author | wenzelm |
Fri, 17 Nov 2006 02:20:03 +0100 | |
changeset 21404 | eb85850d3eb7 |
parent 21141 | f0b5e6254a1f |
child 22719 | c51667189bd3 |
permissions | -rw-r--r-- |
10751 | 1 |
(* Title : Series.thy |
2 |
Author : Jacques D. Fleuriot |
|
3 |
Copyright : 1998 University of Cambridge |
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|
5 |
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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|
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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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more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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definition |
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more robust syntax for definition/abbreviation/notation;
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parents:
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changeset
|
22 |
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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|
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more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21141
diff
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|
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definition |
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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diff
changeset
|
26 |
suminf :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a" where |
20688 | 27 |
"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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|
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|
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lemma sumr_diff_mult_const: |
36 |
"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) |
38 |
||
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lemma real_setsum_nat_ivl_bounded: |
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"(!!p. p < n \<Longrightarrow> f(p) \<le> K) |
|
41 |
\<Longrightarrow> setsum f {0..<n::nat} \<le> real n * K" |
|
42 |
using setsum_bounded[where A = "{0..<n}"] |
|
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by (auto simp:real_of_nat_def) |
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|
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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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|
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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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|
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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) |
|
66 |
done |
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||
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lemma sumr_offset: |
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fixes f :: "nat \<Rightarrow> 'a::ab_group_add" |
70 |
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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|
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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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|
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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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(* |
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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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|
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subsection{* Infinite Sums, by the Properties of Limits*} |
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91 |
||
92 |
(*---------------------- |
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93 |
suminf is the sum |
|
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---------------------*) |
|
95 |
lemma sums_summable: "f sums l ==> summable f" |
|
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by (simp add: sums_def summable_def, blast) |
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97 |
||
98 |
lemma summable_sums: "summable f ==> f sums (suminf f)" |
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apply (simp add: summable_def suminf_def sums_def) |
100 |
apply (blast intro: theI LIMSEQ_unique) |
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done |
102 |
||
103 |
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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(*------------------- |
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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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113 |
done |
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114 |
||
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lemma sums_split_initial_segment: "f sums s ==> |
116 |
(%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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122 |
done |
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123 |
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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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128 |
done |
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129 |
||
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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) |
|
133 |
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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137 |
done |
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138 |
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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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142 |
