author | hoelzl |
Thu, 05 Feb 2009 11:34:42 +0100 | |
changeset 29803 | c56a5571f60a |
parent 29197 | 6d4cb27ed19c |
child 30082 | 43c5b7bfc791 |
permissions | -rw-r--r-- |
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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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||
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header{*Finite Summation and Infinite Series*} |
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theory Series |
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imports SEQ |
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begin |
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|
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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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|
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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 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? *) |
46 |
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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|
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(* FIXME this is an awful lemma! *) |
51 |
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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||
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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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(*------------------- |
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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_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 |
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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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149 |
by auto |
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150 |
qed |
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151 |
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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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|
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proof - |
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from sumSuc[unfolded sums_def] |
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|
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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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Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
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|
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thus ?thesis unfolding sums_def by (rule LIMSEQ_imp_Suc) |
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|
159 |
qed |
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|
160 |
|
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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_def diff_minus[symmetric], safe) |
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apply (rule_tac x = n in exI) |
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apply (clarsimp simp add:setsum_diff[symmetric] cong:setsum_ivl_cong) |
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done |
167 |
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lemma sums_zero: "(\<lambda>n. 0) sums 0" |
169 |
unfolding sums_def by (simp add: LIMSEQ_const) |
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lemma summable_zero: "summable (\<lambda>n. 0)" |
172 |
by (rule sums_zero [THEN sums_summable]) |
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lemma suminf_zero: "suminf (\<lambda>n. 0) = 0" |
175 |
by (rule sums_zero [THEN sums_unique, symmetric]) |
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lemma (in bounded_linear) sums: |
178 |
"(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)" |
|
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unfolding sums_def by (drule LIMSEQ, simp only: setsum) |
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180 |
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181 |
lemma (in bounded_linear) summable: |
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"summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))" |
