author  huffman 
Mon, 03 May 2010 20:42:58 0700  
changeset 36657  f376af79f6b7 
parent 36409  d323e7773aa8 
child 36660  1cc4ab4b7ff7 
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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15131  12 
theory Series 
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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::real_normed_vector) \<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::real_normed_vector) \<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::real_normed_vector) \<Rightarrow> 'a" where 
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"suminf f = (THE s. f sums s)" 
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15546  29 
syntax 
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"_suminf" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a" ("\<Sum>_. _" [0, 10] 10) 
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translations 
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"\<Sum>i. b" == "CONST suminf (%i. b)" 
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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}" 

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by (simp add: diff_minus setsum_addf real_of_nat_def) 
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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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(* 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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(* 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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15543  58 
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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(* 

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lemma sumr_from_1_from_0: "0 < n ==> 

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(\<Sum>n=Suc 0 ..< Suc n. if even(n) then 0 else 

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(( 1) ^ ((n  (Suc 0)) div 2))/(real (fact n))) * a ^ n = 

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(\<Sum>n=0..<Suc n. if even(n) then 0 else 

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(( 1) ^ ((n  (Suc 0)) div 2))/(real (fact n))) * a ^ n" 

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by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto) 

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*) 

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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: "summable f ==> f sums (suminf f)" 

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apply (simp add: summable_def suminf_def sums_def) 
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apply (blast intro: theI LIMSEQ_unique) 

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done 
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lemma summable_sumr_LIMSEQ_suminf: 

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"summable f ==> (%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: "f sums s ==> (s = suminf f)" 

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apply (frule sums_summable [THEN summable_sums]) 

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apply (auto intro!: LIMSEQ_unique simp add: sums_def) 

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done 

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lemma sums_iff: "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: "f sums s ==> 
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(%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 

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done 

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lemma summable_ignore_initial_segment: "summable f ==> 

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summable (%n. f(n + k))" 

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apply (unfold summable_def) 

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apply (auto intro: sums_split_initial_segment) 

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done 

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lemma suminf_minus_initial_segment: "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) 

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apply auto 

146 
done 

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lemma suminf_split_initial_segment: "summable f ==> 

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suminf f = (SUM i = 0..< k. f i) + suminf (%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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lemma sums_Suc: 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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14416  170 
lemma series_zero: 
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"(\<forall>m. n \<le> m > f(m) = 0) ==> f sums (setsum f {0..<n})" 
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apply (simp add: sums_def LIMSEQ_iff diff_minus[symmetric], safe) 
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apply (rule_tac x = n in exI) 
15542  174 
apply (clarsimp simp add:setsum_diff[symmetric] cong:setsum_ivl_cong) 
14416  175 
done 
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lemma sums_zero: "(\<lambda>n. 0) sums 0" 
178 
unfolding sums_def by (simp add: LIMSEQ_const) 

15539  179 

23121  180 
lemma summable_zero: "summable (\<lambda>n. 0)" 
181 
by (rule sums_zero [THEN sums_summable]) 

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23121  183 
lemma suminf_zero: "suminf (\<lambda>n. 0) = 0" 
184 
by (rule sums_zero [THEN sums_unique, symmetric]) 

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23119  186 
lemma (in bounded_linear) sums: 
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"(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)" 

188 
unfolding sums_def by (drule LIMSEQ, simp only: setsum) 

189 

190 
lemma (in bounded_linear) summable: 

191 
"summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))" 

192 
unfolding summable_def by (auto intro: sums) 

193 

194 
lemma (in bounded_linear) suminf: 

195 
"summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))" 

23121  196 
by (intro sums_unique sums summable_sums) 
23119  197 

20692  198 
lemma sums_mult: 
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fixes c :: "'a::real_normed_algebra" 

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

23127  201 
by (rule mult_right.sums) 
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lemma summable_mult: 
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fixes c :: "'a::real_normed_algebra" 

23121  205 
shows "summable f \<Longrightarrow> summable (%n. c * f n)" 
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by (rule mult_right.summable) 
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20692  208 
lemma suminf_mult: 
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fixes c :: "'a::real_normed_algebra" 

210 
shows "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"; 

23127  211 
by (rule mult_right.suminf [symmetric]) 
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lemma sums_mult2: 
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fixes c :: "'a::real_normed_algebra" 

