src/HOL/Hyperreal/Series.thy
author wenzelm
Fri Nov 17 02:20:03 2006 +0100 (2006-11-17)
changeset 21404 eb85850d3eb7
parent 21141 f0b5e6254a1f
child 22719 c51667189bd3
permissions -rw-r--r--
more robust syntax for definition/abbreviation/notation;
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(*  Title       : Series.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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Converted to Isar and polished by lcp
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Converted to setsum and polished yet more by TNN
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Additional contributions by Jeremy Avigad
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*) 
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header{*Finite Summation and Infinite Series*}
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theory Series
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imports SEQ
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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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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? *)
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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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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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(*-------------------
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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 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
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lemma sums_zero: "(%n. 0) sums 0";
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  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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lemma summable_zero: "summable (%n. 0)";
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  apply (rule sums_summable);
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  apply (rule sums_zero);
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done;
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lemma suminf_zero: "suminf (%n. 0) = 0";
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  apply (rule sym);
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  apply (rule sums_unique);
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  apply (rule sums_zero);
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done;
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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)
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lemma summable_mult:
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  fixes c :: "'a::real_normed_algebra"
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  shows "summable f \<Longrightarrow> summable (%n. c * f n)";
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  apply (unfold summable_def);
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  apply (auto intro: sums_mult);
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done;
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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);
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  apply (rule sums_unique);
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  apply (rule sums_mult);
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  apply (erule summable_sums);
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done;
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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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         intro!: LIMSEQ_mult LIMSEQ_const)
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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)"
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  apply (unfold summable_def)
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  apply (auto intro: sums_mult2)
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done
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lemma suminf_mult2:
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  fixes c :: "'a::real_normed_algebra"
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  shows "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
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by (auto intro!: sums_unique sums_mult2 summable_sums)
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lemma sums_divide:
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  fixes c :: "'a::real_normed_field"
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  shows "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
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by (simp add: divide_inverse sums_mult2)
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lemma summable_divide:
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  fixes c :: "'a::real_normed_field"
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  shows "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
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  apply (unfold summable_def);
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  apply (auto intro: sums_divide);
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done;
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lemma suminf_divide:
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  fixes c :: "'a::real_normed_field"
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  shows "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
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  apply (rule sym);
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  apply (rule sums_unique);
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  apply (rule sums_divide);
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  apply (erule summable_sums);
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done;
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lemma sums_add: "[| 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_addf intro: LIMSEQ_add)
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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;
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  apply (rule exI);
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  apply (erule sums_add);
