src/HOL/Series.thy
author paulson
Mon Oct 05 17:27:46 2009 +0100 (2009-10-05)
changeset 32877 6f09346c7c08
parent 32707 836ec9d0a0c8
child 33271 7be66dee1a5a
permissions -rw-r--r--
New lemmas connected with the reals and infinite series
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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 [trans]: "f=g ==> g sums z ==> f sums z"
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  by simp
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lemma sumr_diff_mult_const:
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 "setsum f {0..<n} - (real n*r) = setsum (%i. f i - r) {0..<n::nat}"
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by (simp add: diff_minus setsum_addf real_of_nat_def)
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lemma real_setsum_nat_ivl_bounded:
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     "(!!p. p < n \<Longrightarrow> f(p) \<le> K)
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      \<Longrightarrow> setsum f {0..<n::nat} \<le> real n * K"
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using setsum_bounded[where A = "{0..<n}"]
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by (auto simp:real_of_nat_def)
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(* Generalize from real to some algebraic structure? *)
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lemma sumr_minus_one_realpow_zero [simp]:
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  "(\<Sum>i=0..<2*n. (-1) ^ Suc i) = (0::real)"
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by (induct "n", auto)
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(* FIXME this is an awful lemma! *)
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lemma sumr_one_lb_realpow_zero [simp]:
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  "(\<Sum>n=Suc 0..<n. f(n) * (0::real) ^ n) = 0"
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by (rule setsum_0', simp)
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lemma sumr_group:
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     "(\<Sum>m=0..<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {0 ..< n * k}"
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apply (subgoal_tac "k = 0 | 0 < k", auto)
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apply (induct "n")
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apply (simp_all add: setsum_add_nat_ivl add_commute)
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done
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lemma sumr_offset3:
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  "setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)) + setsum f {0..<k}"
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apply (subst setsum_shift_bounds_nat_ivl [symmetric])
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apply (simp add: setsum_add_nat_ivl add_commute)
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done
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lemma sumr_offset:
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  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
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  shows "(\<Sum>m=0..<n. f(m+k)) = setsum f {0..<n+k} - setsum f {0..<k}"
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by (simp add: sumr_offset3)
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lemma sumr_offset2:
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 "\<forall>f. (\<Sum>m=0..<n::nat. f(m+k)::real) = setsum f {0..<n+k} - setsum f {0..<k}"
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by (simp add: sumr_offset)
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lemma sumr_offset4:
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  "\<forall>n f. setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)::real) + setsum f {0..<k}"
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by (clarify, rule sumr_offset3)
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(*
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lemma sumr_from_1_from_0: "0 < n ==>
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      (\<Sum>n=Suc 0 ..< Suc n. if even(n) then 0 else
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             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n =
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      (\<Sum>n=0..<Suc n. if even(n) then 0 else
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             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n"
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by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto)
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*)
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subsection{* Infinite Sums, by the Properties of Limits*}
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(*----------------------
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   suminf is the sum   
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 ---------------------*)
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lemma sums_summable: "f sums l ==> summable f"
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by (simp add: sums_def summable_def, blast)
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lemma summable_sums: "summable f ==> f sums (suminf f)"
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apply (simp add: summable_def suminf_def sums_def)
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apply (blast intro: theI LIMSEQ_unique)
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done
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lemma summable_sumr_LIMSEQ_suminf: 
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     "summable f ==> (%n. setsum f {0..<n}) ----> (suminf f)"
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by (rule summable_sums [unfolded sums_def])
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lemma suminf_eq_lim: "suminf f = lim (%n. setsum f {0..<n})"
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  by (simp add: suminf_def sums_def lim_def) 
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(*-------------------
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    sum is unique                    
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 ------------------*)
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lemma sums_unique: "f sums s ==> (s = suminf f)"
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apply (frule sums_summable [THEN summable_sums])
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apply (auto intro!: LIMSEQ_unique simp add: sums_def)
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done
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lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
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  by (metis summable_sums sums_summable sums_unique)
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lemma sums_split_initial_segment: "f sums s ==> 
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  (%n. f(n + k)) sums (s - (SUM i = 0..< k. f i))"
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  apply (unfold sums_def);
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  apply (simp add: sumr_offset); 
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  apply (rule LIMSEQ_diff_const)
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  apply (rule LIMSEQ_ignore_initial_segment)
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  apply assumption
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done
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lemma summable_ignore_initial_segment: "summable f ==> 
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    summable (%n. f(n + k))"
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  apply (unfold summable_def)
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  apply (auto intro: sums_split_initial_segment)
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done
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lemma suminf_minus_initial_segment: "summable f ==>
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    suminf f = s ==> suminf (%n. f(n + k)) = s - (SUM i = 0..< k. f i)"
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  apply (frule summable_ignore_initial_segment)
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  apply (rule sums_unique [THEN sym])
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  apply (frule summable_sums)
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  apply (rule sums_split_initial_segment)
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  apply auto
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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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    by auto
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qed
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lemma sums_Suc: assumes sumSuc: "(\<lambda> n. f (Suc n)) sums l" shows "f sums (l + f 0)"
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proof -
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  from sumSuc[unfolded sums_def]
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  have "(\<lambda>i. \<Sum>n = Suc 0..<Suc i. f n) ----> l" unfolding setsum_reindex[OF inj_Suc] image_Suc_atLeastLessThan[symmetric] comp_def .