||
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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 |
149 |
||
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lemma sums_zero: "(%n. 0) sums 0"; |
151 |
apply (unfold sums_def); |
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apply simp; |
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apply (rule LIMSEQ_const); |
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done; |
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|
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lemma summable_zero: "summable (%n. 0)"; |
157 |
apply (rule sums_summable); |
|
158 |
apply (rule sums_zero); |
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159 |
done; |
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160 |
||
161 |
lemma suminf_zero: "suminf (%n. 0) = 0"; |
|
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apply (rule sym); |
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163 |
apply (rule sums_unique); |
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164 |
apply (rule sums_zero); |
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165 |
done; |
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166 |
||
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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 (auto simp add: sums_def setsum_right_distrib [symmetric] |
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intro!: LIMSEQ_mult intro: LIMSEQ_const) |
172 |
||
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lemma summable_mult: |
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fixes c :: "'a::real_normed_algebra" |
|
175 |
shows "summable f \<Longrightarrow> summable (%n. c * f n)"; |
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apply (unfold summable_def); |
177 |
apply (auto intro: sums_mult); |
|
178 |
done; |
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179 |
||
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lemma suminf_mult: |
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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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apply (rule sym); |
184 |
apply (rule sums_unique); |
|
185 |
apply (rule sums_mult); |
|
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apply (erule summable_sums); |
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187 |
done; |
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||
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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)" |
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by (auto simp add: sums_def setsum_left_distrib [symmetric] |
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193 |
intro!: LIMSEQ_mult LIMSEQ_const) |
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|
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lemma summable_mult2: |
196 |
fixes c :: "'a::real_normed_algebra" |
|
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shows "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)" |
|
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apply (unfold summable_def) |
199 |
apply (auto intro: sums_mult2) |
|
200 |
done |
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201 |
||
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lemma suminf_mult2: |
203 |
fixes c :: "'a::real_normed_algebra" |
|
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shows "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)" |
|
205 |
by (auto intro!: sums_unique sums_mult2 summable_sums) |
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|
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lemma sums_divide: |
208 |
fixes c :: "'a::real_normed_field" |
|
209 |
shows "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)" |
|
210 |
by (simp add: divide_inverse sums_mult2) |
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|
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lemma summable_divide: |
213 |
fixes c :: "'a::real_normed_field" |
|
214 |
shows "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)" |
|
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apply (unfold summable_def); |
216 |
apply (auto intro: sums_divide); |
|
217 |
done; |
|
218 |
||
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lemma suminf_divide: |
220 |
fixes c :: "'a::real_normed_field" |
|
221 |
shows "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c" |
|
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apply (rule sym); |
223 |
apply (rule sums_unique); |
|
224 |
apply (rule sums_divide); |
|
225 |
apply (erule summable_sums); |
|
226 |
done; |
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227 |
||
228 |
lemma sums_add: "[| x sums x0; y sums y0 |] ==> (%n. x n + y n) sums (x0+y0)" |
|
229 |
by (auto simp add: sums_def setsum_addf intro: LIMSEQ_add) |
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230 |
||
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lemma summable_add: "summable f ==> summable g ==> summable (%x. f x + g x)"; |
|
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apply (unfold summable_def); |
|
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apply clarify; |
|
234 |
apply (rule exI); |
|
235 |
apply (erule sums_add); |
|
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apply assumption; |
|
237 |
done; |
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238 |
||
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lemma suminf_add: |
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240 |
"[| summable f; summable g |] |
|
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==> suminf f + suminf g = (\<Sum>n. f n + g n)" |
|
