src/HOL/Hyperreal/Series.thy
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new proof of Cauchy product formula for 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 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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   285
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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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parents: 12018
diff changeset
   300
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   301
lemma series_pos_less:
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   302
  fixes f :: "nat \<Rightarrow> real"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   303
  shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} < suminf f"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   304
apply (rule_tac y="setsum f {0..<Suc n}" in order_less_le_trans)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   305
apply simp
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   306
apply (erule series_pos_le)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   307
apply (simp add: order_less_imp_le)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   308
done
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   309
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   310
lemma suminf_gt_zero:
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   311
  fixes f :: "nat \<Rightarrow> real"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   312
  shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   313
by (drule_tac n="0" in series_pos_less, simp_all)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   314
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   315
lemma suminf_ge_zero:
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   316
  fixes f :: "nat \<Rightarrow> real"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   317
  shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 \<le> suminf f"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   318
by (drule_tac n="0" in series_pos_le, simp_all)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   319
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   320
lemma sumr_pos_lt_pair:
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   321
  fixes f :: "nat \<Rightarrow> real"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   322
  shows "\<lbrakk>summable f;
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   323
        \<forall>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   324
      \<Longrightarrow> setsum f {0..<k} < suminf f"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   325
apply (subst suminf_split_initial_segment [where k="k"])
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   326
apply assumption
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   327
apply simp
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   328
apply (drule_tac k="k" in summable_ignore_initial_segment)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   329
apply (drule_tac k="Suc (Suc 0)" in sums_group, simp)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   330
apply simp
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   331
apply (frule sums_unique)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   332
apply (drule sums_summable)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   333
apply simp
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   334
apply (erule suminf_gt_zero)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   335
apply (simp add: add_ac)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   336
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   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
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   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
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   341
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   342
lemma geometric_sums:
22719
c51667189bd3 lemma geometric_sum no longer needs class division_by_zero
huffman
parents: 21404
diff changeset
   343
  fixes x :: "'a::{real_normed_field,recpower}"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   344
  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) sums (1 / (1 - x))"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   345
proof -
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   346
  assume less_1: "norm x < 1"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   347
  hence neq_1: "x \<noteq> 1" by auto
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   348
  hence neq_0: "x - 1 \<noteq> 0" by simp
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   349
  from less_1 have lim_0: "(\<lambda>n. x ^ n) ----> 0"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   350
    by (rule LIMSEQ_power_zero)
22719
c51667189bd3 lemma geometric_sum no longer needs class division_by_zero
huffman
parents: 21404
diff changeset
   351
  hence "(\<lambda>n. x ^ n / (x - 1) - 1 / (x - 1)) ----> 0 / (x - 1) - 1 / (x - 1)"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   352
    using neq_0 by (intro LIMSEQ_divide LIMSEQ_diff LIMSEQ_const)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   353
  hence "(\<lambda>n. (x ^ n - 1) / (x - 1)) ----> 1 / (1 - x)"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   354
    by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   355
  thus "(\<lambda>n. x ^ n) sums (1 / (1 - x))"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   356
    by (simp add: sums_def geometric_sum neq_1)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   357
qed
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   358
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   359
lemma summable_geometric:
22719
c51667189bd3 lemma geometric_sum no longer needs class division_by_zero
huffman
parents: 21404
diff changeset
   360
  fixes x :: "'a::{real_normed_field,recpower}"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   361
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   362
by (rule geometric_sums [THEN sums_summable])
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   363
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   364
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   365
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15537
diff changeset
   366
lemma summable_convergent_sumr_iff:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15537
diff changeset
   367
 "summable f = convergent (%n. setsum f {0..<n})"
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   368
by (simp add: summable_def sums_def convergent_def)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   369
20689
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   370
lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   371
apply (drule summable_convergent_sumr_iff [THEN iffD1])
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   372
apply (drule convergent_Cauchy)
20689
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   373
apply (simp only: Cauchy_def LIMSEQ_def, safe)
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   374
apply (drule_tac x="r" in spec, safe)
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   375
apply (rule_tac x="M" in exI, safe)
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   376
apply (drule_tac x="Suc n" in spec, simp)
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   377
apply (drule_tac x="n" in spec, simp)
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   378
done
4950e45442b8 add proof of summable_LIMSEQ_zero
huffman
parents: 20688
diff changeset
   379
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   380
lemma summable_Cauchy:
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   381
     "summable (f::nat \<Rightarrow> 'a::banach) =  
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   382
      (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. norm (setsum f {m..<n}) < e)"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   383
apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def, safe)
20410
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   384
apply (drule spec, drule (1) mp)
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   385
apply (erule exE, rule_tac x="M" in exI, clarify)
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   386
apply (rule_tac x="m" and y="n" in linorder_le_cases)
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   387
apply (frule (1) order_trans)
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   388
apply (drule_tac x="n" in spec, drule (1) mp)
