src/HOL/Hyperreal/Series.thy

-(*  Title       : Series.thy
-    Author      : Jacques D. Fleuriot
-    Copyright   : 1998  University of Cambridge
-
-Converted to Isar and polished by lcp
-Converted to setsum and polished yet more by TNN
-Additional contributions by Jeremy Avigad
-*) 
-
-header{*Finite Summation and Infinite Series*}
-
-theory Series
-imports SEQ
-begin
-
-definition
-   sums  :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool"
-     (infixr "sums" 80) where
-   "f sums s = (%n. setsum f {0..<n}) ----> s"
-
-definition
-   summable :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> bool" where
-   "summable f = (\<exists>s. f sums s)"
-
-definition
-   suminf   :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a" where
-   "suminf f = (THE s. f sums s)"
-
-syntax
-  "_suminf" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a" ("\<Sum>_. _" [0, 10] 10)
-translations
-  "\<Sum>i. b" == "CONST suminf (%i. b)"
-
-
-lemma sumr_diff_mult_const:
- "setsum f {0..<n} - (real n*r) = setsum (%i. f i - r) {0..<n::nat}"
-by (simp add: diff_minus setsum_addf real_of_nat_def)
-
-lemma real_setsum_nat_ivl_bounded:
-     "(!!p. p < n \<Longrightarrow> f(p) \<le> K)
-      \<Longrightarrow> setsum f {0..<n::nat} \<le> real n * K"
-using setsum_bounded[where A = "{0..<n}"]
-by (auto simp:real_of_nat_def)
-
-(* Generalize from real to some algebraic structure? *)
-lemma sumr_minus_one_realpow_zero [simp]:
-  "(\<Sum>i=0..<2*n. (-1) ^ Suc i) = (0::real)"
-by (induct "n", auto)
-
-(* FIXME this is an awful lemma! *)
-lemma sumr_one_lb_realpow_zero [simp]:
-  "(\<Sum>n=Suc 0..<n. f(n) * (0::real) ^ n) = 0"
-by (rule setsum_0', simp)
-
-lemma sumr_group:
-     "(\<Sum>m=0..<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {0 ..< n * k}"
-apply (subgoal_tac "k = 0 | 0 < k", auto)
-apply (induct "n")
-apply (simp_all add: setsum_add_nat_ivl add_commute)
-done
-
-lemma sumr_offset3:
-  "setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)) + setsum f {0..<k}"
-apply (subst setsum_shift_bounds_nat_ivl [symmetric])
-apply (simp add: setsum_add_nat_ivl add_commute)
-done
-
-lemma sumr_offset:
-  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
-  shows "(\<Sum>m=0..<n. f(m+k)) = setsum f {0..<n+k} - setsum f {0..<k}"
-by (simp add: sumr_offset3)
-
-lemma sumr_offset2:
- "\<forall>f. (\<Sum>m=0..<n::nat. f(m+k)::real) = setsum f {0..<n+k} - setsum f {0..<k}"
-by (simp add: sumr_offset)
-
-lemma sumr_offset4:
-  "\<forall>n f. setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)::real) + setsum f {0..<k}"
-by (clarify, rule sumr_offset3)
-
-(*
-lemma sumr_from_1_from_0: "0 < n ==>
-      (\<Sum>n=Suc 0 ..< Suc n. if even(n) then 0 else
-             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n =
-      (\<Sum>n=0..<Suc n. if even(n) then 0 else
-             ((- 1) ^ ((n - (Suc 0)) div 2))/(real (fact n))) * a ^ n"
-by (rule_tac n1 = 1 in sumr_split_add [THEN subst], auto)
-*)
-
-subsection{* Infinite Sums, by the Properties of Limits*}
-
-(*----------------------
-   suminf is the sum   
- ---------------------*)
-lemma sums_summable: "f sums l ==> summable f"
-by (simp add: sums_def summable_def, blast)
-
-lemma summable_sums: "summable f ==> f sums (suminf f)"
-apply (simp add: summable_def suminf_def sums_def)
-apply (blast intro: theI LIMSEQ_unique)
-done
-
-lemma summable_sumr_LIMSEQ_suminf: 
-     "summable f ==> (%n. setsum f {0..<n}) ----> (suminf f)"