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  from LIMSEQ_add_const[OF this, where b="f 0"] 
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  have "(\<lambda>i. \<Sum>n = 0..<Suc i. f n) ----> l + f 0" unfolding add_commute setsum_head_upt_Suc[OF zero_less_Suc] .
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  thus ?thesis unfolding sums_def by (rule LIMSEQ_imp_Suc)
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qed
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lemma series_zero: 
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     "(\<forall>m. n \<le> m --> f(m) = 0) ==> f sums (setsum f {0..<n})"
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apply (simp add: sums_def LIMSEQ_iff diff_minus[symmetric], safe)
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apply (rule_tac x = n in exI)
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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: "(\<lambda>n. 0) sums 0"
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unfolding sums_def by (simp add: LIMSEQ_const)
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lemma summable_zero: "summable (\<lambda>n. 0)"
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by (rule sums_zero [THEN sums_summable])
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lemma suminf_zero: "suminf (\<lambda>n. 0) = 0"
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by (rule sums_zero [THEN sums_unique, symmetric])
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lemma (in bounded_linear) sums:
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  "(\<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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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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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)
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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:
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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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by (rule mult_right.summable)
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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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by (rule mult_right.suminf [symmetric])
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lemma sums_mult2:
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  fixes c :: "'a::real_normed_algebra"
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  shows "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
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by (rule mult_left.sums)
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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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by (rule mult_left.summable)
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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 (rule mult_left.suminf)
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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 (rule divide.sums)
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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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by (rule divide.summable)
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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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by (rule divide.suminf [symmetric])
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lemma sums_add: "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) sums (a + b)"
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unfolding sums_def by (simp add: setsum_addf LIMSEQ_add)
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lemma summable_add: "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n + Y n)"
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unfolding summable_def by (auto intro: sums_add)
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lemma suminf_add:
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  "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X + suminf Y = (\<Sum>n. X n + Y n)"
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by (intro sums_unique sums_add summable_sums)
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lemma sums_diff: "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) sums (a - b)"
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unfolding sums_def by (simp add: setsum_subtractf LIMSEQ_diff)
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lemma summable_diff: "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n - Y n)"
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unfolding summable_def by (auto intro: sums_diff)
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lemma suminf_diff:
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  "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X - suminf Y = (\<Sum>n. X n - Y n)"
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by (intro sums_unique sums_diff summable_sums)
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lemma sums_minus: "X sums a ==> (\<lambda>n. - X n) sums (- a)"
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unfolding sums_def by (simp add: setsum_negf LIMSEQ_minus)
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lemma summable_minus: "summable X \<Longrightarrow> summable (\<lambda>n. - X n)"
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unfolding summable_def by (auto intro: sums_minus)
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lemma suminf_minus: "summable X \<Longrightarrow> (\<Sum>n. - X n) = - (\<Sum>n. X n)"
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by (intro sums_unique [symmetric] sums_minus summable_sums)
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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_iff, 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)
huffman@20692
   304
apply simp
huffman@20692
   305
apply (erule series_pos_le)
huffman@20692
   306
apply (simp add: order_less_imp_le)
huffman@20692
   307
done
huffman@20692
   308
huffman@20692
   309
lemma suminf_gt_zero:
huffman@20692
   310
  fixes f :: "nat \<Rightarrow> real"
huffman@20692
   311
  shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f"
huffman@20692
   312
by (drule_tac n="0" in series_pos_less, simp_all)
huffman@20692
   313
huffman@20692
   314
lemma suminf_ge_zero:
huffman@20692
   315
  fixes f :: "nat \<Rightarrow> real"
huffman@20692
   316
  shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 \<le> suminf f"
huffman@20692
   317
by (drule_tac n="0" in series_pos_le, simp_all)
huffman@20692
   318
huffman@20692
   319
lemma sumr_pos_lt_pair:
huffman@20692
   320
  fixes f :: "nat \<Rightarrow> real"
huffman@20692
   321
  shows "\<lbrakk>summable f;
huffman@20692
   322
        \<forall>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
huffman@20692
   323
      \<Longrightarrow> setsum f {0..<k} < suminf f"
huffman@30082
   324
unfolding One_nat_def
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:
haftmann@31017
   343
  fixes x :: "'a::{real_normed_field}"
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@22719
   351
  hence "(\<lambda>n. x ^ n / (x - 1) - 1 / (x - 1)) ----> 0 / (x - 1) - 1 / (x - 1)"
huffman@20692
   352
    using neq_0 by (intro LIMSEQ_divide LIMSEQ_diff LIMSEQ_const)
huffman@20692
   353
  hence "(\<lambda>n. (x ^ n - 1) / (x - 1)) ----> 1 / (1 - x)"
huffman@20692
   354
    by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
huffman@20692
   355
  thus "(\<lambda>n. x ^ n) sums (1 / (1 - x))"
huffman@20692
   356
    by (simp add: sums_def geometric_sum neq_1)
huffman@20692
   357
qed
huffman@20692
   358
huffman@20692
   359
lemma summable_geometric:
haftmann@31017
   360
  fixes x :: "'a::{real_normed_field}"
huffman@20692
   361
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
huffman@20692
   362
by (rule geometric_sums [THEN sums_summable])
paulson@14416
   363
paulson@15085
   364
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
paulson@15085
   365
nipkow@15539
   366
lemma summable_convergent_sumr_iff:
nipkow@15539
   367
 "summable f = convergent (%n. setsum f {0..<n})"
paulson@14416
   368
by (simp add: summable_def sums_def convergent_def)
paulson@14416
   369
huffman@20689
   370
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
huffman@20689
   371
apply (drule summable_convergent_sumr_iff [THEN iffD1])
huffman@20692
   372
apply (drule convergent_Cauchy)
huffman@31336
   373
apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
huffman@20689
   374
apply (drule_tac x="r" in spec, safe)
huffman@20689
   375
apply (rule_tac x="M" in exI, safe)
huffman@20689
   376
apply (drule_tac x="Suc n" in spec, simp)
huffman@20689
   377
apply (drule_tac x="n" in spec, simp)
huffman@20689
   378
done
huffman@20689
   379
paulson@32707
   380
lemma suminf_le:
paulson@32707
   381
  fixes x :: real
paulson@32707
   382
  shows "summable f \<Longrightarrow> (!!n. setsum f {0..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
paulson@32707
   383
  by (simp add: summable_convergent_sumr_iff suminf_eq_lim lim_le) 
paulson@32707
   384
paulson@14416
   385
lemma summable_Cauchy:
huffman@20848
   386
     "summable (f::nat \<Rightarrow> 'a::banach) =  
huffman@20848
   387
      (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. norm (setsum f {m..<n}) < e)"
huffman@31336
   388
apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_iff, safe)
huffman@20410
   389
apply (drule spec, drule (1) mp)
huffman@20410
   390
apply (erule exE, rule_tac x="M" in exI, clarify)
huffman@20410
   391
apply (rule_tac x="m" and y="n" in linorder_le_cases)
huffman@20410
   392
apply (frule (1) order_trans)
huffman@20410
   393
apply (drule_tac x="n" in spec, drule (1) mp)
huffman@20410
   394