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   389
apply (drule_tac x="m" in spec, drule (1) mp)
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   390
apply (simp add: setsum_diff [symmetric])
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   391
apply simp
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   392
apply (drule spec, drule (1) mp)
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   393
apply (erule exE, rule_tac x="N" in exI, clarify)
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   394
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
   395
apply (subst norm_minus_commute)
20410
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   396
apply (simp add: setsum_diff [symmetric])
4bd5cd97c547 speed up proof of summable_Cauchy
huffman
parents: 20254
diff changeset
   397
apply (simp add: setsum_diff [symmetric])
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   398
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   399
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   400
text{*Comparison test*}
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   401
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   402
lemma norm_setsum:
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   403
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   404
  shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   405
apply (case_tac "finite A")
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   406
apply (erule finite_induct)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   407
apply simp
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   408
apply simp
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   409
apply (erule order_trans [OF norm_triangle_ineq add_left_mono])
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   410
apply simp
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   411
done
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   412
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   413
lemma summable_comparison_test:
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   414
  fixes f :: "nat \<Rightarrow> 'a::banach"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   415
  shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   416
apply (simp add: summable_Cauchy, safe)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   417
apply (drule_tac x="e" in spec, safe)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   418
apply (rule_tac x = "N + Na" in exI, safe)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   419
apply (rotate_tac 2)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   420
apply (drule_tac x = m in spec)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   421
apply (auto, rotate_tac 2, drule_tac x = n in spec)
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   422
apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   423
apply (rule norm_setsum)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15537
diff changeset
   424
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22959
diff changeset
   425
apply (auto intro: setsum_mono simp add: abs_less_iff)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   426
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   427
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   428
lemma summable_norm_comparison_test:
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   429
  fixes f :: "nat \<Rightarrow> 'a::banach"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   430
  shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk>
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   431
         \<Longrightarrow> summable (\<lambda>n. norm (f n))"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   432
apply (rule summable_comparison_test)
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   433
apply (auto)
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   434
done
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   435
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   436
lemma summable_rabs_comparison_test:
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   437
  fixes f :: "nat \<Rightarrow> real"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   438
  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
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   439
apply (rule summable_comparison_test)
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   440
apply (auto)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   441
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   442
23084
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   443
text{*Summability of geometric series for real algebras*}
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   444
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   445
lemma complete_algebra_summable_geometric:
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   446
  fixes x :: "'a::{real_normed_algebra_1,banach,recpower}"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   447
  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   448
proof (rule summable_comparison_test)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   449
  show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   450
    by (simp add: norm_power_ineq)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   451
  show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)"
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   452
    by (simp add: summable_geometric)
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   453
qed
bc000fc64fce add lemma complete_algebra_summable_geometric
huffman
parents: 22998
diff changeset
   454
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   455
text{*Limit comparison property for series (c.f. jrh)*}
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   456
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   457
lemma summable_le:
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   458
  fixes f g :: "nat \<Rightarrow> real"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   459
  shows "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   460
apply (drule summable_sums)+
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   461
apply (simp only: sums_def, erule (1) LIMSEQ_le)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   462
apply (rule exI)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15537
diff changeset
   463
apply (auto intro!: setsum_mono)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   464
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   465
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   466
lemma summable_le2:
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   467
  fixes f g :: "nat \<Rightarrow> real"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   468
  shows "\<lbrakk>\<forall>n. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f \<and> suminf f \<le> suminf g"
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   469
apply (subgoal_tac "summable f")
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   470
apply (auto intro!: summable_le)
22998
97e1f9c2cc46 avoid using redundant lemmas from RealDef.thy
huffman
parents: 22959
diff changeset
   471
apply (simp add: abs_le_iff)
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   472
apply (rule_tac g="g" in summable_comparison_test, simp_all)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   473
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   474
19106
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 17149
diff changeset
   475
(* 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
   476
lemma suminf_0_le:
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 17149
diff changeset
   477
  fixes f::"nat\<Rightarrow>real"
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 17149
diff changeset
   478
  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
   479
  shows "0 \<le> suminf f"
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 17149
diff changeset
   480
proof -
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 17149
diff changeset
   481
  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
   482
  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
   483
  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
   484
  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
   485
  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
   486
  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
   487
qed 
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 17149
diff changeset
   488
6e6b5b1fdc06 * added Library/ASeries (sum of arithmetic series with instantiation to nat and int)