|
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unfolding summable_def by (auto intro: sums) |
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184 |
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lemma (in bounded_linear) suminf: |
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"summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))" |
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by (intro sums_unique sums summable_sums) |
23119 | 188 |
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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)" |
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by (rule mult_right.sums) |
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lemma summable_mult: |
195 |
fixes c :: "'a::real_normed_algebra" |
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shows "summable f \<Longrightarrow> summable (%n. c * f n)" |
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by (rule mult_right.summable) |
16819 | 198 |
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lemma suminf_mult: |
200 |
fixes c :: "'a::real_normed_algebra" |
|
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shows "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"; |
|
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by (rule mult_right.suminf [symmetric]) |
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lemma sums_mult2: |
205 |
fixes c :: "'a::real_normed_algebra" |
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shows "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)" |
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by (rule mult_left.sums) |
16819 | 208 |
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20692 | 209 |
lemma summable_mult2: |
210 |
fixes c :: "'a::real_normed_algebra" |
|
211 |
shows "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)" |
|
23127 | 212 |
by (rule mult_left.summable) |
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20692 | 214 |
lemma suminf_mult2: |
215 |
fixes c :: "'a::real_normed_algebra" |
|
216 |
shows "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)" |
|
23127 | 217 |
by (rule mult_left.suminf) |
16819 | 218 |
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20692 | 219 |
lemma sums_divide: |
220 |
fixes c :: "'a::real_normed_field" |
|
221 |
shows "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)" |
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23127 | 222 |
by (rule divide.sums) |
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20692 | 224 |
lemma summable_divide: |
225 |
fixes c :: "'a::real_normed_field" |
|
226 |
shows "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)" |
|
23127 | 227 |
by (rule divide.summable) |
16819 | 228 |
|
20692 | 229 |
lemma suminf_divide: |
230 |
fixes c :: "'a::real_normed_field" |
|
231 |
shows "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c" |
|
23127 | 232 |
by (rule divide.suminf [symmetric]) |
16819 | 233 |
|
23121 | 234 |
lemma sums_add: "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) sums (a + b)" |
235 |
unfolding sums_def by (simp add: setsum_addf LIMSEQ_add) |
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16819 | 236 |
|
23121 | 237 |
lemma summable_add: "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n + Y n)" |
238 |
unfolding summable_def by (auto intro: sums_add) |
|
16819 | 239 |
|
240 |
lemma suminf_add: |
|
23121 | 241 |
"\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X + suminf Y = (\<Sum>n. X n + Y n)" |
242 |
by (intro sums_unique sums_add summable_sums) |
|
14416 | 243 |
|
23121 | 244 |
lemma sums_diff: "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) sums (a - b)" |
245 |
unfolding sums_def by (simp add: setsum_subtractf LIMSEQ_diff) |
|
246 |
||
247 |
lemma summable_diff: "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n - Y n)" |
|
248 |
unfolding summable_def by (auto intro: sums_diff) |
|
14416 | 249 |
|
250 |
lemma suminf_diff: |
|
23121 | 251 |
"\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X - suminf Y = (\<Sum>n. X n - Y n)" |
252 |
by (intro sums_unique sums_diff summable_sums) |
|
14416 | 253 |
|
23121 | 254 |
lemma sums_minus: "X sums a ==> (\<lambda>n. - X n) sums (- a)" |
255 |