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  apply assumption;
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done;
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lemma suminf_add:
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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_add sums_unique summable_sums)
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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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lemma summable_diff: "summable f ==> summable g ==> summable (%x. f x - g x)";
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  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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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)
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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);
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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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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)
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apply (unfold LIMSEQ_def, safe)
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apply (drule_tac x="r" in spec, safe)
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apply (rule_tac x="no" in exI, safe)
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apply (drule_tac x="n*k" in spec)
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apply (erule mp)
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apply (erule order_trans)
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apply simp
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done
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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.*}
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lemma series_pos_le:
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  fixes f :: "nat \<Rightarrow> real"
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  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)
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apply (simp add: sums_def)
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apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const)
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apply (erule LIMSEQ_le, blast)
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apply (rule_tac x="n" in exI, clarify)
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apply (rule setsum_mono2)
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apply auto
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done
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lemma series_pos_less:
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  fixes f :: "nat \<Rightarrow> real"
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  shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} < suminf f"
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apply (rule_tac y="setsum f {0..<Suc n}" in order_less_le_trans)
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apply simp
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apply (erule series_pos_le)
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apply (simp add: order_less_imp_le)
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done
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lemma suminf_gt_zero:
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  fixes f :: "nat \<Rightarrow> real"
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  shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f"
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by (drule_tac n="0" in series_pos_less, simp_all)
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lemma suminf_ge_zero:
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  fixes f :: "nat \<Rightarrow> real"
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  shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 \<le> suminf f"
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by (drule_tac n="0" in series_pos_le, simp_all)
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lemma sumr_pos_lt_pair:
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  fixes f :: "nat \<Rightarrow> real"
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  shows "\<lbrakk>summable f;
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        \<forall>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
huffman@20692
   324
      \<Longrightarrow> setsum f {0..<k} < suminf f"
huffman@20692
   325
apply (subst suminf_split_initial_segment [where k="k"])
huffman@20692
   326
apply assumption
huffman@20692
   327
apply simp
huffman@20692
   328
apply (drule_tac k="k" in summable_ignore_initial_segment)
huffman@20692
   329
apply (drule_tac k="Suc (Suc 0)" in sums_group, simp)
huffman@20692
   330
apply simp
huffman@20692
   331
apply (frule sums_unique)
huffman@20692
   332
apply (drule sums_summable)
huffman@20692
   333
apply simp
huffman@20692
   334
apply (erule suminf_gt_zero)
huffman@20692
   335
apply (simp add: add_ac)
paulson@14416
   336
done
paulson@14416
   337
paulson@15085
   338
text{*Sum of a geometric progression.*}
paulson@14416
   339
ballarin@17149
   340
lemmas sumr_geometric = geometric_sum [where 'a = real]
paulson@14416
   341
huffman@20692
   342
lemma geometric_sums:
huffman@20692
   343
  fixes x :: "'a::{real_normed_field,recpower,division_by_zero}"
huffman@20692
   344
  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) sums (1 / (1 - x))"
huffman@20692
   345
proof -
huffman@20692
   346
  assume less_1: "norm x < 1"
huffman@20692
   347
  hence neq_1: "x \<noteq> 1" by auto
huffman@20692
   348
  hence neq_0: "x - 1 \<noteq> 0" by simp
huffman@20692
   349
  from less_1 have lim_0: "(\<lambda>n. x ^ n) ----> 0"
huffman@20692
   350
    by (rule LIMSEQ_power_zero)
huffman@20692
   351
  hence "(\<lambda>n. x ^ n / (x - 1) - 1 / (x - 1)) ----> 0 / (x - 1) - 1 / (x
huffman@20692
   352
- 1)"
huffman@20692
   353
    using neq_0 by (intro LIMSEQ_divide LIMSEQ_diff LIMSEQ_const)
huffman@20692
   354
  hence "(\<lambda>n. (x ^ n - 1) / (x - 1)) ----> 1 / (1 - x)"
huffman@20692
   355
    by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
huffman@20692
   356