-by (rule summable_sums [unfolded sums_def])
-
-(*-------------------
-    sum is unique                    
- ------------------*)
-lemma sums_unique: "f sums s ==> (s = suminf f)"
-apply (frule sums_summable [THEN summable_sums])
-apply (auto intro!: LIMSEQ_unique simp add: sums_def)
-done
-
-lemma sums_split_initial_segment: "f sums s ==> 
-  (%n. f(n + k)) sums (s - (SUM i = 0..< k. f i))"
-  apply (unfold sums_def);
-  apply (simp add: sumr_offset); 
-  apply (rule LIMSEQ_diff_const)
-  apply (rule LIMSEQ_ignore_initial_segment)
-  apply assumption
-done
-
-lemma summable_ignore_initial_segment: "summable f ==> 
-    summable (%n. f(n + k))"
-  apply (unfold summable_def)
-  apply (auto intro: sums_split_initial_segment)
-done
-
-lemma suminf_minus_initial_segment: "summable f ==>
-    suminf f = s ==> suminf (%n. f(n + k)) = s - (SUM i = 0..< k. f i)"
-  apply (frule summable_ignore_initial_segment)
-  apply (rule sums_unique [THEN sym])
-  apply (frule summable_sums)
-  apply (rule sums_split_initial_segment)
-  apply auto
-done
-
-lemma suminf_split_initial_segment: "summable f ==> 
-    suminf f = (SUM i = 0..< k. f i) + suminf (%n. f(n + k))"
-by (auto simp add: suminf_minus_initial_segment)
-
-lemma series_zero: 
-     "(\<forall>m. n \<le> m --> f(m) = 0) ==> f sums (setsum f {0..<n})"
-apply (simp add: sums_def LIMSEQ_def diff_minus[symmetric], safe)
-apply (rule_tac x = n in exI)
-apply (clarsimp simp add:setsum_diff[symmetric] cong:setsum_ivl_cong)
-done
-
-lemma sums_zero: "(\<lambda>n. 0) sums 0"
-unfolding sums_def by (simp add: LIMSEQ_const)
-
-lemma summable_zero: "summable (\<lambda>n. 0)"
-by (rule sums_zero [THEN sums_summable])
-
-lemma suminf_zero: "suminf (\<lambda>n. 0) = 0"
-by (rule sums_zero [THEN sums_unique, symmetric])
-  
-lemma (in bounded_linear) sums:
-  "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
-unfolding sums_def by (drule LIMSEQ, simp only: setsum)
-
-lemma (in bounded_linear) summable:
-  "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
-unfolding summable_def by (auto intro: sums)
-
-lemma (in bounded_linear) suminf:
-  "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
-by (intro sums_unique sums summable_sums)
-
-lemma sums_mult:
-  fixes c :: "'a::real_normed_algebra"
-  shows "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
-by (rule mult_right.sums)
-
-lemma summable_mult:
-  fixes c :: "'a::real_normed_algebra"
-  shows "summable f \<Longrightarrow> summable (%n. c * f n)"
-by (rule mult_right.summable)
-
-lemma suminf_mult:
-  fixes c :: "'a::real_normed_algebra"
-  shows "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f";
-by (rule mult_right.suminf [symmetric])
-
-lemma sums_mult2:
-  fixes c :: "'a::real_normed_algebra"
-  shows "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
-by (rule mult_left.sums)
-
-lemma summable_mult2:
-  fixes c :: "'a::real_normed_algebra"
-  shows "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
-by (rule mult_left.summable)
-
-lemma suminf_mult2:
-  fixes c :: "'a::real_normed_algebra"
-  shows "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
-by (rule mult_left.suminf)
-
-lemma sums_divide:
-  fixes c :: "'a::real_normed_field"
-  shows "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
-by (rule divide.sums)
-
-lemma summable_divide:
-  fixes c :: "'a::real_normed_field"
-  shows "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