unfolding sums_def by (simp add: setsum_negf LIMSEQ_minus) |
|
16819 | 256 |
|
23121 | 257 |
lemma summable_minus: "summable X \<Longrightarrow> summable (\<lambda>n. - X n)" |
258 |
unfolding summable_def by (auto intro: sums_minus) |
|
16819 | 259 |
|
23121 | 260 |
lemma suminf_minus: "summable X \<Longrightarrow> (\<Sum>n. - X n) = - (\<Sum>n. X n)" |
261 |
by (intro sums_unique [symmetric] sums_minus summable_sums) |
|
14416 | 262 |
|
263 |
lemma sums_group: |
|
15539 | 264 |
"[|summable f; 0 < k |] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)" |
14416 | 265 |
apply (drule summable_sums) |
20692 | 266 |
apply (simp only: sums_def sumr_group) |
267 |
apply (unfold LIMSEQ_def, safe) |
|
268 |
apply (drule_tac x="r" in spec, safe) |
|
269 |
apply (rule_tac x="no" in exI, safe) |
|
270 |
apply (drule_tac x="n*k" in spec) |
|
271 |
apply (erule mp) |
|
272 |
apply (erule order_trans) |
|
273 |
apply simp |
|
14416 | 274 |
done |
275 |
||
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
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|
276 |
text{*A summable series of positive terms has limit that is at least as |
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
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|
277 |
great as any partial sum.*} |
14416 | 278 |
|
20692 | 279 |
lemma series_pos_le: |
280 |
fixes f :: "nat \<Rightarrow> real" |
|
281 |
shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 \<le> f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} \<le> suminf f" |
|
14416 | 282 |
apply (drule summable_sums) |
283 |
apply (simp add: sums_def) |
|
15539 | 284 |
apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const) |
285 |
apply (erule LIMSEQ_le, blast) |
|
20692 | 286 |
apply (rule_tac x="n" in exI, clarify) |
15539 | 287 |
apply (rule setsum_mono2) |
288 |
apply auto |
|
14416 | 289 |
done |
290 |
||
291 |
lemma series_pos_less: |
|
20692 | 292 |
fixes f :: "nat \<Rightarrow> real" |
293 |
shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} < suminf f" |
|
294 |
apply (rule_tac y="setsum f {0..<Suc n}" in order_less_le_trans) |
|
295 |
apply simp |
|
296 |
apply (erule series_pos_le) |
|
297 |
apply (simp add: order_less_imp_le) |
|
298 |
done |
|
299 |
||
300 |
lemma suminf_gt_zero: |
|
301 |
fixes f :: "nat \<Rightarrow> real" |
|
302 |
shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f" |
|
303 |
by (drule_tac n="0" in series_pos_less, simp_all) |
|
304 |
||
305 |
lemma suminf_ge_zero: |
|
306 |
fixes f :: "nat \<Rightarrow> real" |
|
307 |
shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 \<le> suminf f" |
|
308 |
by (drule_tac n="0" in series_pos_le, simp_all) |
|
309 |
||
310 |
lemma sumr_pos_lt_pair: |
|
311 |
fixes f :: "nat \<Rightarrow> real" |
|
312 |
shows "\<lbrakk>summable f; |
|
313 |
\<forall>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk> |
|
314 |
\<Longrightarrow> setsum f {0..<k} < suminf f" |
|
315 |
apply (subst suminf_split_initial_segment [where k="k"]) |
|
316 |
apply assumption |
|
317 |
apply simp |
|
318 |
apply (drule_tac k="k" in summable_ignore_initial_segment) |
|
319 |
apply (drule_tac k="Suc (Suc 0)" in sums_group, simp) |
|
320 |
apply simp |
|
321 |
apply (frule sums_unique) |
|
322 |
apply (drule sums_summable) |
|
323 |
apply simp |
|
324 |
apply (erule suminf_gt_zero) |
|
325 |
apply (simp add: add_ac) |
|
14416 | 326 |
done |
327 |
||
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
328 |
text{*Sum of a geometric progression.*} |
14416 | 329 |
|
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16819
diff
changeset
|
330 |
lemmas sumr_geometric = geometric_sum [where 'a = real] |
14416 | 331 |
|
20692 | 332 |
lemma geometric_sums: |
22719
c51667189bd3
lemma geometric_sum no longer needs class division_by_zero
huffman
parents:
21404
diff
changeset
|
333 |
fixes x :: "'a::{real_normed_field,recpower}" |
20692 | 334 |
shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) sums (1 / (1 - x))" |
335 |
proof - |
|
336 |
assume less_1: "norm x < 1" |
|
337 |
hence neq_1: "x \<noteq> 1" by auto |
|
338 |
hence neq_0: "x - 1 \<noteq> 0" by simp |
|
339 |
from less_1 have lim_0: "(\<lambda>n. x ^ n) ----> 0" |
|
340 |
by (rule LIMSEQ_power_zero) |
|
22719
c51667189bd3