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by (auto intro!: sums_add sums_unique summable_sums) |
|
243 |
||
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lemma sums_diff: "[| x sums x0; y sums y0 |] ==> (%n. x n - y n) sums (x0-y0)" |
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by (auto simp add: sums_def setsum_subtractf intro: LIMSEQ_diff) |
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|
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lemma summable_diff: "summable f ==> summable g ==> summable (%x. f x - g x)"; |
248 |
apply (unfold summable_def); |
|
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apply clarify; |
|
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apply (rule exI); |
|
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apply (erule sums_diff); |
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apply assumption; |
|
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done; |
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|
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lemma suminf_diff: |
|
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"[| summable f; summable g |] |
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==> suminf f - suminf g = (\<Sum>n. f n - g n)" |
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by (auto intro!: sums_diff sums_unique summable_sums) |
259 |
||
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lemma sums_minus: "f sums s ==> (%x. - f x) sums (- s)"; |
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by (simp add: sums_def setsum_negf LIMSEQ_minus); |
|
262 |
||
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lemma summable_minus: "summable f ==> summable (%x. - f x)"; |
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by (auto simp add: summable_def intro: sums_minus); |
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lemma suminf_minus: "summable f ==> suminf (%x. - f x) = - (suminf f)"; |
|
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apply (rule sym); |
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apply (rule sums_unique); |
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apply (rule sums_minus); |
|
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apply (erule summable_sums); |
|
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done; |
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|
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lemma sums_group: |
|
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"[|summable f; 0 < k |] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)" |
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apply (drule summable_sums) |
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apply (simp only: sums_def sumr_group) |
277 |
apply (unfold LIMSEQ_def, safe) |
|
278 |
apply (drule_tac x="r" in spec, safe) |
|
279 |
apply (rule_tac x="no" in exI, safe) |
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280 |
apply (drule_tac x="n*k" in spec) |
|
281 |
apply (erule mp) |
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282 |
apply (erule order_trans) |
|
283 |
apply simp |
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done |
285 |
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15085
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286 |
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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|
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great as any partial sum.*} |
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|
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lemma series_pos_le: |
290 |
fixes f :: "nat \<Rightarrow> real" |
|
291 |
shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 \<le> f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} \<le> suminf f" |
|
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apply (drule summable_sums) |
293 |
apply (simp add: sums_def) |
|
15539 | 294 |
apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const) |
295 |
apply (erule LIMSEQ_le, blast) |
|
20692 | 296 |
apply (rule_tac x="n" in exI, clarify) |
15539 | 297 |
apply (rule setsum_mono2) |
298 |
apply auto |
|
14416 | 299 |
done |
300 |
||
301 |
lemma series_pos_less: |
|
20692 | 302 |
fixes f :: "nat \<Rightarrow> real" |
303 |
shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} < suminf f" |
|
304 |
apply (rule_tac y="setsum f {0..<Suc n}" in order_less_le_trans) |
|
305 |
apply simp |
|
306 |
apply (erule series_pos_le) |
|
307 |
apply (simp add: order_less_imp_le) |
|
308 |
done |
|
309 |
||
310 |
lemma suminf_gt_zero: |
|
311 |
fixes f :: "nat \<Rightarrow> real" |
|
312 |
shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f" |
|
313 |
by (drule_tac n="0" in series_pos_less, simp_all) |
|
314 |
||
315 |
lemma suminf_ge_zero: |
|
316 |
fixes f :: "nat \<Rightarrow> real" |
|
317 |
shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 \<le> suminf f" |
|
318 |
by (drule_tac n="0" in series_pos_le, simp_all) |
|
319 |
||
320 |
lemma sumr_pos_lt_pair: |
|
321 |
fixes f :: "nat \<Rightarrow> real" |
|
322 |
shows "\<lbrakk>summable f; |
|
323 |
\<forall>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk> |
|
324 |
\<Longrightarrow> setsum f {0..<k} < suminf f" |
|
325 |
apply (subst suminf_split_initial_segment [where k="k"]) |
|
326 |
apply assumption |
|
327 |
apply simp |
|
328 |
apply (drule_tac k="k" in summable_ignore_initial_segment) |
|
329 |
apply (drule_tac k="Suc (Suc 0)" in sums_group, simp) |
|
330 |
apply simp |
|
331 |
apply (frule sums_unique) |
|
332 |
apply (drule sums_summable) |
|
333 |
apply simp |
|
334 |
apply (erule suminf_gt_zero) |
|
335 |
apply (simp add: add_ac) |
|
14416 | 336 |
done |
337 |
||
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
338 |