-by (rule divide.summable)
-
-lemma suminf_divide:
-  fixes c :: "'a::real_normed_field"
-  shows "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
-by (rule divide.suminf [symmetric])
-
-lemma sums_add: "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) sums (a + b)"
-unfolding sums_def by (simp add: setsum_addf LIMSEQ_add)
-
-lemma summable_add: "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n + Y n)"
-unfolding summable_def by (auto intro: sums_add)
-
-lemma suminf_add:
-  "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X + suminf Y = (\<Sum>n. X n + Y n)"
-by (intro sums_unique sums_add summable_sums)
-
-lemma sums_diff: "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) sums (a - b)"
-unfolding sums_def by (simp add: setsum_subtractf LIMSEQ_diff)
-
-lemma summable_diff: "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n - Y n)"
-unfolding summable_def by (auto intro: sums_diff)
-
-lemma suminf_diff:
-  "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X - suminf Y = (\<Sum>n. X n - Y n)"
-by (intro sums_unique sums_diff summable_sums)
-
-lemma sums_minus: "X sums a ==> (\<lambda>n. - X n) sums (- a)"
-unfolding sums_def by (simp add: setsum_negf LIMSEQ_minus)
-
-lemma summable_minus: "summable X \<Longrightarrow> summable (\<lambda>n. - X n)"
-unfolding summable_def by (auto intro: sums_minus)
-
-lemma suminf_minus: "summable X \<Longrightarrow> (\<Sum>n. - X n) = - (\<Sum>n. X n)"
-by (intro sums_unique [symmetric] sums_minus summable_sums)
-
-lemma sums_group:
-     "[|summable f; 0 < k |] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)"
-apply (drule summable_sums)
-apply (simp only: sums_def sumr_group)
-apply (unfold LIMSEQ_def, safe)
-apply (drule_tac x="r" in spec, safe)
-apply (rule_tac x="no" in exI, safe)
-apply (drule_tac x="n*k" in spec)
-apply (erule mp)
-apply (erule order_trans)
-apply simp
-done
-
-text{*A summable series of positive terms has limit that is at least as
-great as any partial sum.*}
-
-lemma series_pos_le:
-  fixes f :: "nat \<Rightarrow> real"
-  shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 \<le> f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} \<le> suminf f"
-apply (drule summable_sums)
-apply (simp add: sums_def)
-apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const)
-apply (erule LIMSEQ_le, blast)
-apply (rule_tac x="n" in exI, clarify)
-apply (rule setsum_mono2)
-apply auto
-done
-
-lemma series_pos_less:
-  fixes f :: "nat \<Rightarrow> real"
-  shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} < suminf f"
-apply (rule_tac y="setsum f {0..<Suc n}" in order_less_le_trans)
-apply simp
-apply (erule series_pos_le)
-apply (simp add: order_less_imp_le)
-done
-
-lemma suminf_gt_zero:
-  fixes f :: "nat \<Rightarrow> real"
-  shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f"
-by (drule_tac n="0" in series_pos_less, simp_all)
-
-lemma suminf_ge_zero:
-  fixes f :: "nat \<Rightarrow> real"
-  shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 \<le> suminf f"
-by (drule_tac n="0" in series_pos_le, simp_all)
-
-lemma sumr_pos_lt_pair:
-  fixes f :: "nat \<Rightarrow> real"
-  shows "\<lbrakk>summable f;
-        \<forall>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
-      \<Longrightarrow> setsum f {0..<k} < suminf f"
-apply (subst suminf_split_initial_segment [where k="k"])
-apply assumption
-apply simp
-apply (drule_tac k="k" in summable_ignore_initial_segment)