lemma geometric_sum no longer needs class division_by_zero
huffman
parents:
21404
diff
changeset
|
341 |
hence "(\<lambda>n. x ^ n / (x - 1) - 1 / (x - 1)) ----> 0 / (x - 1) - 1 / (x - 1)" |
20692 | 342 |
using neq_0 by (intro LIMSEQ_divide LIMSEQ_diff LIMSEQ_const) |
343 |
hence "(\<lambda>n. (x ^ n - 1) / (x - 1)) ----> 1 / (1 - x)" |
|
344 |
by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib) |
|
345 |
thus "(\<lambda>n. x ^ n) sums (1 / (1 - x))" |
|
346 |
by (simp add: sums_def geometric_sum neq_1) |
|
347 |
qed |
|
348 |
||
349 |
lemma summable_geometric: |
|
22719
c51667189bd3
lemma geometric_sum no longer needs class division_by_zero
huffman
parents:
21404
diff
changeset
|
350 |
fixes x :: "'a::{real_normed_field,recpower}" |
20692 | 351 |
shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)" |
352 |
by (rule geometric_sums [THEN sums_summable]) |
|
14416 | 353 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
354 |
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*} |
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
355 |
|
15539 | 356 |
lemma summable_convergent_sumr_iff: |
357 |
"summable f = convergent (%n. setsum f {0..<n})" |
|
14416 | 358 |
by (simp add: summable_def sums_def convergent_def) |
359 |
||
20689 | 360 |
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0" |
361 |
apply (drule summable_convergent_sumr_iff [THEN iffD1]) |
|
20692 | 362 |
apply (drule convergent_Cauchy) |
20689 | 363 |
apply (simp only: Cauchy_def LIMSEQ_def, safe) |
364 |
apply (drule_tac x="r" in spec, safe) |
|
365 |
apply (rule_tac x="M" in exI, safe) |
|
366 |
apply (drule_tac x="Suc n" in spec, simp) |
|
367 |
apply (drule_tac x="n" in spec, simp) |
|
368 |
done |
|
369 |
||
14416 | 370 |
lemma summable_Cauchy: |
20848 | 371 |
"summable (f::nat \<Rightarrow> 'a::banach) = |
372 |
(\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. norm (setsum f {m..<n}) < e)" |
|
373 |
apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def, safe) |
|
20410 | 374 |
apply (drule spec, drule (1) mp) |
375 |
apply (erule exE, rule_tac x="M" in exI, clarify) |
|
376 |
apply (rule_tac x="m" and y="n" in linorder_le_cases) |
|
377 |
apply (frule (1) order_trans) |
|
378 |
apply (drule_tac x="n" in spec, drule (1) mp) |
|
379 |
apply (drule_tac x="m" in spec, drule (1) mp) |
|
380 |
apply (simp add: setsum_diff [symmetric]) |
|
381 |
apply simp |
|
382 |
apply (drule spec, drule (1) mp) |
|
383 |
apply (erule exE, rule_tac x="N" in exI, clarify) |
|
384 |
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
|
385 |
apply (subst norm_minus_commute) |
20410 | 386 |
apply (simp add: setsum_diff [symmetric]) |
387 |
apply (simp add: setsum_diff [symmetric]) |
|
14416 | 388 |
done |
389 |
||
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
390 |
text{*Comparison test*} |
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
391 |
|
20692 | 392 |
lemma norm_setsum: |
393 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
|
394 |
shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))" |
|
395 |
apply (case_tac "finite A") |
|
396 |
apply (erule finite_induct) |
|
397 |
apply simp |
|
398 |
apply simp |
|
399 |
apply (erule order_trans [OF norm_triangle_ineq add_left_mono]) |
|
400 |
apply simp |
|
401 |
done |
|
402 |
||
14416 | 403 |
lemma summable_comparison_test: |
20848 | 404 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
405 |
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f" |
|
20692 | 406 |
apply (simp add: summable_Cauchy, safe) |
407 |
apply (drule_tac x="e" in spec, safe) |
|
408 |
apply (rule_tac x = "N + Na" in exI, safe) |
|
14416 | 409 |
apply (rotate_tac 2) |
410 |
apply (drule_tac x = m in spec) |
|
411 |
apply (auto, rotate_tac 2, drule_tac x = n in spec) |
|
20848 | 412 |
apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans) |
413 |
apply (rule norm_setsum) |
|
15539 | 414 |
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans) |
22998 | 415 |
apply (auto intro: setsum_mono simp add: abs_less_iff) |
14416 | 416 |
done |
417 |
||
20848 | 418 |
lemma summable_norm_comparison_test: |
419 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
|
420 |
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> |
|
421 |
\<Longrightarrow> summable (\<lambda>n. norm (f n))" |