kleing
parents: 17149
diff changeset
   489
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   490
text{*Absolute convergence imples normal convergence*}
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   491
lemma summable_norm_cancel:
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   492
  fixes f :: "nat \<Rightarrow> 'a::banach"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   493
  shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   494
apply (simp only: summable_Cauchy, safe)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   495
apply (drule_tac x="e" in spec, safe)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   496
apply (rule_tac x="N" in exI, safe)
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   497
apply (drule_tac x="m" in spec, safe)
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   498
apply (rule order_le_less_trans [OF norm_setsum])
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   499
apply (rule order_le_less_trans [OF abs_ge_self])
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   500
apply simp
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   501
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   502
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   503
lemma summable_rabs_cancel:
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   504
  fixes f :: "nat \<Rightarrow> real"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   505
  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   506
by (rule summable_norm_cancel, simp)
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   507
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15053
diff changeset
   508
text{*Absolute convergence of series*}
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   509
lemma summable_norm:
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   510
  fixes f :: "nat \<Rightarrow> 'a::banach"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   511
  shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   512
by (auto intro: LIMSEQ_le LIMSEQ_norm summable_norm_cancel
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   513
                summable_sumr_LIMSEQ_suminf norm_setsum)
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   514
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   515
lemma summable_rabs:
20692
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   516
  fixes f :: "nat \<Rightarrow> real"
6df83a636e67 generalized types of sums, summable, and suminf
huffman
parents: 20689
diff changeset
   517
  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   518
by (fold real_norm_def, rule summable_norm)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   519
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   520
subsection{* The Ratio Test*}
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   521
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   522
lemma norm_ratiotest_lemma:
22852
2490d4b4671a clean up RealVector classes
huffman
parents: 22719
diff changeset
   523
  fixes x y :: "'a::real_normed_vector"
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   524
  shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   525
apply (subgoal_tac "norm x \<le> 0", simp)
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   526
apply (erule order_trans)
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   527
apply (simp add: mult_le_0_iff)
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   528
done
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   529
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   530
lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   531
by (erule norm_ratiotest_lemma, simp)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   532
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   533
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   534
apply (drule le_imp_less_or_eq)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   535
apply (auto dest: less_imp_Suc_add)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   536
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   537
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   538
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   539
by (auto simp add: le_Suc_ex)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   540
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   541
(*All this trouble just to get 0<c *)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   542
lemma ratio_test_lemma2:
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   543
  fixes f :: "nat \<Rightarrow> 'a::banach"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   544
  shows "\<lbrakk>\<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> 0 < c \<or> summable f"
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   545
apply (simp (no_asm) add: linorder_not_le [symmetric])
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   546
apply (simp add: summable_Cauchy)
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   547
apply (safe, subgoal_tac "\<forall>n. N < n --> f (n) = 0")
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   548
 prefer 2
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   549
 apply clarify
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   550
 apply(erule_tac x = "n - 1" in allE)
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   551
 apply (simp add:diff_Suc split:nat.splits)
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   552
 apply (blast intro: norm_ratiotest_lemma)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   553
apply (rule_tac x = "Suc N" in exI, clarify)
15543
0024472afce7 more setsum tuning
nipkow
parents: 15542
diff changeset
   554
apply(simp cong:setsum_ivl_cong)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   555
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   556
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   557
lemma ratio_test:
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   558
  fixes f :: "nat \<Rightarrow> 'a::banach"
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   559
  shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f"
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   560
apply (frule ratio_test_lemma2, auto)
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   561
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
   562
       in summable_comparison_test)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   563
apply (rule_tac x = N in exI, safe)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   564
apply (drule le_Suc_ex_iff [THEN iffD1])
22959
07a7c2900877 remove redundant lemmas
huffman
parents: 22852
diff changeset
   565
apply (auto simp add: power_add field_power_not_zero)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15537
diff changeset
   566
apply (induct_tac "na", auto)
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   567
apply (rule_tac y = "c * norm (f (N + n))" in order_trans)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   568
apply (auto intro: mult_right_mono simp add: summable_def)
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   569
apply (simp add: mult_ac)
20848
27a09c3eca1f generalize summability lemmas using class banach
huffman
parents: 20792
diff changeset
   570
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
   571
apply (rule sums_divide) 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   572
apply (rule sums_mult) 
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   573
apply (auto intro!: geometric_sums)
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   574
done
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   575
23111
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   576
subsection {* Cauchy Product Formula *}
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   577
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   578
(* Proof based on Analysis WebNotes: Chapter 07, Class 41
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   579
http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm *)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   580
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   581
lemma setsum_triangle_reindex:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   582
  fixes n :: nat
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   583
  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))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   584
proof -
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   585
  have "(\<Sum>(i, j)\<in>{(i, j). i + j < n}. f i j) =