-apply (drule_tac k="Suc (Suc 0)" in sums_group, simp)
-apply simp
-apply (frule sums_unique)
-apply (drule sums_summable)
-apply simp
-apply (erule suminf_gt_zero)
-apply (simp add: add_ac)
-done
-
-text{*Sum of a geometric progression.*}
-
-lemmas sumr_geometric = geometric_sum [where 'a = real]
-
-lemma geometric_sums:
-  fixes x :: "'a::{real_normed_field,recpower}"
-  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) sums (1 / (1 - x))"
-proof -
-  assume less_1: "norm x < 1"
-  hence neq_1: "x \<noteq> 1" by auto
-  hence neq_0: "x - 1 \<noteq> 0" by simp
-  from less_1 have lim_0: "(\<lambda>n. x ^ n) ----> 0"
-    by (rule LIMSEQ_power_zero)
-  hence "(\<lambda>n. x ^ n / (x - 1) - 1 / (x - 1)) ----> 0 / (x - 1) - 1 / (x - 1)"
-    using neq_0 by (intro LIMSEQ_divide LIMSEQ_diff LIMSEQ_const)
-  hence "(\<lambda>n. (x ^ n - 1) / (x - 1)) ----> 1 / (1 - x)"
-    by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
-  thus "(\<lambda>n. x ^ n) sums (1 / (1 - x))"
-    by (simp add: sums_def geometric_sum neq_1)
-qed
-
-lemma summable_geometric:
-  fixes x :: "'a::{real_normed_field,recpower}"
-  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
-by (rule geometric_sums [THEN sums_summable])
-
-text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
-
-lemma summable_convergent_sumr_iff:
- "summable f = convergent (%n. setsum f {0..<n})"
-by (simp add: summable_def sums_def convergent_def)
-
-lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
-apply (drule summable_convergent_sumr_iff [THEN iffD1])
-apply (drule convergent_Cauchy)
-apply (simp only: Cauchy_def LIMSEQ_def, safe)
-apply (drule_tac x="r" in spec, safe)
-apply (rule_tac x="M" in exI, safe)
-apply (drule_tac x="Suc n" in spec, simp)
-apply (drule_tac x="n" in spec, simp)
-done
-
-lemma summable_Cauchy:
-     "summable (f::nat \<Rightarrow> 'a::banach) =  
-      (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. norm (setsum f {m..<n}) < e)"
-apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def, safe)
-apply (drule spec, drule (1) mp)
-apply (erule exE, rule_tac x="M" in exI, clarify)
-apply (rule_tac x="m" and y="n" in linorder_le_cases)
-apply (frule (1) order_trans)
-apply (drule_tac x="n" in spec, drule (1) mp)
-apply (drule_tac x="m" in spec, drule (1) mp)
-apply (simp add: setsum_diff [symmetric])
-apply simp
-apply (drule spec, drule (1) mp)
-apply (erule exE, rule_tac x="N" in exI, clarify)
-apply (rule_tac x="m" and y="n" in linorder_le_cases)
-apply (subst norm_minus_commute)
-apply (simp add: setsum_diff [symmetric])
-apply (simp add: setsum_diff [symmetric])
-done
-
-text{*Comparison test*}
-
-lemma norm_setsum:
-  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
-  shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
-apply (case_tac "finite A")
-apply (erule finite_induct)
-apply simp
-apply simp
-apply (erule order_trans [OF norm_triangle_ineq add_left_mono])
-apply simp
-done
-
-lemma summable_comparison_test:
-  fixes f :: "nat \<Rightarrow> 'a::banach"
-  shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f"
-apply (simp add: summable_Cauchy, safe)
-apply (drule_tac x="e" in spec, safe)
-apply (rule_tac x = "N + Na" in exI, safe)
-apply (rotate_tac 2)
-apply (drule_tac x = m in spec)
-apply (auto, rotate_tac 2, drule_tac x = n in spec)
-apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
-apply (rule norm_setsum)
-apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