|
422 |
apply (rule summable_comparison_test) |
|
423 |
apply (auto) |
|
424 |
done |
|
425 |
||
14416 | 426 |
lemma summable_rabs_comparison_test: |
20692 | 427 |
fixes f :: "nat \<Rightarrow> real" |
428 |
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 | 429 |
apply (rule summable_comparison_test) |
15543 | 430 |
apply (auto) |
14416 | 431 |
done |
432 |
||
23084 | 433 |
text{*Summability of geometric series for real algebras*} |
434 |
||
435 |
lemma complete_algebra_summable_geometric: |
|
436 |
fixes x :: "'a::{real_normed_algebra_1,banach,recpower}" |
|
437 |
shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)" |
|
438 |
proof (rule summable_comparison_test) |
|
439 |
show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n" |
|
440 |
by (simp add: norm_power_ineq) |
|
441 |
show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)" |
|
442 |
by (simp add: summable_geometric) |
|
443 |
qed |
|
444 |
||
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
445 |
text{*Limit comparison property for series (c.f. jrh)*} |
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
446 |
|
14416 | 447 |
lemma summable_le: |
20692 | 448 |
fixes f g :: "nat \<Rightarrow> real" |
449 |
shows "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g" |
|
14416 | 450 |
apply (drule summable_sums)+ |
20692 | 451 |
apply (simp only: sums_def, erule (1) LIMSEQ_le) |
14416 | 452 |
apply (rule exI) |
15539 | 453 |
apply (auto intro!: setsum_mono) |
14416 | 454 |
done |
455 |
||
456 |
lemma summable_le2: |
|
20692 | 457 |
fixes f g :: "nat \<Rightarrow> real" |
458 |
shows "\<lbrakk>\<forall>n. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f \<and> suminf f \<le> suminf g" |
|
20848 | 459 |
apply (subgoal_tac "summable f") |
460 |
apply (auto intro!: summable_le) |
|
22998 | 461 |
apply (simp add: abs_le_iff) |
20848 | 462 |
apply (rule_tac g="g" in summable_comparison_test, simp_all) |
14416 | 463 |
done |
464 |
||
19106
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
465 |
(* 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
|
466 |
lemma suminf_0_le: |
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
467 |
fixes f::"nat\<Rightarrow>real" |
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
468 |
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
|
469 |
shows "0 \<le> suminf f" |
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
470 |
proof - |
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
471 |
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
|
472 |
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
|
473 |
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
|
474 |
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
|
475 |
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
|
476 |
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
|
477 |
qed |
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
478 |
|
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
479 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
480 |
text{*Absolute convergence imples normal convergence*} |
20848 | 481 |
lemma summable_norm_cancel: |
482 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
|
483 |
shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f" |
|
20692 | 484 |
apply (simp only: summable_Cauchy, safe) |
485 |
apply (drule_tac x="e" in spec, safe) |
|
486 |
apply (rule_tac x="N" in exI, safe) |
|
487 |
apply (drule_tac x="m" in spec, safe) |
|
20848 | 488 |
apply (rule order_le_less_trans [OF norm_setsum]) |
489 |
apply (rule order_le_less_trans [OF abs_ge_self]) |
|
20692 | 490 |
apply simp |
14416 | 491 |
done |
492 |
||
20848 | 493 |
lemma summable_rabs_cancel: |
494 |
fixes f :: "nat \<Rightarrow> real" |
|
495 |
shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f" |
|
496 |
by (rule summable_norm_cancel, simp) |
|
497 |
||
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
498 |
text{*Absolute convergence of series*} |
20848 | 499 |
lemma summable_norm: |
500 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
|
501 |
shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))" |
|
502 |
by (auto intro: LIMSEQ_le LIMSEQ_norm summable_norm_cancel |
|
503 |
summable_sumr_LIMSEQ_suminf norm_setsum) |
|