text{*Sum of a geometric progression.*} |
14416 | 339 |
|
17149
e2b19c92ef51
Lemmas on dvd, power and finite summation added or strengthened.
ballarin
parents:
16819
diff
changeset
|
340 |
lemmas sumr_geometric = geometric_sum [where 'a = real] |
14416 | 341 |
|
20692 | 342 |
lemma geometric_sums: |
343 |
fixes x :: "'a::{real_normed_field,recpower,division_by_zero}" |
|
344 |
shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) sums (1 / (1 - x))" |
|
345 |
proof - |
|
346 |
assume less_1: "norm x < 1" |
|
347 |
hence neq_1: "x \<noteq> 1" by auto |
|
348 |
hence neq_0: "x - 1 \<noteq> 0" by simp |
|
349 |
from less_1 have lim_0: "(\<lambda>n. x ^ n) ----> 0" |
|
350 |
by (rule LIMSEQ_power_zero) |
|
351 |
hence "(\<lambda>n. x ^ n / (x - 1) - 1 / (x - 1)) ----> 0 / (x - 1) - 1 / (x |
|
352 |
- 1)" |
|
353 |
using neq_0 by (intro LIMSEQ_divide LIMSEQ_diff LIMSEQ_const) |
|
354 |
hence "(\<lambda>n. (x ^ n - 1) / (x - 1)) ----> 1 / (1 - x)" |
|
355 |
by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib) |
|
356 |
thus "(\<lambda>n. x ^ n) sums (1 / (1 - x))" |
|
357 |
by (simp add: sums_def geometric_sum neq_1) |
|
358 |
qed |
|
359 |
||
360 |
lemma summable_geometric: |
|
361 |
fixes x :: "'a::{real_normed_field,recpower,division_by_zero}" |
|
362 |
shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)" |
|
363 |
by (rule geometric_sums [THEN sums_summable]) |
|
14416 | 364 |
|
15085
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paulson
parents:
15053
diff
changeset
|
365 |
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*} |
5693a977a767
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paulson
parents:
15053
diff
changeset
|
366 |
|
15539 | 367 |
lemma summable_convergent_sumr_iff: |
368 |
"summable f = convergent (%n. setsum f {0..<n})" |
|
14416 | 369 |
by (simp add: summable_def sums_def convergent_def) |
370 |
||
20689 | 371 |
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0" |
372 |
apply (drule summable_convergent_sumr_iff [THEN iffD1]) |
|
20692 | 373 |
apply (drule convergent_Cauchy) |
20689 | 374 |
apply (simp only: Cauchy_def LIMSEQ_def, safe) |
375 |
apply (drule_tac x="r" in spec, safe) |
|
376 |
apply (rule_tac x="M" in exI, safe) |
|
377 |
apply (drule_tac x="Suc n" in spec, simp) |
|
378 |
apply (drule_tac x="n" in spec, simp) |
|
379 |
done |
|
380 |
||
14416 | 381 |
lemma summable_Cauchy: |
20848 | 382 |
"summable (f::nat \<Rightarrow> 'a::banach) = |
383 |
(\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. norm (setsum f {m..<n}) < e)" |
|
384 |
apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def, safe) |
|
20410 | 385 |
apply (drule spec, drule (1) mp) |
386 |
apply (erule exE, rule_tac x="M" in exI, clarify) |
|
387 |
apply (rule_tac x="m" and y="n" in linorder_le_cases) |
|
388 |
apply (frule (1) order_trans) |
|
389 |
apply (drule_tac x="n" in spec, drule (1) mp) |
|
390 |
apply (drule_tac x="m" in spec, drule (1) mp) |
|
391 |
apply (simp add: setsum_diff [symmetric]) |
|
392 |
apply simp |
|
393 |
apply (drule spec, drule (1) mp) |
|
394 |
apply (erule exE, rule_tac x="N" in exI, clarify) |
|
395 |
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
|
396 |
apply (subst norm_minus_commute) |
20410 | 397 |
apply (simp add: setsum_diff [symmetric]) |
398 |
apply (simp add: setsum_diff [symmetric]) |
|
14416 | 399 |
done |
400 |
||
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
401 |
text{*Comparison test*} |
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
402 |
|
20692 | 403 |
lemma norm_setsum: |
404 |
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" |
|
405 |
shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))" |
|
406 |
apply (case_tac "finite A") |
|
407 |
apply (erule finite_induct) |
|
408 |
apply simp |
|
409 |
apply simp |
|
410 |
apply (erule order_trans [OF norm_triangle_ineq add_left_mono]) |
|
411 |
apply simp |
|
412 |
done |
|
413 |
||
14416 | 414 |
lemma summable_comparison_test: |
20848 | 415 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
416 |
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f" |
|
20692 | 417 |
apply (simp add: summable_Cauchy, safe) |
418 |
apply (drule_tac x="e" in spec, safe) |
|
419 |
apply (rule_tac x = "N + Na" in exI, safe) |
|
14416 | 420 |
apply (rotate_tac 2) |
421 |
apply (drule_tac x = m in spec) |
|
422 |
apply (auto, rotate_tac 2, drule_tac x = n in spec) |
|
20848 | 423 |
apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans) |
424 |
apply (rule norm_setsum) |
|
15539 | 425 |
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans) |
426 |
apply (auto intro: setsum_mono simp add: abs_interval_iff) |
|
14416 | 427 |
done |
428 |
||
20848 | 429 |
lemma summable_norm_comparison_test: |
430 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
|
431 |
shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> |
|
432 |
\<Longrightarrow> summable (\<lambda>n. norm (f n))" |
|
433 |
apply (rule summable_comparison_test) |
|
434 |
apply (auto) |
|
435 |
done |
|
436 |
||
14416 | 437 |
lemma summable_rabs_comparison_test: |
20692 | 438 |
fixes f :: "nat \<Rightarrow> real" |
439 |
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 | 440 |
apply (rule summable_comparison_test) |
15543 | 441 |
apply (auto) |
14416 | 442 |
done |
443 |
||
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
444 |
text{*Limit comparison property for series (c.f. jrh)*} |
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
445 |
|
14416 | 446 |
lemma summable_le: |
20692 | 447 |
fixes f g :: "nat \<Rightarrow> real" |
448 |
shows "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g" |
|
14416 | 449 |
apply (drule summable_sums)+ |
20692 | 450 |
apply (simp only: sums_def, erule (1) LIMSEQ_le) |
14416 | 451 |
apply (rule exI) |
15539 | 452 |
apply (auto intro!: setsum_mono) |