  thus "(\<lambda>n. x ^ n) sums (1 / (1 - x))"
huffman@20692
   357
    by (simp add: sums_def geometric_sum neq_1)
huffman@20692
   358
qed
huffman@20692
   359
huffman@20692
   360
lemma summable_geometric:
huffman@20692
   361
  fixes x :: "'a::{real_normed_field,recpower,division_by_zero}"
huffman@20692
   362
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
huffman@20692
   363
by (rule geometric_sums [THEN sums_summable])
paulson@14416
   364
paulson@15085
   365
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
paulson@15085
   366
nipkow@15539
   367
lemma summable_convergent_sumr_iff:
nipkow@15539
   368
 "summable f = convergent (%n. setsum f {0..<n})"
paulson@14416
   369
by (simp add: summable_def sums_def convergent_def)
paulson@14416
   370
huffman@20689
   371
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
huffman@20689
   372
apply (drule summable_convergent_sumr_iff [THEN iffD1])
huffman@20692
   373
apply (drule convergent_Cauchy)
huffman@20689
   374
apply (simp only: Cauchy_def LIMSEQ_def, safe)
huffman@20689
   375
apply (drule_tac x="r" in spec, safe)
huffman@20689
   376
apply (rule_tac x="M" in exI, safe)
huffman@20689
   377
apply (drule_tac x="Suc n" in spec, simp)
huffman@20689
   378
apply (drule_tac x="n" in spec, simp)
huffman@20689
   379
done
huffman@20689
   380
paulson@14416
   381
lemma summable_Cauchy:
huffman@20848
   382
     "summable (f::nat \<Rightarrow> 'a::banach) =  
huffman@20848
   383
      (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. norm (setsum f {m..<n}) < e)"
huffman@20848
   384
apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def, safe)
huffman@20410
   385
apply (drule spec, drule (1) mp)
huffman@20410
   386
apply (erule exE, rule_tac x="M" in exI, clarify)
huffman@20410
   387
apply (rule_tac x="m" and y="n" in linorder_le_cases)
huffman@20410
   388
apply (frule (1) order_trans)
huffman@20410
   389
apply (drule_tac x="n" in spec, drule (1) mp)
huffman@20410
   390
apply (drule_tac x="m" in spec, drule (1) mp)
huffman@20410
   391
apply (simp add: setsum_diff [symmetric])
huffman@20410
   392
apply simp
huffman@20410
   393
apply (drule spec, drule (1) mp)
huffman@20410
   394
apply (erule exE, rule_tac x="N" in exI, clarify)
huffman@20410
   395
apply (rule_tac x="m" and y="n" in linorder_le_cases)
huffman@20552
   396
apply (subst norm_minus_commute)
huffman@20410
   397
apply (simp add: setsum_diff [symmetric])
huffman@20410
   398
apply (simp add: setsum_diff [symmetric])
paulson@14416
   399
done
paulson@14416
   400
paulson@15085
   401
text{*Comparison test*}
paulson@15085
   402
huffman@20692
   403
lemma norm_setsum:
huffman@20692
   404
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
huffman@20692
   405
  shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
huffman@20692
   406
apply (case_tac "finite A")
huffman@20692
   407
apply (erule finite_induct)
huffman@20692
   408
apply simp
huffman@20692
   409
apply simp
huffman@20692
   410
apply (erule order_trans [OF norm_triangle_ineq add_left_mono])
huffman@20692
   411
apply simp
huffman@20692
   412
done
huffman@20692
   413
paulson@14416
   414
lemma summable_comparison_test:
huffman@20848
   415
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   416
  shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f"
huffman@20692
   417
apply (simp add: summable_Cauchy, safe)
huffman@20692
   418
apply (drule_tac x="e" in spec, safe)
huffman@20692
   419
apply (rule_tac x = "N + Na" in exI, safe)
paulson@14416
   420
apply (rotate_tac 2)
paulson@14416
   421
apply (drule_tac x = m in spec)
paulson@14416
   422
apply (auto, rotate_tac 2, drule_tac x = n in spec)
huffman@20848
   423
apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
huffman@20848
   424
apply (rule norm_setsum)
nipkow@15539
   425
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
nipkow@15539
   426
apply (auto intro: setsum_mono simp add: abs_interval_iff)
paulson@14416
   427
done
paulson@14416
   428
huffman@20848
   429
lemma summable_norm_comparison_test:
huffman@20848
   430
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   431
  shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk>
huffman@20848
   432
         \<Longrightarrow> summable (\<lambda>n. norm (f n))"
huffman@20848
   433
apply (rule summable_comparison_test)
huffman@20848
   434
apply (auto)
huffman@20848
   435
done
huffman@20848
   436
paulson@14416
   437
lemma summable_rabs_comparison_test:
huffman@20692
   438
  fixes f :: "nat \<Rightarrow> real"
huffman@20692
   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>)"
paulson@14416
   440
apply (rule summable_comparison_test)
nipkow@15543
   441
apply (auto)
paulson@14416
   442
done
paulson@14416
   443
paulson@15085
   444
text{*Limit comparison property for series (c.f. jrh)*}
paulson@15085
   445
paulson@14416
   446
lemma summable_le:
huffman@20692
   447
  fixes f g :: "nat \<Rightarrow> real"
huffman@20692
   448
  shows "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
paulson@14416
   449
apply (drule summable_sums)+
huffman@20692
   450
apply (simp only: sums_def, erule (1) LIMSEQ_le)
paulson@14416
   451
apply (rule exI)
nipkow@15539
   452
apply (auto intro!: setsum_mono)
paulson@14416
   453
done
paulson@14416
   454
paulson@14416
   455
lemma summable_le2:
huffman@20692
   456
  fixes f g :: "nat \<Rightarrow> real"
huffman@20692
   457
  shows "\<lbrakk>\<forall>n. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f \<and> suminf f \<le> suminf g"
huffman@20848
   458
apply (subgoal_tac "summable f")
huffman@20848
   459
apply (auto intro!: summable_le)
paulson@14416
   460
apply (simp add: abs_le_interval_iff)
huffman@20848
   461
apply (rule_tac g="g" in summable_comparison_test, simp_all)
paulson@14416
   462
done
paulson@14416
   463
kleing@19106
   464
(* specialisation for the common 0 case *)
kleing@19106
   465
lemma suminf_0_le:
kleing@19106
   466
  fixes f::"nat\<Rightarrow>real"
kleing@19106
   467
  assumes gt0: "\<forall>n. 0 \<le> f n" and sm: "summable f"
kleing@19106
   468
  shows "0 \<le> suminf f"
kleing@19106
   469
proof -
kleing@19106
   470
  let ?g = "(\<lambda>n. (0::real))"
kleing@19106
   471
  from gt0 have "\<forall>n. ?g n \<le> f n" by simp
kleing@19106
   472
  moreover have "summable ?g" by (rule summable_zero)
kleing@19106
   473
  moreover from sm have "summable f" .