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

23127  216 
by (rule mult_left.sums) 
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20692  218 
lemma summable_mult2: 
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fixes c :: "'a::real_normed_algebra" 

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

23127  221 
by (rule mult_left.summable) 
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lemma suminf_mult2: 
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fixes c :: "'a::real_normed_algebra" 

225 
shows "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)" 

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by (rule mult_left.suminf) 
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lemma sums_divide: 
229 
fixes c :: "'a::real_normed_field" 

230 
shows "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)" 

23127  231 
by (rule divide.sums) 
14416  232 

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

235 
shows "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)" 

23127  236 
by (rule divide.summable) 
16819  237 

20692  238 
lemma suminf_divide: 
239 
fixes c :: "'a::real_normed_field" 

240 
shows "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c" 

23127  241 
by (rule divide.suminf [symmetric]) 
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23121  243 
lemma sums_add: "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) sums (a + b)" 
244 
unfolding sums_def by (simp add: setsum_addf LIMSEQ_add) 

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23121  246 
lemma summable_add: "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n + Y n)" 
247 
unfolding summable_def by (auto intro: sums_add) 

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249 
lemma suminf_add: 

23121  250 
"\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X + suminf Y = (\<Sum>n. X n + Y n)" 
251 
by (intro sums_unique sums_add summable_sums) 

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23121  253 
lemma sums_diff: "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n  Y n) sums (a  b)" 
254 
unfolding sums_def by (simp add: setsum_subtractf LIMSEQ_diff) 

255 

256 
lemma summable_diff: "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n  Y n)" 

257 
unfolding summable_def by (auto intro: sums_diff) 

14416  258 

259 
lemma suminf_diff: 

23121  260 
"\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X  suminf Y = (\<Sum>n. X n  Y n)" 
261 
by (intro sums_unique sums_diff summable_sums) 

14416  262 

23121  263 
lemma sums_minus: "X sums a ==> (\<lambda>n.  X n) sums ( a)" 
264 
unfolding sums_def by (simp add: setsum_negf LIMSEQ_minus) 

16819  265 

23121  266 
lemma summable_minus: "summable X \<Longrightarrow> summable (\<lambda>n.  X n)" 
267 
unfolding summable_def by (auto intro: sums_minus) 

16819  268 

23121  269 
lemma suminf_minus: "summable X \<Longrightarrow> (\<Sum>n.  X n) =  (\<Sum>n. X n)" 
270 
by (intro sums_unique [symmetric] sums_minus summable_sums) 

14416  271 

272 
lemma sums_group: 

15539  273 
"[summable f; 0 < k ] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)" 
14416  274 
apply (drule summable_sums) 
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apply (simp only: sums_def sumr_group) 
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apply (unfold LIMSEQ_iff, safe) 
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apply (drule_tac x="r" in spec, safe) 
278 
apply (rule_tac x="no" in exI, safe) 

279 
apply (drule_tac x="n*k" in spec) 

280 
apply (erule mp) 

281 
apply (erule order_trans) 

282 
apply simp 

14416  283 
done 
284 

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text{*A summable series of positive terms has limit that is at least as 
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great as any partial sum.*} 
14416  287 

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lemma pos_summable: 
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fixes f:: "nat \<Rightarrow> real" 
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assumes pos: "!!n. 0 \<le> f n" and le: "!!n. setsum f {0..<n} \<le> x" 
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291 
shows "summable f" 
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292 
proof  
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have "convergent (\<lambda>n. setsum f {0..<n})" 
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proof (rule Bseq_mono_convergent) 
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295 
show "Bseq (\<lambda>n. setsum f {0..<n})" 
33536  296 
by (rule f_inc_g_dec_Beq_f [of "(\<lambda>n. setsum f {0..<n})" "\<lambda>n. x"]) 
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(auto simp add: le pos) 
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next 
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show "\<forall>m n. m \<le> n \<longrightarrow> setsum f {0..<m} \<le> setsum f {0..<n}" 
33536  300 
by (auto intro: setsum_mono2 pos) 
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301 
qed 
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then obtain L where "(%n. setsum f {0..<n}) > L" 
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303 
by (blast dest: convergentD) 
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New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

304 
thus ?thesis 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

305 
by (force simp add: summable_def sums_def) 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