apply (drule_tac x="m" in spec, drule (1) mp)
huffman@20410
   395
apply (simp add: setsum_diff [symmetric])
huffman@20410
   396
apply simp
huffman@20410
   397
apply (drule spec, drule (1) mp)
huffman@20410
   398
apply (erule exE, rule_tac x="N" in exI, clarify)
huffman@20410
   399
apply (rule_tac x="m" and y="n" in linorder_le_cases)
huffman@20552
   400
apply (subst norm_minus_commute)
huffman@20410
   401
apply (simp add: setsum_diff [symmetric])
huffman@20410
   402
apply (simp add: setsum_diff [symmetric])
paulson@14416
   403
done
paulson@14416
   404
paulson@15085
   405
text{*Comparison test*}
paulson@15085
   406
huffman@20692
   407
lemma norm_setsum:
huffman@20692
   408
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
huffman@20692
   409
  shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
huffman@20692
   410
apply (case_tac "finite A")
huffman@20692
   411
apply (erule finite_induct)
huffman@20692
   412
apply simp
huffman@20692
   413
apply simp
huffman@20692
   414
apply (erule order_trans [OF norm_triangle_ineq add_left_mono])
huffman@20692
   415
apply simp
huffman@20692
   416
done
huffman@20692
   417
paulson@14416
   418
lemma summable_comparison_test:
huffman@20848
   419
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   420
  shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f"
huffman@20692
   421
apply (simp add: summable_Cauchy, safe)
huffman@20692
   422
apply (drule_tac x="e" in spec, safe)
huffman@20692
   423
apply (rule_tac x = "N + Na" in exI, safe)
paulson@14416
   424
apply (rotate_tac 2)
paulson@14416
   425
apply (drule_tac x = m in spec)
paulson@14416
   426
apply (auto, rotate_tac 2, drule_tac x = n in spec)
huffman@20848
   427
apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
huffman@20848
   428
apply (rule norm_setsum)
nipkow@15539
   429
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
huffman@22998
   430
apply (auto intro: setsum_mono simp add: abs_less_iff)
paulson@14416
   431
done
paulson@14416
   432
huffman@20848
   433
lemma summable_norm_comparison_test:
huffman@20848
   434
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   435
  shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk>
huffman@20848
   436
         \<Longrightarrow> summable (\<lambda>n. norm (f n))"
huffman@20848
   437
apply (rule summable_comparison_test)
huffman@20848
   438
apply (auto)
huffman@20848
   439
done
huffman@20848
   440
paulson@14416
   441
lemma summable_rabs_comparison_test:
huffman@20692
   442
  fixes f :: "nat \<Rightarrow> real"
huffman@20692
   443
  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
   444
apply (rule summable_comparison_test)
nipkow@15543
   445
apply (auto)
paulson@14416
   446
done
paulson@14416
   447
huffman@23084
   448
text{*Summability of geometric series for real algebras*}
huffman@23084
   449
huffman@23084
   450
lemma complete_algebra_summable_geometric:
haftmann@31017
   451
  fixes x :: "'a::{real_normed_algebra_1,banach}"
huffman@23084
   452
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
huffman@23084
   453
proof (rule summable_comparison_test)
huffman@23084
   454
  show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
huffman@23084
   455
    by (simp add: norm_power_ineq)
huffman@23084
   456
  show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)"
huffman@23084
   457
    by (simp add: summable_geometric)
huffman@23084
   458
qed
huffman@23084
   459
paulson@15085
   460
text{*Limit comparison property for series (c.f. jrh)*}
paulson@15085
   461
paulson@14416
   462
lemma summable_le:
huffman@20692
   463
  fixes f g :: "nat \<Rightarrow> real"
huffman@20692
   464
  shows "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
paulson@14416
   465
apply (drule summable_sums)+
huffman@20692
   466
apply (simp only: sums_def, erule (1) LIMSEQ_le)
paulson@14416
   467
apply (rule exI)
nipkow@15539
   468
apply (auto intro!: setsum_mono)
paulson@14416
   469
done
paulson@14416
   470
paulson@14416
   471
lemma summable_le2:
huffman@20692
   472
  fixes f g :: "nat \<Rightarrow> real"
huffman@20692
   473
  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
   474
apply (subgoal_tac "summable f")
huffman@20848
   475
apply (auto intro!: summable_le)
huffman@22998
   476
apply (simp add: abs_le_iff)
huffman@20848
   477
apply (rule_tac g="g" in summable_comparison_test, simp_all)
paulson@14416
   478
done
paulson@14416
   479
kleing@19106
   480
(* specialisation for the common 0 case *)
kleing@19106
   481
lemma suminf_0_le:
kleing@19106
   482
  fixes f::"nat\<Rightarrow>real"
kleing@19106
   483
  assumes gt0: "\<forall>n. 0 \<le> f n" and sm: "summable f"
kleing@19106
   484
  shows "0 \<le> suminf f"
kleing@19106
   485
proof -
kleing@19106
   486
  let ?g = "(\<lambda>n. (0::real))"
kleing@19106
   487
  from gt0 have "\<forall>n. ?g n \<le> f n" by simp
kleing@19106
   488
  moreover have "summable ?g" by (rule summable_zero)
kleing@19106
   489
  moreover from sm have "summable f" .