kleing@19106
   474
  ultimately have "suminf ?g \<le> suminf f" by (rule summable_le)
kleing@19106
   475
  then show "0 \<le> suminf f" by (simp add: suminf_zero)
kleing@19106
   476
qed 
kleing@19106
   477
kleing@19106
   478
paulson@15085
   479
text{*Absolute convergence imples normal convergence*}
huffman@20848
   480
lemma summable_norm_cancel:
huffman@20848
   481
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   482
  shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
huffman@20692
   483
apply (simp only: summable_Cauchy, safe)
huffman@20692
   484
apply (drule_tac x="e" in spec, safe)
huffman@20692
   485
apply (rule_tac x="N" in exI, safe)
huffman@20692
   486
apply (drule_tac x="m" in spec, safe)
huffman@20848
   487
apply (rule order_le_less_trans [OF norm_setsum])
huffman@20848
   488
apply (rule order_le_less_trans [OF abs_ge_self])
huffman@20692
   489
apply simp
paulson@14416
   490
done
paulson@14416
   491
huffman@20848
   492
lemma summable_rabs_cancel:
huffman@20848
   493
  fixes f :: "nat \<Rightarrow> real"
huffman@20848
   494
  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f"
huffman@20848
   495
by (rule summable_norm_cancel, simp)
huffman@20848
   496
paulson@15085
   497
text{*Absolute convergence of series*}
huffman@20848
   498
lemma summable_norm:
huffman@20848
   499
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   500
  shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
huffman@20848
   501
by (auto intro: LIMSEQ_le LIMSEQ_norm summable_norm_cancel
huffman@20848
   502
                summable_sumr_LIMSEQ_suminf norm_setsum)
huffman@20848
   503
paulson@14416
   504
lemma summable_rabs:
huffman@20692
   505
  fixes f :: "nat \<Rightarrow> real"
huffman@20692
   506
  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
huffman@20848
   507
by (fold real_norm_def, rule summable_norm)
paulson@14416
   508
paulson@14416
   509
subsection{* The Ratio Test*}
paulson@14416
   510
huffman@20848
   511
lemma norm_ratiotest_lemma:
huffman@20848
   512
  fixes x y :: "'a::normed"
huffman@20848
   513
  shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0"
huffman@20848
   514
apply (subgoal_tac "norm x \<le> 0", simp)
huffman@20848
   515
apply (erule order_trans)
huffman@20848
   516
apply (simp add: mult_le_0_iff)
huffman@20848
   517
done
huffman@20848
   518
paulson@14416
   519
lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
huffman@20848
   520
by (erule norm_ratiotest_lemma, simp)
paulson@14416
   521
paulson@14416
   522
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"
paulson@14416
   523
apply (drule le_imp_less_or_eq)
paulson@14416
   524
apply (auto dest: less_imp_Suc_add)
paulson@14416
   525
done
paulson@14416
   526
paulson@14416
   527
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"
paulson@14416
   528
by (auto simp add: le_Suc_ex)
paulson@14416
   529
paulson@14416
   530
(*All this trouble just to get 0<c *)
paulson@14416
   531
lemma ratio_test_lemma2:
huffman@20848
   532
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   533
  shows "\<lbrakk>\<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> 0 < c \<or> summable f"
paulson@14416
   534
apply (simp (no_asm) add: linorder_not_le [symmetric])
paulson@14416
   535
apply (simp add: summable_Cauchy)
nipkow@15543
   536
apply (safe, subgoal_tac "\<forall>n. N < n --> f (n) = 0")
nipkow@15543
   537
 prefer 2
nipkow@15543
   538
 apply clarify
nipkow@15543
   539
 apply(erule_tac x = "n - 1" in allE)
nipkow@15543
   540
 apply (simp add:diff_Suc split:nat.splits)
huffman@20848
   541
 apply (blast intro: norm_ratiotest_lemma)
paulson@14416
   542
apply (rule_tac x = "Suc N" in exI, clarify)
nipkow@15543
   543
apply(simp cong:setsum_ivl_cong)
paulson@14416
   544
done
paulson@14416
   545
paulson@14416
   546
lemma ratio_test:
huffman@20848
   547
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   548
  shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f"
paulson@14416
   549
apply (frule ratio_test_lemma2, auto)
huffman@20848
   550
apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n" 
paulson@15234
   551
       in summable_comparison_test)
paulson@14416
   552
apply (rule_tac x = N in exI, safe)
paulson@14416
   553
apply (drule le_Suc_ex_iff [THEN iffD1])
paulson@14416
   554
apply (auto simp add: power_add realpow_not_zero)
nipkow@15539
   555
apply (induct_tac "na", auto)
huffman@20848
   556
apply (rule_tac y = "c * norm (f (N + n))" in order_trans)
paulson@14416
   557
apply (auto intro: mult_right_mono simp add: summable_def)
paulson@14416
   558
apply (simp add: mult_ac)
huffman@20848
   559
apply (rule_tac x = "norm (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
paulson@15234
   560
apply (rule sums_divide) 
paulson@15234
   561
apply (rule sums_mult) 
paulson@15234
   562
apply (auto intro!: geometric_sums)
paulson@14416
   563
done
paulson@14416
   564
paulson@14416
   565
end