306 
qed 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

307 

20692  308 
lemma series_pos_le: 
309 
fixes f :: "nat \<Rightarrow> real" 

310 
shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 \<le> f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} \<le> suminf f" 

14416  311 
apply (drule summable_sums) 
312 
apply (simp add: sums_def) 

15539  313 
apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const) 
314 
apply (erule LIMSEQ_le, blast) 

20692  315 
apply (rule_tac x="n" in exI, clarify) 
15539  316 
apply (rule setsum_mono2) 
317 
apply auto 

14416  318 
done 
319 

320 
lemma series_pos_less: 

20692  321 
fixes f :: "nat \<Rightarrow> real" 
322 
shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} < suminf f" 

323 
apply (rule_tac y="setsum f {0..<Suc n}" in order_less_le_trans) 

324 
apply simp 

325 
apply (erule series_pos_le) 

326 
apply (simp add: order_less_imp_le) 

327 
done 

328 

329 
lemma suminf_gt_zero: 

330 
fixes f :: "nat \<Rightarrow> real" 

331 
shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f" 

332 
by (drule_tac n="0" in series_pos_less, simp_all) 

333 

334 
lemma suminf_ge_zero: 

335 
fixes f :: "nat \<Rightarrow> real" 

336 
shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 \<le> suminf f" 

337 
by (drule_tac n="0" in series_pos_le, simp_all) 

338 

339 
lemma sumr_pos_lt_pair: 

340 
fixes f :: "nat \<Rightarrow> real" 

341 
shows "\<lbrakk>summable f; 

342 
\<forall>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk> 

343 
\<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

344 
unfolding One_nat_def 
20692  345 
apply (subst suminf_split_initial_segment [where k="k"]) 
346 
apply assumption 

347 
apply simp 

348 
apply (drule_tac k="k" in summable_ignore_initial_segment) 

349 
apply (drule_tac k="Suc (Suc 0)" in sums_group, simp) 

350 
apply simp 

351 
apply (frule sums_unique) 

352 
apply (drule sums_summable) 

353 
apply simp 

354 
apply (erule suminf_gt_zero) 

355 
apply (simp add: add_ac) 

14416  356 
done 
357 

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

358 
text{*Sum of a geometric progression.*} 
14416  359 

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

360 
lemmas sumr_geometric = geometric_sum [where 'a = real] 
14416  361 

20692  362 
lemma geometric_sums: 
31017  363 
fixes x :: "'a::{real_normed_field}" 
20692  364 
shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) sums (1 / (1  x))" 
365 
proof  

366 
assume less_1: "norm x < 1" 

367 
hence neq_1: "x \<noteq> 1" by auto 

368 
hence neq_0: "x  1 \<noteq> 0" by simp 

369 
from less_1 have lim_0: "(\<lambda>n. x ^ n) > 0" 

370 
by (rule LIMSEQ_power_zero) 

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

371 
hence "(\<lambda>n. x ^ n / (x  1)  1 / (x  1)) > 0 / (x  1)  1 / (x  1)" 
20692  372 
using neq_0 by (intro LIMSEQ_divide LIMSEQ_diff LIMSEQ_const) 
373 
hence "(\<lambda>n. (x ^ n  1) / (x  1)) > 1 / (1  x)" 

374 
by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib) 

375 
thus "(\<lambda>n. x ^ n) sums (1 / (1  x))" 

376 
by (simp add: sums_def geometric_sum neq_1) 

377 
qed 

378 

379 
lemma summable_geometric: 

31017  380 
fixes x :: "'a::{real_normed_field}" 
20692  381 
shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)" 
382 
by (rule geometric_sums [THEN sums_summable]) 

14416  383 

36409  384 
lemma half: "0 < 1 / (2::'a::{number_ring,linordered_field_inverse_zero})" 
33271
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

385 
by arith 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

386 

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

387 
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

388 
proof  
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

389 
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

390 
by auto 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

391 
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

392 
by simp 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

393 
thus ?thesis using divide.sums [OF 2, of 2] 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

394 
by simp 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

395 
qed 
7be66dee1a5a
New theory Probability, which contains a development of measure theory
paulson
parents:
32877
diff
changeset