-apply (auto intro: setsum_mono simp add: abs_less_iff)
-done
-
-lemma summable_norm_comparison_test:
-  fixes f :: "nat \<Rightarrow> 'a::banach"
-  shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk>
-         \<Longrightarrow> summable (\<lambda>n. norm (f n))"
-apply (rule summable_comparison_test)
-apply (auto)
-done
-
-lemma summable_rabs_comparison_test:
-  fixes f :: "nat \<Rightarrow> real"
-  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>)"
-apply (rule summable_comparison_test)
-apply (auto)
-done
-
-text{*Summability of geometric series for real algebras*}
-
-lemma complete_algebra_summable_geometric:
-  fixes x :: "'a::{real_normed_algebra_1,banach,recpower}"
-  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
-proof (rule summable_comparison_test)
-  show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
-    by (simp add: norm_power_ineq)
-  show "norm x < 1 \<Longrightarrow> summable (\<lambda>n. norm x ^ n)"
-    by (simp add: summable_geometric)
-qed
-
-text{*Limit comparison property for series (c.f. jrh)*}
-
-lemma summable_le:
-  fixes f g :: "nat \<Rightarrow> real"
-  shows "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
-apply (drule summable_sums)+
-apply (simp only: sums_def, erule (1) LIMSEQ_le)
-apply (rule exI)
-apply (auto intro!: setsum_mono)
-done
-
-lemma summable_le2:
-  fixes f g :: "nat \<Rightarrow> real"
-  shows "\<lbrakk>\<forall>n. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f \<and> suminf f \<le> suminf g"
-apply (subgoal_tac "summable f")
-apply (auto intro!: summable_le)
-apply (simp add: abs_le_iff)
-apply (rule_tac g="g" in summable_comparison_test, simp_all)
-done
-
-(* specialisation for the common 0 case *)
-lemma suminf_0_le:
-  fixes f::"nat\<Rightarrow>real"
-  assumes gt0: "\<forall>n. 0 \<le> f n" and sm: "summable f"
-  shows "0 \<le> suminf f"
-proof -
-  let ?g = "(\<lambda>n. (0::real))"
-  from gt0 have "\<forall>n. ?g n \<le> f n" by simp
-  moreover have "summable ?g" by (rule summable_zero)
-  moreover from sm have "summable f" .
-  ultimately have "suminf ?g \<le> suminf f" by (rule summable_le)
-  then show "0 \<le> suminf f" by (simp add: suminf_zero)
-qed 
-
-
-text{*Absolute convergence imples normal convergence*}
-lemma summable_norm_cancel:
-  fixes f :: "nat \<Rightarrow> 'a::banach"
-  shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
-apply (simp only: summable_Cauchy, safe)
-apply (drule_tac x="e" in spec, safe)
-apply (rule_tac x="N" in exI, safe)
-apply (drule_tac x="m" in spec, safe)
-apply (rule order_le_less_trans [OF norm_setsum])
-apply (rule order_le_less_trans [OF abs_ge_self])
-apply simp
-done
-
-lemma summable_rabs_cancel:
-  fixes f :: "nat \<Rightarrow> real"
-  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f"
-by (rule summable_norm_cancel, simp)
-
-text{*Absolute convergence of series*}
-lemma summable_norm:
-  fixes f :: "nat \<Rightarrow> 'a::banach"
-  shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
-by (auto intro: LIMSEQ_le LIMSEQ_norm summable_norm_cancel
-                summable_sumr_LIMSEQ_suminf norm_setsum)
-
-lemma summable_rabs:
-  fixes f :: "nat \<Rightarrow> real"
-  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
-by (fold real_norm_def, rule summable_norm)
-
-subsection{* The Ratio Test*}
-
-lemma norm_ratiotest_lemma:
-  fixes x y :: "'a::real_normed_vector"
-  shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0"