kleing@19106
   490
  ultimately have "suminf ?g \<le> suminf f" by (rule summable_le)
kleing@19106
   491
  then show "0 \<le> suminf f" by (simp add: suminf_zero)
kleing@19106
   492
qed 
kleing@19106
   493
kleing@19106
   494
paulson@15085
   495
text{*Absolute convergence imples normal convergence*}
huffman@20848
   496
lemma summable_norm_cancel:
huffman@20848
   497
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   498
  shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
huffman@20692
   499
apply (simp only: summable_Cauchy, safe)
huffman@20692
   500
apply (drule_tac x="e" in spec, safe)
huffman@20692
   501
apply (rule_tac x="N" in exI, safe)
huffman@20692
   502
apply (drule_tac x="m" in spec, safe)
huffman@20848
   503
apply (rule order_le_less_trans [OF norm_setsum])
huffman@20848
   504
apply (rule order_le_less_trans [OF abs_ge_self])
huffman@20692
   505
apply simp
paulson@14416
   506
done
paulson@14416
   507
huffman@20848
   508
lemma summable_rabs_cancel:
huffman@20848
   509
  fixes f :: "nat \<Rightarrow> real"
huffman@20848
   510
  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f"
huffman@20848
   511
by (rule summable_norm_cancel, simp)
huffman@20848
   512
paulson@15085
   513
text{*Absolute convergence of series*}
huffman@20848
   514
lemma summable_norm:
huffman@20848
   515
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   516
  shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
huffman@20848
   517
by (auto intro: LIMSEQ_le LIMSEQ_norm summable_norm_cancel
huffman@20848
   518
                summable_sumr_LIMSEQ_suminf norm_setsum)
huffman@20848
   519
paulson@14416
   520
lemma summable_rabs:
huffman@20692
   521
  fixes f :: "nat \<Rightarrow> real"
huffman@20692
   522
  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
huffman@20848
   523
by (fold real_norm_def, rule summable_norm)
paulson@14416
   524
paulson@14416
   525
subsection{* The Ratio Test*}
paulson@14416
   526
huffman@20848
   527
lemma norm_ratiotest_lemma:
huffman@22852
   528
  fixes x y :: "'a::real_normed_vector"
huffman@20848
   529
  shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0"
huffman@20848
   530
apply (subgoal_tac "norm x \<le> 0", simp)
huffman@20848
   531
apply (erule order_trans)
huffman@20848
   532
apply (simp add: mult_le_0_iff)
huffman@20848
   533
done
huffman@20848
   534
paulson@14416
   535
lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
huffman@20848
   536
by (erule norm_ratiotest_lemma, simp)
paulson@14416
   537
paulson@14416
   538
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"
paulson@14416
   539
apply (drule le_imp_less_or_eq)
paulson@14416
   540
apply (auto dest: less_imp_Suc_add)
paulson@14416
   541
done
paulson@14416
   542
paulson@14416
   543
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"
paulson@14416
   544
by (auto simp add: le_Suc_ex)
paulson@14416
   545
paulson@14416
   546
(*All this trouble just to get 0<c *)
paulson@14416
   547
lemma ratio_test_lemma2:
huffman@20848
   548
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   549
  shows "\<lbrakk>\<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> 0 < c \<or> summable f"
paulson@14416
   550
apply (simp (no_asm) add: linorder_not_le [symmetric])
paulson@14416
   551
apply (simp add: summable_Cauchy)
nipkow@15543
   552
apply (safe, subgoal_tac "\<forall>n. N < n --> f (n) = 0")
nipkow@15543
   553
 prefer 2
nipkow@15543
   554
 apply clarify
huffman@30082
   555
 apply(erule_tac x = "n - Suc 0" in allE)
nipkow@15543
   556
 apply (simp add:diff_Suc split:nat.splits)
huffman@20848
   557
 apply (blast intro: norm_ratiotest_lemma)
paulson@14416
   558
apply (rule_tac x = "Suc N" in exI, clarify)
nipkow@15543
   559
apply(simp cong:setsum_ivl_cong)
paulson@14416
   560
done
paulson@14416
   561
paulson@14416
   562
lemma ratio_test:
huffman@20848
   563
  fixes f :: "nat \<Rightarrow> 'a::banach"
huffman@20848
   564
  shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f"
paulson@14416
   565
apply (frule ratio_test_lemma2, auto)
huffman@20848
   566
apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n" 
paulson@15234
   567
       in summable_comparison_test)
paulson@14416
   568
apply (rule_tac x = N in exI, safe)
paulson@14416
   569
apply (drule le_Suc_ex_iff [THEN iffD1])
huffman@22959
   570
apply (auto simp add: power_add field_power_not_zero)
nipkow@15539
   571
apply (induct_tac "na", auto)
huffman@20848
   572
apply (rule_tac y = "c * norm (f (N + n))" in order_trans)
paulson@14416
   573
apply (auto intro: mult_right_mono simp add: summable_def)
huffman@20848
   574
apply (rule_tac x = "norm (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
paulson@15234
   575
apply (rule sums_divide) 
haftmann@27108
   576
apply (rule sums_mult)
paulson@15234
   577
apply (auto intro!: geometric_sums)
paulson@14416
   578
done
paulson@14416
   579
huffman@23111
   580
subsection {* Cauchy Product Formula *}
huffman@23111
   581
huffman@23111
   582
(* Proof based on Analysis WebNotes: Chapter 07, Class 41
huffman@23111
   583
http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm *)
huffman@23111
   584
huffman@23111
   585
lemma setsum_triangle_reindex:
huffman@23111
   586
  fixes n :: nat
huffman@23111
   587
  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))"
huffman@23111
   588
proof -
huffman@23111
   589
  have "(\<Sum>(i, j)\<in>{(i, j). i + j < n}. f i j) =
huffman@23111
   590
    (\<Sum>(k, i)\<in>(SIGMA k:{0..<n}. {0..k}). f i (k - i))"
huffman@23111
   591
  proof (rule setsum_reindex_cong)
huffman@23111
   592
    show "inj_on (\<lambda>(k,i). (i, k - i)) (SIGMA k:{0..<n}. {0..k})"
huffman@23111
   593
      by (rule inj_on_inverseI [where g="\<lambda>(i,j). (i+j, i)"], auto)
huffman@23111
   594
    show "{(i,j). i + j < n} = (\<lambda>(k,i). (i, k - i)) ` (SIGMA k:{0..<n}. {0..k})"
huffman@23111
   595
      by (safe, rule_tac x="(a+b,a)" in image_eqI, auto)
huffman@23111
   596
    show "\<And>a. (\<lambda>(k, i). f i (k - i)) a = split f ((\<lambda>(k, i). (i, k - i)) a)"
huffman@23111
   597
      by clarify
huffman@23111
   598
  qed
huffman@23111
   599
  thus ?thesis by (simp add: setsum_Sigma)
huffman@23111
   600
qed
huffman@23111
   601
huffman@23111
   602
lemma Cauchy_product_sums:
huffman@23111
   603
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
huffman@23111
   604
  assumes a: "summable (\<lambda>k. norm (a k))"
huffman@23111
   605
  assumes b: "summable (\<lambda>k. norm (b k))"
huffman@23111
   606
  shows "(\<lambda>k. \<Sum>i=0..k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
huffman@23111
   607
proof -
huffman@23111
   608
  let ?S1 = "\<lambda>n::nat. {0..<n} \<times> {0..<n}"
huffman@23111
   609
  let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
huffman@23111
   610
  have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
huffman@23111
   611
  have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
huffman@23111
   612
  have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
huffman@23111
   613
  have finite_S1: "\<And>n. finite (?S1 n)" by simp
huffman@23111
   614
  with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
huffman@23111
   615
huffman@23111
   616
  let ?g = "\<lambda>(i,j). a i * b j"
huffman@23111
   617
  let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
huffman@23111
   618
  have f_nonneg: "\<And>x. 0 \<le> ?f x"
huffman@23111
   619
    by (auto simp add: mult_nonneg_nonneg)
huffman@23111
   620
  hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A"
huffman@23111
   621
    unfolding real_norm_def
huffman@23111
   622
    by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
huffman@23111
   623
huffman@23111
   624
  have "(\<lambda>n. (\<Sum>k=0..<n. a k) * (\<Sum>k=0..<n. b k))
huffman@23111
   625
           ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
huffman@23111