396 

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

397 
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

398 

15539  399 
lemma summable_convergent_sumr_iff: 
400 
"summable f = convergent (%n. setsum f {0..<n})" 

14416  401 
by (simp add: summable_def sums_def convergent_def) 
402 

20689  403 
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f > 0" 
404 
apply (drule summable_convergent_sumr_iff [THEN iffD1]) 

20692  405 
apply (drule convergent_Cauchy) 
31336
e17f13cd1280
generalize constants in SEQ.thy to class metric_space
huffman
parents:
31017
diff
changeset

406 
apply (simp only: Cauchy_iff LIMSEQ_iff, safe) 
20689  407 
apply (drule_tac x="r" in spec, safe) 
408 
apply (rule_tac x="M" in exI, safe) 

409 
apply (drule_tac x="Suc n" in spec, simp) 

410 
apply (drule_tac x="n" in spec, simp) 

411 
done 

412 

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

413 
lemma suminf_le: 
836ec9d0a0c8
New lemmas involving the real numbers, especially limits and series
paulson
parents:
31336
diff
changeset

414 
fixes x :: real 
836ec9d0a0c8
New lemmas involving the real numbers, especially limits and series
paulson
parents:
31336
diff
changeset

415 
shows "summable f \<Longrightarrow> (!!n. setsum f {0..<n} \<le> x) \<Longrightarrow> suminf f \<le> x" 
836ec9d0a0c8
New lemmas involving the real numbers, especially limits and series
paulson
parents:
31336
diff
changeset

416 
by (simp add: summable_convergent_sumr_iff suminf_eq_lim lim_le) 
836ec9d0a0c8
New lemmas involving the real numbers, especially limits and series
paulson
parents:
31336
diff
changeset

417 

14416  418 
lemma summable_Cauchy: 
20848  419 
"summable (f::nat \<Rightarrow> 'a::banach) = 
420 
(\<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

421 
apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_iff, safe) 
20410  422 
apply (drule spec, drule (1) mp) 
423 
apply (erule exE, rule_tac x="M" in exI, clarify) 

424 
apply (rule_tac x="m" and y="n" in linorder_le_cases) 

425 
apply (frule (1) order_trans) 

426 
apply (drule_tac x="n" in spec, drule (1) mp) 

427 
apply (drule_tac x="m" in spec, drule (1) mp) 

428 
apply (simp add: setsum_diff [symmetric]) 

429 
apply simp 

430 
apply (drule spec, drule (1) mp) 

431 
apply (erule exE, rule_tac x="N" in exI, clarify) 

432 
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

433 
apply (subst norm_minus_commute) 
20410  434 
apply (simp add: setsum_diff [symmetric]) 
435 
apply (simp add: setsum_diff [symmetric]) 

14416  436 
done 
437 

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

438 
text{*Comparison test*} 
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

439 

20692  440 
lemma norm_setsum: 
441 
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" 

442 
shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))" 

443 
apply (case_tac "finite A") 

444 
apply (erule finite_induct) 

445 
apply simp 

446 
apply simp 

447 
apply (erule order_trans [OF norm_triangle_ineq add_left_mono]) 

448 
apply simp 

449 
done 

450 

14416  451 
lemma summable_comparison_test: 
20848  452 
fixes f :: "nat \<Rightarrow> 'a::banach" 
453 
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f" 

20692  454 
apply (simp add: summable_Cauchy, safe) 
455 
apply (drule_tac x="e" in spec, safe) 

456 
apply (rule_tac x = "N + Na" in exI, safe) 

14416  457 
apply (rotate_tac 2) 
458 
apply (drule_tac x = m in spec) 

459 
apply (auto, rotate_tac 2, drule_tac x = n in spec) 

20848  460 
apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans) 
461 
apply (rule norm_setsum) 

15539  462 
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans) 
22998  463 
apply (auto intro: setsum_mono simp add: abs_less_iff) 
14416  464 
done 
465 

20848  466 
lemma summable_norm_comparison_test: 
467 
fixes f :: "nat \<Rightarrow> 'a::banach" 

468 
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> 

469 
\<Longrightarrow> summable (\<lambda>n. norm (f n))" 

470 
apply (rule summable_comparison_test) 

471 
apply (auto) 