504 |
||
14416 | 505 |
lemma summable_rabs: |
20692 | 506 |
fixes f :: "nat \<Rightarrow> real" |
507 |
shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)" |
|
20848 | 508 |
by (fold real_norm_def, rule summable_norm) |
14416 | 509 |
|
510 |
subsection{* The Ratio Test*} |
|
511 |
||
20848 | 512 |
lemma norm_ratiotest_lemma: |
22852 | 513 |
fixes x y :: "'a::real_normed_vector" |
20848 | 514 |
shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0" |
515 |
apply (subgoal_tac "norm x \<le> 0", simp) |
|
516 |
apply (erule order_trans) |
|
517 |
apply (simp add: mult_le_0_iff) |
|
518 |
done |
|
519 |
||
14416 | 520 |
lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)" |
20848 | 521 |
by (erule norm_ratiotest_lemma, simp) |
14416 | 522 |
|
523 |
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)" |
|
524 |
apply (drule le_imp_less_or_eq) |
|
525 |
apply (auto dest: less_imp_Suc_add) |
|
526 |
done |
|
527 |
||
528 |
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)" |
|
529 |
by (auto simp add: le_Suc_ex) |
|
530 |
||
531 |
(*All this trouble just to get 0<c *) |
|
532 |
lemma ratio_test_lemma2: |
|
20848 | 533 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
534 |
shows "\<lbrakk>\<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> 0 < c \<or> summable f" |
|
14416 | 535 |
apply (simp (no_asm) add: linorder_not_le [symmetric]) |
536 |
apply (simp add: summable_Cauchy) |
|
15543 | 537 |
apply (safe, subgoal_tac "\<forall>n. N < n --> f (n) = 0") |
538 |
prefer 2 |
|
539 |
apply clarify |
|
540 |
apply(erule_tac x = "n - 1" in allE) |
|
541 |
apply (simp add:diff_Suc split:nat.splits) |
|
20848 | 542 |
apply (blast intro: norm_ratiotest_lemma) |
14416 | 543 |
apply (rule_tac x = "Suc N" in exI, clarify) |
15543 | 544 |
apply(simp cong:setsum_ivl_cong) |
14416 | 545 |
done |
546 |
||
547 |
lemma ratio_test: |
|
20848 | 548 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
549 |
shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f" |
|
14416 | 550 |
apply (frule ratio_test_lemma2, auto) |
20848 | 551 |
apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n" |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
552 |
in summable_comparison_test) |
14416 | 553 |
apply (rule_tac x = N in exI, safe) |
554 |
apply (drule le_Suc_ex_iff [THEN iffD1]) |
|
22959 | 555 |
apply (auto simp add: power_add field_power_not_zero) |
15539 | 556 |
apply (induct_tac "na", auto) |
20848 | 557 |
apply (rule_tac y = "c * norm (f (N + n))" in order_trans) |
14416 | 558 |
apply (auto intro: mult_right_mono simp add: summable_def) |
559 |
apply (simp add: mult_ac) |
|
20848 | 560 |
apply (rule_tac x = "norm (f N) * (1/ (1 - c)) / (c ^ N)" in exI) |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
561 |
apply (rule sums_divide) |
27108 | 562 |
apply (rule sums_mult) |
15234
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
563 |
apply (auto intro!: geometric_sums) |
14416 | 564 |
done |
565 |
||
23111 | 566 |
subsection {* Cauchy Product Formula *} |
567 |
||
568 |
(* Proof based on Analysis WebNotes: Chapter 07, Class 41 |
|
569 |
http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm *) |
|
570 |
||
571 |
lemma setsum_triangle_reindex: |
|
572 |
fixes n :: nat |
|
573 |
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))" |
|
574 |
proof - |
|
575 |
have "(\<Sum>(i, j)\<in>{(i, j). i + j < n}. f i j) = |
|
576 |
(\<Sum>(k, i)\<in>(SIGMA k:{0..<n}. {0..k}). f i (k - i))" |
|
577 |
proof (rule setsum_reindex_cong) |
|
578 |
show "inj_on (\<lambda>(k,i). (i, k - i)) (SIGMA k:{0..<n}. {0..k})" |
|
579 |
by (rule inj_on_inverseI [where g="\<lambda>(i,j). (i+j, i)"], auto) |
|
580 |
show "{(i,j). i + j < n} = (\<lambda>(k,i). (i, k - i)) ` (SIGMA k:{0..<n}. {0..k})" |
|
581 |
by (safe, rule_tac x="(a+b,a)" in image_eqI, auto) |
|
582 |
show "\<And>a. (\<lambda>(k, i). f i (k - i)) a = split f ((\<lambda>(k, i). (i, k - i)) a)" |
|
583 |
by clarify |
|
584 |
qed |
|
585 |
thus ?thesis by (simp add: setsum_Sigma) |
|
586 |
qed |
|
587 |
||
588 |
lemma Cauchy_product_sums: |
|
589 |
fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" |
|
590 |
assumes a: "summable (\<lambda>k. norm (a k))" |
|
591 |
assumes b: "summable (\<lambda>k. norm (b k))" |
|
592 |