-apply (subgoal_tac "norm x \<le> 0", simp)
-apply (erule order_trans)
-apply (simp add: mult_le_0_iff)
-done
-
-lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
-by (erule norm_ratiotest_lemma, simp)
-
-lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"
-apply (drule le_imp_less_or_eq)
-apply (auto dest: less_imp_Suc_add)
-done
-
-lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"
-by (auto simp add: le_Suc_ex)
-
-(*All this trouble just to get 0<c *)
-lemma ratio_test_lemma2:
-  fixes f :: "nat \<Rightarrow> 'a::banach"
-  shows "\<lbrakk>\<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> 0 < c \<or> summable f"
-apply (simp (no_asm) add: linorder_not_le [symmetric])
-apply (simp add: summable_Cauchy)
-apply (safe, subgoal_tac "\<forall>n. N < n --> f (n) = 0")
- prefer 2
- apply clarify
- apply(erule_tac x = "n - 1" in allE)
- apply (simp add:diff_Suc split:nat.splits)
- apply (blast intro: norm_ratiotest_lemma)
-apply (rule_tac x = "Suc N" in exI, clarify)
-apply(simp cong:setsum_ivl_cong)
-done
-
-lemma ratio_test:
-  fixes f :: "nat \<Rightarrow> 'a::banach"
-  shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f"
-apply (frule ratio_test_lemma2, auto)
-apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n" 
-       in summable_comparison_test)
-apply (rule_tac x = N in exI, safe)
-apply (drule le_Suc_ex_iff [THEN iffD1])
-apply (auto simp add: power_add field_power_not_zero)
-apply (induct_tac "na", auto)
-apply (rule_tac y = "c * norm (f (N + n))" in order_trans)
-apply (auto intro: mult_right_mono simp add: summable_def)
-apply (simp add: mult_ac)
-apply (rule_tac x = "norm (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
-apply (rule sums_divide) 
-apply (rule sums_mult)
-apply (auto intro!: geometric_sums)
-done
-
-subsection {* Cauchy Product Formula *}
-
-(* Proof based on Analysis WebNotes: Chapter 07, Class 41
-http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm *)
-
-lemma setsum_triangle_reindex:
-  fixes n :: nat
-  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))"
-proof -
-  have "(\<Sum>(i, j)\<in>{(i, j). i + j < n}. f i j) =
-    (\<Sum>(k, i)\<in>(SIGMA k:{0..<n}. {0..k}). f i (k - i))"
-  proof (rule setsum_reindex_cong)
-    show "inj_on (\<lambda>(k,i). (i, k - i)) (SIGMA k:{0..<n}. {0..k})"
-      by (rule inj_on_inverseI [where g="\<lambda>(i,j). (i+j, i)"], auto)
-    show "{(i,j). i + j < n} = (\<lambda>(k,i). (i, k - i)) ` (SIGMA k:{0..<n}. {0..k})"
-      by (safe, rule_tac x="(a+b,a)" in image_eqI, auto)
-    show "\<And>a. (\<lambda>(k, i). f i (k - i)) a = split f ((\<lambda>(k, i). (i, k - i)) a)"
-      by clarify
-  qed
-  thus ?thesis by (simp add: setsum_Sigma)
-qed
-
-lemma Cauchy_product_sums:
-  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
-  assumes a: "summable (\<lambda>k. norm (a k))"
-  assumes b: "summable (\<lambda>k. norm (b k))"
-  shows "(\<lambda>k. \<Sum>i=0..k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
-proof -
-  let ?S1 = "\<lambda>n::nat. {0..<n} \<times> {0..<n}"
-  let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
-  have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
-  have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
-  have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
-  have finite_S1: "\<And>n. finite (?S1 n)" by simp
-  with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
-
-  let ?g = "\<lambda>(i,j). a i * b j"