472 
done 

473 

14416  474 
lemma summable_rabs_comparison_test: 
20692  475 
fixes f :: "nat \<Rightarrow> real" 
476 
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  477 
apply (rule summable_comparison_test) 
15543  478 
apply (auto) 
14416  479 
done 
480 

23084  481 
text{*Summability of geometric series for real algebras*} 
482 

483 
lemma complete_algebra_summable_geometric: 

31017  484 
fixes x :: "'a::{real_normed_algebra_1,banach}" 
23084  485 
shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)" 
486 
proof (rule summable_comparison_test) 

487 
show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n" 

488 
by (simp add: norm_power_ineq) 

489 
show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)" 

490 
by (simp add: summable_geometric) 

491 
qed 

492 

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

493 
text{*Limit comparison property for series (c.f. jrh)*} 
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset

494 

14416  495 
lemma summable_le: 
20692  496 
fixes f g :: "nat \<Rightarrow> real" 
497 
shows "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g" 

14416  498 
apply (drule summable_sums)+ 
20692  499 
apply (simp only: sums_def, erule (1) LIMSEQ_le) 
14416  500 
apply (rule exI) 
15539  501 
apply (auto intro!: setsum_mono) 
14416  502 
done 
503 

504 
lemma summable_le2: 

20692  505 
fixes f g :: "nat \<Rightarrow> real" 
506 
shows "\<lbrakk>\<forall>n. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f \<and> suminf f \<le> suminf g" 

20848  507 
apply (subgoal_tac "summable f") 
508 
apply (auto intro!: summable_le) 

22998  509 
apply (simp add: abs_le_iff) 
20848  510 
apply (rule_tac g="g" in summable_comparison_test, simp_all) 
14416  511 
done 
512 

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

513 
(* 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

514 
lemma suminf_0_le: 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

515 
fixes f::"nat\<Rightarrow>real" 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

516 
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

517 
shows "0 \<le> suminf f" 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

518 
proof  
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

519 
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

520 
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

521 
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

522 
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

523 
ultimately have "suminf ?g \<le> suminf f" by (rule summable_le) 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

524 
then show "0 \<le> suminf f" by (simp add: suminf_zero) 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

525 
qed 
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset

526 

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

527 

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

528 
text{*Absolute convergence imples normal convergence*} 
20848  529 
lemma summable_norm_cancel: 
530 
fixes f :: "nat \<Rightarrow> 'a::banach" 

531 
shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f" 

20692  532 
apply (simp only: summable_Cauchy, safe) 
533 
apply (drule_tac x="e" in spec, safe) 

534 
apply (rule_tac x="N" in exI, safe) 

535 
apply (drule_tac x="m" in spec, safe) 

20848  536 
apply (rule order_le_less_trans [OF norm_setsum]) 
537 
apply (rule order_le_less_trans [OF abs_ge_self]) 

20692  538 
apply simp 
14416  539 
done 
540 

20848  541 
lemma summable_rabs_cancel: 
542 
fixes f :: "nat \<Rightarrow> real" 

543 
shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f" 

544 
by (rule summable_norm_cancel, simp) 

545 

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

546 
text{*Absolute convergence of series*} 
20848  547 
lemma summable_norm: 
548 
fixes f :: "nat \<Rightarrow> 'a::banach" 

549 
shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))" 

550 
by (auto intro: LIMSEQ_le LIMSEQ_norm summable_norm_cancel 

551 
summable_sumr_LIMSEQ_suminf norm_setsum) 

552 

14416  553 
lemma summable_rabs: 
20692  554 
fixes f :: "nat \<Rightarrow> real" 
555 
shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)" 

20848  556 
by (fold real_norm_def, rule summable_norm) 
14416  557 

558 
subsection{* The Ratio Test*} 

559 

20848  560 
lemma norm_ratiotest_lemma: 
22852  561 
fixes x y :: "'a::real_normed_vector" 
20848  562 
shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0" 
563 
apply (subgoal_tac "norm x \<le> 0", simp) 

564 
apply (erule order_trans) 

565 
apply (simp add: mult_le_0_iff) 

566 
done 

567 

14416  568 
lemma rabs_ratiotest_lemma: "[ c \<le> 0; abs x \<le> c * abs y ] ==> x = (0::real)" 
20848  569 
by (erule norm_ratiotest_lemma, simp) 
14416  570 

571 
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)" 