shows "(\<lambda>k. \<Sum>i=0..k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))" |
|
593 |
proof - |
|
594 |
let ?S1 = "\<lambda>n::nat. {0..<n} \<times> {0..<n}" |
|
595 |
let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}" |
|
596 |
have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto |
|
597 |
have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto |
|
598 |
have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto |
|
599 |
have finite_S1: "\<And>n. finite (?S1 n)" by simp |
|
600 |
with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset) |
|
601 |
||
602 |
let ?g = "\<lambda>(i,j). a i * b j" |
|
603 |
let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)" |
|
604 |
have f_nonneg: "\<And>x. 0 \<le> ?f x" |
|
605 |
by (auto simp add: mult_nonneg_nonneg) |
|
606 |
hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A" |
|
607 |
unfolding real_norm_def |
|
608 |
by (simp only: abs_of_nonneg setsum_nonneg [rule_format]) |
|
609 |
||
610 |
have "(\<lambda>n. (\<Sum>k=0..<n. a k) * (\<Sum>k=0..<n. b k)) |
|
611 |
----> (\<Sum>k. a k) * (\<Sum>k. b k)" |
|
612 |
by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf |
|
613 |
summable_norm_cancel [OF a] summable_norm_cancel [OF b]) |
|
614 |
hence 1: "(\<lambda>n. setsum ?g (?S1 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)" |
|
615 |
by (simp only: setsum_product setsum_Sigma [rule_format] |
|
616 |
finite_atLeastLessThan) |
|
617 |
||
618 |
have "(\<lambda>n. (\<Sum>k=0..<n. norm (a k)) * (\<Sum>k=0..<n. norm (b k))) |
|
619 |
----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))" |
|
620 |
using a b by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf) |
|
621 |
hence "(\<lambda>n. setsum ?f (?S1 n)) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))" |
|
622 |
by (simp only: setsum_product setsum_Sigma [rule_format] |
|
623 |
finite_atLeastLessThan) |
|
624 |
hence "convergent (\<lambda>n. setsum ?f (?S1 n))" |
|
625 |
by (rule convergentI) |
|
626 |
hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))" |
|
627 |
by (rule convergent_Cauchy) |
|
628 |
have "Zseq (\<lambda>n. setsum ?f (?S1 n - ?S2 n))" |
|
629 |
proof (rule ZseqI, simp only: norm_setsum_f) |
|
630 |
fix r :: real |
|
631 |
assume r: "0 < r" |
|
632 |
from CauchyD [OF Cauchy r] obtain N |
|
633 |
where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" .. |
|
634 |
hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r" |
|
635 |
by (simp only: setsum_diff finite_S1 S1_mono) |
|
636 |
hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r" |
|
637 |
by (simp only: norm_setsum_f) |
|
638 |
show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r" |
|
639 |
proof (intro exI allI impI) |
|
640 |
fix n assume "2 * N \<le> n" |
|
641 |
hence n: "N \<le> n div 2" by simp |
|
642 |
have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))" |
|
643 |
by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg |
|
644 |
Diff_mono subset_refl S1_le_S2) |
|
645 |
also have "\<dots> < r" |
|
646 |
using n div_le_dividend by (rule N) |
|
647 |
finally show "setsum ?f (?S1 n - ?S2 n) < r" . |
|
648 |
qed |
|
649 |
qed |
|
650 |
hence "Zseq (\<lambda>n. setsum ?g (?S1 n - ?S2 n))" |
|
651 |
apply (rule Zseq_le [rule_format]) |
|
652 |
apply (simp only: norm_setsum_f) |
|
653 |
apply (rule order_trans [OF norm_setsum setsum_mono]) |
|
654 |
apply (auto simp add: norm_mult_ineq) |
|
655 |
done |
|
656 |
hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) ----> 0" |
|
657 |
by (simp only: LIMSEQ_Zseq_iff setsum_diff finite_S1 S2_le_S1 diff_0_right) |
|
658 |
||
659 |
with 1 have "(\<lambda>n. setsum ?g (?S2 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)" |
|
660 |
by (rule LIMSEQ_diff_approach_zero2) |
|
661 |
thus ?thesis by (simp only: sums_def setsum_triangle_reindex) |
|
662 |
qed |
|
663 |
||
664 |
lemma Cauchy_product: |
|
665 |
fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}" |
|
666 |
assumes a: "summable (\<lambda>k. norm (a k))" |
|
667 |
assumes b: "summable (\<lambda>k. norm (b k))" |
|
668 |
shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i=0..k. a i * b (k - i))" |
|
23441 | 669 |
using a b |
23111 | 670 |
by (rule Cauchy_product_sums [THEN sums_unique]) |
671 |
||
14416 | 672 |
end |