-  let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
-  have f_nonneg: "\<And>x. 0 \<le> ?f x"
-    by (auto simp add: mult_nonneg_nonneg)
-  hence norm_setsum_f: "\<And>A. norm (setsum ?f A) = setsum ?f A"
-    unfolding real_norm_def
-    by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
-
-  have "(\<lambda>n. (\<Sum>k=0..<n. a k) * (\<Sum>k=0..<n. b k))
-           ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
-    by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf
-        summable_norm_cancel [OF a] summable_norm_cancel [OF b])
-  hence 1: "(\<lambda>n. setsum ?g (?S1 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
-    by (simp only: setsum_product setsum_Sigma [rule_format]
-                   finite_atLeastLessThan)
-
-  have "(\<lambda>n. (\<Sum>k=0..<n. norm (a k)) * (\<Sum>k=0..<n. norm (b k)))
-       ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
-    using a b by (intro LIMSEQ_mult summable_sumr_LIMSEQ_suminf)
-  hence "(\<lambda>n. setsum ?f (?S1 n)) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
-    by (simp only: setsum_product setsum_Sigma [rule_format]
-                   finite_atLeastLessThan)
-  hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
-    by (rule convergentI)
-  hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"
-    by (rule convergent_Cauchy)
-  have "Zseq (\<lambda>n. setsum ?f (?S1 n - ?S2 n))"
-  proof (rule ZseqI, simp only: norm_setsum_f)
-    fix r :: real
-    assume r: "0 < r"
-    from CauchyD [OF Cauchy r] obtain N
-    where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (setsum ?f (?S1 m) - setsum ?f (?S1 n)) < r" ..
-    hence "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> norm (setsum ?f (?S1 m - ?S1 n)) < r"
-      by (simp only: setsum_diff finite_S1 S1_mono)
-    hence N: "\<And>m n. \<lbrakk>N \<le> n; n \<le> m\<rbrakk> \<Longrightarrow> setsum ?f (?S1 m - ?S1 n) < r"
-      by (simp only: norm_setsum_f)
-    show "\<exists>N. \<forall>n\<ge>N. setsum ?f (?S1 n - ?S2 n) < r"
-    proof (intro exI allI impI)
-      fix n assume "2 * N \<le> n"
-      hence n: "N \<le> n div 2" by simp
-      have "setsum ?f (?S1 n - ?S2 n) \<le> setsum ?f (?S1 n - ?S1 (n div 2))"
-        by (intro setsum_mono2 finite_Diff finite_S1 f_nonneg
-                  Diff_mono subset_refl S1_le_S2)
-      also have "\<dots> < r"
-        using n div_le_dividend by (rule N)
-      finally show "setsum ?f (?S1 n - ?S2 n) < r" .
-    qed
-  qed
-  hence "Zseq (\<lambda>n. setsum ?g (?S1 n - ?S2 n))"
-    apply (rule Zseq_le [rule_format])
-    apply (simp only: norm_setsum_f)
-    apply (rule order_trans [OF norm_setsum setsum_mono])
-    apply (auto simp add: norm_mult_ineq)
-    done
-  hence 2: "(\<lambda>n. setsum ?g (?S1 n) - setsum ?g (?S2 n)) ----> 0"
-    by (simp only: LIMSEQ_Zseq_iff setsum_diff finite_S1 S2_le_S1 diff_0_right)
-
-  with 1 have "(\<lambda>n. setsum ?g (?S2 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
-    by (rule LIMSEQ_diff_approach_zero2)
-  thus ?thesis by (simp only: sums_def setsum_triangle_reindex)
-qed
-
-lemma Cauchy_product:
-  fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
-  assumes a: "summable (\<lambda>k. norm (a k))"
-  assumes b: "summable (\<lambda>k. norm (b k))"
-  shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i=0..k. a i * b (k - i))"
-using a b
-by (rule Cauchy_product_sums [THEN sums_unique])
-
-end