572 
apply (drule le_imp_less_or_eq) 

573 
apply (auto dest: less_imp_Suc_add) 

574 
done 

575 

576 
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)" 

577 
by (auto simp add: le_Suc_ex) 

578 

579 
(*All this trouble just to get 0<c *) 

580 
lemma ratio_test_lemma2: 

20848  581 
fixes f :: "nat \<Rightarrow> 'a::banach" 
582 
shows "\<lbrakk>\<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> 0 < c \<or> summable f" 

14416  583 
apply (simp (no_asm) add: linorder_not_le [symmetric]) 
584 
apply (simp add: summable_Cauchy) 

15543  585 
apply (safe, subgoal_tac "\<forall>n. N < n > f (n) = 0") 
586 
prefer 2 

587 
apply clarify 

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

588 
apply(erule_tac x = "n  Suc 0" in allE) 
15543  589 
apply (simp add:diff_Suc split:nat.splits) 
20848  590 
apply (blast intro: norm_ratiotest_lemma) 
14416  591 
apply (rule_tac x = "Suc N" in exI, clarify) 
15543  592 
apply(simp cong:setsum_ivl_cong) 
14416  593 
done 
594 

595 
lemma ratio_test: 

20848  596 
fixes f :: "nat \<Rightarrow> 'a::banach" 
597 
shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f" 

14416  598 
apply (frule ratio_test_lemma2, auto) 
20848  599 
apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n" 
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600 
in summable_comparison_test) 
14416  601 
apply (rule_tac x = N in exI, safe) 
602 
apply (drule le_Suc_ex_iff [THEN iffD1]) 

22959  603 
apply (auto simp add: power_add field_power_not_zero) 
15539  604 
apply (induct_tac "na", auto) 
20848  605 
apply (rule_tac y = "c * norm (f (N + n))" in order_trans) 
14416  606 
apply (auto intro: mult_right_mono simp add: summable_def) 
20848  607 
apply (rule_tac x = "norm (f N) * (1/ (1  c)) / (c ^ N)" in exI) 
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608 
apply (rule sums_divide) 
27108  609 
apply (rule sums_mult) 
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610 
apply (auto intro!: geometric_sums) 
14416  611 
done 
612 

23111  613 
subsection {* Cauchy Product Formula *} 
614 

615 
(* Proof based on Analysis WebNotes: Chapter 07, Class 41 

616 
http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm *) 

617 

618 
lemma setsum_triangle_reindex: 

619 
fixes n :: nat 

620 
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))" 

621 
proof  

622 
have "(\<Sum>(i, j)\<in>{(i, j). i + j < n}. f i j) = 

623 
(\<Sum>(k, i)\<in>(SIGMA k:{0..<n}. {0..k}). f i (k  i))" 

624 
proof (rule setsum_reindex_cong) 

625 
show "inj_on (\<lambda>(k,i). (i, k  i)) (SIGMA k:{0..<n}. {0..k})" 

626 
by (rule inj_on_inverseI [where g="\<lambda>(i,j). (i+j, i)"], auto) 

627 
show "{(i,j). i + j < n} = (\<lambda>(k,i). (i, k  i)) ` (SIGMA k:{0..<n}. {0..k})" 

628 
by (safe, rule_tac x="(a+b,a)" in image_eqI, auto) 

629 
show "\<And>a. (\<lambda>(k, i). f i (k  i)) a = split f ((\<lambda>(k, i). (i, k  i)) a)" 

630 
by clarify 

631 
qed 

632 
thus ?thesis by (simp add: setsum_Sigma) 

633 
qed 

634 

635 
lemma Cauchy_product_sums: 

636 
fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" 

637 
assumes a: "summable (\<lambda>k. norm (a k))" 

638 
assumes b: "summable (\<lambda>k. norm (b k))" 

639 
shows "(\<lambda>k. \<Sum>i=0..k. a i * b (k  i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))" 

640 
proof  

641 
let ?S1 = "\<lambda>n::nat. {0..<n} \<times> {0..<n}" 

642 
let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}" 

643 
have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto 

644 
have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto 

645 
have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto 

646 
have finite_S1: "\<And>n. finite (?S1 n)" by simp 

647 
with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset) 

648 

649 
let ?g = "\<lambda>(i,j). a i * b j" 