14416 | 453 |
done |
454 |
||
455 |
lemma summable_le2: |
|
20692 | 456 |
fixes f g :: "nat \<Rightarrow> real" |
457 |
shows "\<lbrakk>\<forall>n. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f \<and> suminf f \<le> suminf g" |
|
20848 | 458 |
apply (subgoal_tac "summable f") |
459 |
apply (auto intro!: summable_le) |
|
14416 | 460 |
apply (simp add: abs_le_interval_iff) |
20848 | 461 |
apply (rule_tac g="g" in summable_comparison_test, simp_all) |
14416 | 462 |
done |
463 |
||
19106
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
464 |
(* 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
|
465 |
lemma suminf_0_le: |
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
466 |
fixes f::"nat\<Rightarrow>real" |
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
467 |
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
|
468 |
shows "0 \<le> suminf f" |
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
469 |
proof - |
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
470 |
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
|
471 |
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
|
472 |
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
|
473 |
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
|
474 |
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
|
475 |
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
|
476 |
qed |
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
477 |
|
6e6b5b1fdc06
* added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents:
17149
diff
changeset
|
478 |
|
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
479 |
text{*Absolute convergence imples normal convergence*} |
20848 | 480 |
lemma summable_norm_cancel: |
481 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
|
482 |
shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f" |
|
20692 | 483 |
apply (simp only: summable_Cauchy, safe) |
484 |
apply (drule_tac x="e" in spec, safe) |
|
485 |
apply (rule_tac x="N" in exI, safe) |
|
486 |
apply (drule_tac x="m" in spec, safe) |
|
20848 | 487 |
apply (rule order_le_less_trans [OF norm_setsum]) |
488 |
apply (rule order_le_less_trans [OF abs_ge_self]) |
|
20692 | 489 |
apply simp |
14416 | 490 |
done |
491 |
||
20848 | 492 |
lemma summable_rabs_cancel: |
493 |
fixes f :: "nat \<Rightarrow> real" |
|
494 |
shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f" |
|
495 |
by (rule summable_norm_cancel, simp) |
|
496 |
||
15085
5693a977a767
removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents:
15053
diff
changeset
|
497 |
text{*Absolute convergence of series*} |
20848 | 498 |
lemma summable_norm: |
499 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
|
500 |
shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))" |
|
501 |
by (auto intro: LIMSEQ_le LIMSEQ_norm summable_norm_cancel |
|
502 |
summable_sumr_LIMSEQ_suminf norm_setsum) |
|
503 |
||
14416 | 504 |
lemma summable_rabs: |
20692 | 505 |
fixes f :: "nat \<Rightarrow> real" |
506 |
shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)" |
|
20848 | 507 |
by (fold real_norm_def, rule summable_norm) |
14416 | 508 |
|
509 |
subsection{* The Ratio Test*} |
|
510 |
||
20848 | 511 |
lemma norm_ratiotest_lemma: |
512 |
fixes x y :: "'a::normed" |
|
513 |
shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0" |
|
514 |
apply (subgoal_tac "norm x \<le> 0", simp) |
|
515 |
apply (erule order_trans) |
|
516 |
apply (simp add: mult_le_0_iff) |
|
517 |
done |
|
518 |
||
14416 | 519 |
lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)" |
20848 | 520 |
by (erule norm_ratiotest_lemma, simp) |
14416 | 521 |
|
522 |
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)" |
|
523 |
apply (drule le_imp_less_or_eq) |
|
524 |
apply (auto dest: less_imp_Suc_add) |
|
525 |
done |
|
526 |
||
527 |
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)" |
|
528 |
by (auto simp add: le_Suc_ex) |
|
529 |
||
530 |
(*All this trouble just to get 0<c *) |
|
531 |
lemma ratio_test_lemma2: |
|
20848 | 532 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
533 |
shows "\<lbrakk>\<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> 0 < c \<or> summable f" |
|
14416 | 534 |
apply (simp (no_asm) add: linorder_not_le [symmetric]) |
535 |
apply (simp add: summable_Cauchy) |
|
15543 | 536 |
apply (safe, subgoal_tac "\<forall>n. N < n --> f (n) = 0") |
537 |
prefer 2 |
|
538 |
apply clarify |
|
539 |
apply(erule_tac x = "n - 1" in allE) |
|
540 |
apply (simp add:diff_Suc split:nat.splits) |
|
20848 | 541 |
apply (blast intro: norm_ratiotest_lemma) |
14416 | 542 |
apply (rule_tac x = "Suc N" in exI, clarify) |
15543 | 543 |
apply(simp cong:setsum_ivl_cong) |
14416 | 544 |
done |
545 |
||
546 |
lemma ratio_test: |
|
20848 | 547 |
fixes f :: "nat \<Rightarrow> 'a::banach" |
548 |
shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f" |
|
14416 | 549 |
apply (frule ratio_test_lemma2, auto) |
20848 | 550 |
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
|
551 |
in summable_comparison_test) |
14416 | 552 |
apply (rule_tac x = N in exI, safe) |
553 |
apply (drule le_Suc_ex_iff [THEN iffD1]) |
|
554 |
apply (auto simp add: power_add realpow_not_zero) |
|
15539 | 555 |
apply (induct_tac "na", auto) |
20848 | 556 |
apply (rule_tac y = "c * norm (f (N + n))" in order_trans) |
14416 | 557 |
apply (auto intro: mult_right_mono simp add: summable_def) |
558 |
apply (simp add: mult_ac) |
|
20848 | 559 |
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
|
560 |
apply (rule sums_divide) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
561 |
apply (rule sums_mult) |
ec91a90c604e
simplification tweaks for better arithmetic reasoning
paulson
parents:
15229
diff
changeset
|
562 |
apply (auto intro!: geometric_sums) |
14416 | 563 |
done |
564 |
||
565 |
end |