sums_mult) 
15234
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simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset

646 
apply (auto intro!: geometric_sums) 
14416  647 
done 
648 

23111  649 
subsection {* Cauchy Product Formula *} 
650 

651 
(* Proof based on Analysis WebNotes: Chapter 07, Class 41 

652 
http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm *) 

653 

654 
lemma setsum_triangle_reindex: 

655 
fixes n :: nat 

656 
shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k=0..<n. \<Sum>i=0..k. f i (k  i))" 

657 
proof  

658 
have "(\<Sum>(i, j)\<in>{(i, j). i + j < n}. f i j) = 

659 
(\<Sum>(k, i)\<in>(SIGMA k:{0..<n}. {0..k}). f i (k  i))" 

660 
proof (rule setsum_reindex_cong) 

661 
show "inj_on (\<lambda>(k,i). (i, k  i)) (SIGMA k:{0..<n}. {0..k})" 

662 
by (rule inj_on_inverseI [where g="\<lambda>(i,j). (i+j, i)"], auto) 

663 
show "{(i,j). i + j < n} = (\<lambda>(k,i). (i, k  i)) ` (SIGMA k:{0..<n}. {0..k})" 

664 
by (safe, rule_tac x="(a+b,a)" in image_eqI, auto) 

665 
show "\<And>a. (\<lambda>(k, i). f i (k  i)) a = split f ((\<lambda>(k, i). (i, k  i)) a)" 

666 
by clarify 

667 
qed 

668 
thus ?thesis by (simp add: setsum_Sigma) 

669 
qed 

670 

671 
lemma Cauchy_product_sums: 

672 
fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" 

673 
assumes a: "summable (\<lambda>k. norm (a k))" 

674 
assumes b: "summable (\<lambda>k. norm (b k))" 

675 
shows "(\<lambda>k. \<Sum>i=0..k. a i * b (k  i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))" 

676 
proof  

677 
let ?S1 = "\<lambda>n::nat. {0..<n} \<times> {0..<n}" 

678 
let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}" 

679 
have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto 

680 
have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto 

681 
have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto 

682 
have finite_S1: "\<And>n. finite (?S1 n)" by simp 

683 
with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset) 

684 

685 
let ?g = "\<lambda>(i,j). a i * b j" 

686 
let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)" 

687 
have f_nonneg: "\<And>x. 0 \<le> ?f x" 

688 
by (auto simp add: mult_nonneg_nonneg) 

689 
hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A" 

690 
unfolding real_norm_def 

691 
by (simp only: abs_of_nonneg setsum_nonneg [rule_format]) 

692 

693 
have "(\<lambda>n. (\<Sum>k=0..<n. a k) * (\<Sum>k=0..<n. b k)) 

694 
> (\<Sum>k. a k) * (\<Sum>k. b k)" 

44568
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discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44289
diff
changeset

695 
by (intro tendsto_mult summable_sumr_LIMSEQ_suminf 
23111  696 
summable_norm_cancel [OF a] summable_norm_cancel [OF b]) 
697 
hence 1: "(\<lambda>n. setsum ?g (?S1 n)) > (\<Sum>k. a k) * (\<Sum>k. b k)" 

698 
by (simp only: setsum_product setsum_Sigma [rule_format] 

699 
finite_atLeastLessThan) 

700 

701 
have "(\<lambda>n. (\<Sum>k=0..<n. norm (a k)) * (\<Sum>k=0..<n. norm (b k))) 

702 
> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))" 

44568
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discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents:
44289
diff
changeset

703 
using a b by (intro tendsto_mult summable_sumr_LIMSEQ_suminf) 
23111  704 
hence "(\<lambda>n. setsum ?f (?S1 n)) > (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))" 
705 
by (simp only: setsum_product setsum_Sigma [rule_format] 

706 
finite_atLeastLessThan) 

707 
hence "convergent (\<lambda>n. setsum ?f (?S1 n))" 

708 
by (rule convergentI) 

709 
hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))" 

710 
by (rule convergent_Cauchy) 

36657  711 
have "Zfun (\<lambda>n. setsum ?f (?S1 n  ?S2 n)) sequentially" 
712 
proof (rule ZfunI, simp only: eventually_sequentially norm_setsum_f) 

23111  713 
fix r :: real 
714 
assume r: "0 < r" 

715 
from CauchyD [OF Cauchy r] obtain N 

716 
where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m)  setsum ?f (?S1 n)) < r" .. 

717 
hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m  ?S1 n)) < r" 

718 
by (simp only: setsum_diff finite_S1 S1_mono) 

719 
hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m  ?S1 n) < r" 

720 
by (simp only: norm_setsum_f) 

721 
show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n  ?S2 n) < r" 

722 
proof (intro exI allI impI) 

723 
fix n assume "2 * N \<le> n" 

724 
hence n: "N \<le> n div 2" by simp 

725 
have "setsum ?f (?S1 n  ?S2 n) \<le> setsum ?f (?S1 n  ?S1 (n div 2))" 

726 
by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg 

727 
Diff_mono subset_refl S1_le_S2) 

728 
also have "\<dots> < r" 

729 
using n div_le_dividend by (rule N) 

730 
finally show "setsum ?f (?S1 n  ?S2 n) < r" . 

731 
qed 

732 
qed 

36657  733 
hence "Zfun (\<lambda>n. setsum ?g (?S1 n  ?S2 n)) sequentially" 
734 
apply (rule Zfun_le [rule_format]) 

23111  735 
apply (simp only: norm_setsum_f) 
736 
apply (rule order_trans [OF norm_setsum setsum_mono]) 

737 
apply (auto simp add: norm_mult_ineq) 

738 
done 

739 
hence 2: "(\<lambda>n. setsum ?g (?S1 n)  setsum ?g (?S2 n)) > 0" 

36660
1cc4ab4b7ff7
make (X > L) an abbreviation for (X > L) sequentially
huffman
parents:
36657
diff
changeset

740 
unfolding tendsto_Zfun_iff diff_0_right 
36657  741 
by (simp only: setsum_diff finite_S1 S2_le_S1) 
23111  742 

743 
with 1 have "(\<lambda>n. setsum ?g (?S2 n)) > (\<Sum>k. a k) * (\<Sum>k. b k)" 

744 
by (rule LIMSEQ_diff_approach_zero2) 

745 
thus ?thesis by (simp only: sums_def setsum_triangle_reindex) 

746 
qed 

747 

748 
lemma Cauchy_product: 

749 
fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" 

750 
assumes a: "summable (\<lambda>k. norm (a k))" 

751 
assumes b: "summable (\<lambda>k. norm (b k))" 

752 
shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i=0..k. a i * b (k  i))" 

23441  753 
using a b 
23111  754 
by (rule Cauchy_product_sums [THEN sums_unique]) 
755 

14416  756 
end 