650 
let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)" 

651 
have f_nonneg: "\<And>x. 0 \<le> ?f x" 

652 
by (auto simp add: mult_nonneg_nonneg) 

653 
hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A" 

654 
unfolding real_norm_def 

655 
by (simp only: abs_of_nonneg setsum_nonneg [rule_format]) 

656 

657 
have "(\<lambda>n. (\<Sum>k=0..<n. a k) * (\<Sum>k=0..<n. b k)) 

658 
> (\<Sum>k. a k) * (\<Sum>k. b k)" 

659 
by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf 

660 
summable_norm_cancel [OF a] summable_norm_cancel [OF b]) 

661 
hence 1: "(\<lambda>n. setsum ?g (?S1 n)) > (\<Sum>k. a k) * (\<Sum>k. b k)" 

662 
by (simp only: setsum_product setsum_Sigma [rule_format] 

663 
finite_atLeastLessThan) 

664 

665 
have "(\<lambda>n. (\<Sum>k=0..<n. norm (a k)) * (\<Sum>k=0..<n. norm (b k))) 

666 
> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))" 

667 
using a b by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf) 

668 
hence "(\<lambda>n. setsum ?f (?S1 n)) > (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))" 

669 
by (simp only: setsum_product setsum_Sigma [rule_format] 

670 
finite_atLeastLessThan) 

671 
hence "convergent (\<lambda>n. setsum ?f (?S1 n))" 

672 
by (rule convergentI) 

673 
hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))" 

674 
by (rule convergent_Cauchy) 

36657  675 
have "Zfun (\<lambda>n. setsum ?f (?S1 n  ?S2 n)) sequentially" 
676 
proof (rule ZfunI, simp only: eventually_sequentially norm_setsum_f) 

23111  677 
fix r :: real 
678 
assume r: "0 < r" 

679 
from CauchyD [OF Cauchy r] obtain N 

680 
where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m)  setsum ?f (?S1 n)) < r" .. 

681 
hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m  ?S1 n)) < r" 

682 
by (simp only: setsum_diff finite_S1 S1_mono) 

683 
hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m  ?S1 n) < r" 

684 
by (simp only: norm_setsum_f) 

685 
show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n  ?S2 n) < r" 

686 
proof (intro exI allI impI) 

687 
fix n assume "2 * N \<le> n" 

688 
hence n: "N \<le> n div 2" by simp 

689 
have "setsum ?f (?S1 n  ?S2 n) \<le> setsum ?f (?S1 n  ?S1 (n div 2))" 

690 
by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg 

691 
Diff_mono subset_refl S1_le_S2) 

692 
also have "\<dots> < r" 

693 
using n div_le_dividend by (rule N) 

694 
finally show "setsum ?f (?S1 n  ?S2 n) < r" . 

695 
qed 

696 
qed 

36657  697 
hence "Zfun (\<lambda>n. setsum ?g (?S1 n  ?S2 n)) sequentially" 
698 
apply (rule Zfun_le [rule_format]) 

23111  699 
apply (simp only: norm_setsum_f) 
700 
apply (rule order_trans [OF norm_setsum setsum_mono]) 

701 
apply (auto simp add: norm_mult_ineq) 

702 
done 

703 
hence 2: "(\<lambda>n. setsum ?g (?S1 n)  setsum ?g (?S2 n)) > 0" 

36657  704 
unfolding LIMSEQ_conv_tendsto tendsto_Zfun_iff diff_0_right 
705 
by (simp only: setsum_diff finite_S1 S2_le_S1) 

23111  706 

707 
with 1 have "(\<lambda>n. setsum ?g (?S2 n)) > (\<Sum>k. a k) * (\<Sum>k. b k)" 

708 
by (rule LIMSEQ_diff_approach_zero2) 

709 
thus ?thesis by (simp only: sums_def setsum_triangle_reindex) 

710 
qed 

711 

712 
lemma Cauchy_product: 

713 
fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" 

714 
assumes a: "summable (\<lambda>k. norm (a k))" 

715 
assumes b: "summable (\<lambda>k. norm (b k))" 

716 
shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i=0..k. a i * b (k  i))" 

23441  717 
using a b 
23111  718 
by (rule Cauchy_product_sums [THEN sums_unique]) 
719 

14416  720 
end 