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   586
    (\<Sum>(k, i)\<in>(SIGMA k:{0..<n}. {0..k}). f i (k - i))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   587
  proof (rule setsum_reindex_cong)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   588
    show "inj_on (\<lambda>(k,i). (i, k - i)) (SIGMA k:{0..<n}. {0..k})"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   589
      by (rule inj_on_inverseI [where g="\<lambda>(i,j). (i+j, i)"], auto)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   590
    show "{(i,j). i + j < n} = (\<lambda>(k,i). (i, k - i)) ` (SIGMA k:{0..<n}. {0..k})"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   591
      by (safe, rule_tac x="(a+b,a)" in image_eqI, auto)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   592
    show "\<And>a. (\<lambda>(k, i). f i (k - i)) a = split f ((\<lambda>(k, i). (i, k - i)) a)"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   593
      by clarify
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   594
  qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   595
  thus ?thesis by (simp add: setsum_Sigma)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   596
qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   597
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   598
lemma Cauchy_product_sums:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   599
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   600
  assumes a: "summable (\<lambda>k. norm (a k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   601
  assumes b: "summable (\<lambda>k. norm (b k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   602
  shows "(\<lambda>k. \<Sum>i=0..k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   603
proof -
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   604
  let ?S1 = "\<lambda>n::nat. {0..<n} \<times> {0..<n}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   605
  let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   606
  have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   607
  have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   608
  have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   609
  have finite_S1: "\<And>n. finite (?S1 n)" by simp
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   610
  with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   611
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   612
  let ?g = "\<lambda>(i,j). a i * b j"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   613
  let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   614
  have f_nonneg: "\<And>x. 0 \<le> ?f x"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   615
    by (auto simp add: mult_nonneg_nonneg)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   616
  hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   617
    unfolding real_norm_def
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   618
    by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   619
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   620
  have "(\<lambda>n. (\<Sum>k=0..<n. a k) * (\<Sum>k=0..<n. b k))
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   621
           ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   622
    by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   623
        summable_norm_cancel [OF a] summable_norm_cancel [OF b])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   624
  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   625
    by (simp only: setsum_product setsum_Sigma [rule_format]
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   626
                   finite_atLeastLessThan)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   627
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   628
  have "(\<lambda>n. (\<Sum>k=0..<n. norm (a k)) * (\<Sum>k=0..<n. norm (b k)))
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   629
       ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   630
    using a b by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   631
  hence "(\<lambda>n. setsum ?f (?S1 n)) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   632
    by (simp only: setsum_product setsum_Sigma [rule_format]
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   633
                   finite_atLeastLessThan)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   634
  hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   635
    by (rule convergentI)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   636
  hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   637
    by (rule convergent_Cauchy)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   638
  have "Zseq (\<lambda>n. setsum ?f (?S1 n - ?S2 n))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   639
  proof (rule ZseqI, simp only: norm_setsum_f)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   640
    fix r :: real
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   641
    assume r: "0 < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   642
    from CauchyD [OF Cauchy r] obtain N
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   643
    where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" ..
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   644
    hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   645
      by (simp only: setsum_diff finite_S1 S1_mono)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   646
    hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   647
      by (simp only: norm_setsum_f)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   648
    show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   649
    proof (intro exI allI impI)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   650
      fix n assume "2 * N \<le> n"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   651
      hence n: "N \<le> n div 2" by simp
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   652
      have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   653
        by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   654
                  Diff_mono subset_refl S1_le_S2)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   655
      also have "\<dots> < r"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   656
        using n div_le_dividend by (rule N)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   657
      finally show "setsum ?f (?S1 n - ?S2 n) < r" .
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   658
    qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   659
  qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   660
  hence "Zseq (\<lambda>n. setsum ?g (?S1 n - ?S2 n))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   661
    apply (rule Zseq_le [rule_format])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   662
    apply (simp only: norm_setsum_f)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   663
    apply (rule order_trans [OF norm_setsum setsum_mono])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   664
    apply (auto simp add: norm_mult_ineq)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   665
    done
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   666
  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) ----> 0"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   667
    by (simp only: LIMSEQ_Zseq_iff setsum_diff finite_S1 S2_le_S1 diff_0_right)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   668
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   669
  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   670
    by (rule LIMSEQ_diff_approach_zero2)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   671
  thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   672
qed
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   673
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   674
lemma Cauchy_product:
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   675
  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   676
  assumes a: "summable (\<lambda>k. norm (a k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   677
  assumes b: "summable (\<lambda>k. norm (b k))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   678
  shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i=0..k. a i * b (k - i))"
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   679
by (rule Cauchy_product_sums [THEN sums_unique])
f8583c2a491a new proof of Cauchy product formula for series
huffman
parents: 23084
diff changeset
   680
14416
1f256287d4f0 converted Hyperreal/Series to Isar script
paulson
parents: 12018
diff changeset
   681
end