   626
    by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf
huffman@23111
   627
        summable_norm_cancel [OF a] summable_norm_cancel [OF b])
huffman@23111
   628
  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
huffman@23111
   629
    by (simp only: setsum_product setsum_Sigma [rule_format]
huffman@23111
   630
                   finite_atLeastLessThan)
huffman@23111
   631
huffman@23111
   632
  have "(\<lambda>n. (\<Sum>k=0..<n. norm (a k)) * (\<Sum>k=0..<n. norm (b k)))
huffman@23111
   633
       ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
huffman@23111
   634
    using a b by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf)
huffman@23111
   635
  hence "(\<lambda>n. setsum ?f (?S1 n)) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
huffman@23111
   636
    by (simp only: setsum_product setsum_Sigma [rule_format]
huffman@23111
   637
                   finite_atLeastLessThan)
huffman@23111
   638
  hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
huffman@23111
   639
    by (rule convergentI)
huffman@23111
   640
  hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"
huffman@23111
   641
    by (rule convergent_Cauchy)
huffman@23111
   642
  have "Zseq (\<lambda>n. setsum ?f (?S1 n - ?S2 n))"
huffman@23111
   643
  proof (rule ZseqI, simp only: norm_setsum_f)
huffman@23111
   644
    fix r :: real
huffman@23111
   645
    assume r: "0 < r"
huffman@23111
   646
    from CauchyD [OF Cauchy r] obtain N
huffman@23111
   647
    where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" ..
huffman@23111
   648
    hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r"
huffman@23111
   649
      by (simp only: setsum_diff finite_S1 S1_mono)
huffman@23111
   650
    hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r"
huffman@23111
   651
      by (simp only: norm_setsum_f)
huffman@23111
   652
    show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r"
huffman@23111
   653
    proof (intro exI allI impI)
huffman@23111
   654
      fix n assume "2 * N \<le> n"
huffman@23111
   655
      hence n: "N \<le> n div 2" by simp
huffman@23111
   656
      have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))"
huffman@23111
   657
        by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg
huffman@23111
   658
                  Diff_mono subset_refl S1_le_S2)
huffman@23111
   659
      also have "\<dots> < r"
huffman@23111
   660
        using n div_le_dividend by (rule N)
huffman@23111
   661
      finally show "setsum ?f (?S1 n - ?S2 n) < r" .
huffman@23111
   662
    qed
huffman@23111
   663
  qed
huffman@23111
   664
  hence "Zseq (\<lambda>n. setsum ?g (?S1 n - ?S2 n))"
huffman@23111
   665
    apply (rule Zseq_le [rule_format])
huffman@23111
   666
    apply (simp only: norm_setsum_f)
huffman@23111
   667
    apply (rule order_trans [OF norm_setsum setsum_mono])
huffman@23111
   668
    apply (auto simp add: norm_mult_ineq)
huffman@23111
   669
    done
huffman@23111
   670
  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) ----> 0"
huffman@23111
   671
    by (simp only: LIMSEQ_Zseq_iff setsum_diff finite_S1 S2_le_S1 diff_0_right)
huffman@23111
   672
huffman@23111
   673
  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
huffman@23111
   674
    by (rule LIMSEQ_diff_approach_zero2)
huffman@23111
   675
  thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
huffman@23111
   676
qed
huffman@23111
   677
huffman@23111
   678
lemma Cauchy_product:
huffman@23111
   679
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
huffman@23111
   680
  assumes a: "summable (\<lambda>k. norm (a k))"
huffman@23111
   681
  assumes b: "summable (\<lambda>k. norm (b k))"
huffman@23111
   682
  shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i=0..k. a i * b (k - i))"
huffman@23441
   683
using a b
huffman@23111
   684
by (rule Cauchy_product_sums [THEN sums_unique])
huffman@23111
   685
paulson@14416
   686
end