src/HOL/Series.thy

 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*}
+header {* Finite Summation and Infinite Series *}
 
 theory Series
 imports Limits
 begin
 
+(* TODO: MOVE *)
+lemma Suc_less_iff: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
+  by (cases m) auto
+
+(* TODO: MOVE *)
+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
+
+(* TODO: MOVE *)
+lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
+by (erule norm_ratiotest_lemma, simp)
+
+(* TODO: MOVE *)
+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
+
+(* MOVE *)
+lemma setsum_even_minus_one [simp]: "(\<Sum>i<2 * n. (-1) ^ Suc i) = (0::'a::ring_1)"
+  by (induct "n") auto
+
+(* MOVE *)
+lemma setsum_nat_group: "(\<Sum>m<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {..< n * k}"
+  apply (subgoal_tac "k = 0 | 0 < k", auto)
+  apply (induct "n")
+  apply (simp_all add: setsum_add_nat_ivl add_commute atLeast0LessThan[symmetric])
+  done
+
+(* MOVE *)
+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
+
+(* MOVE *)
+lemma norm_bound_subset:
+  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+  assumes "finite s" "t \<subseteq> s"
+  assumes le: "(\<And>x. x \<in> s \<Longrightarrow> norm(f x) \<le> g x)"
+  shows "norm (setsum f t) \<le> setsum g s"
+proof -
+  have "norm (setsum f t) \<le> (\<Sum>i\<in>t. norm (f i))"
+    by (rule norm_setsum)
+  also have "\<dots> \<le> (\<Sum>i\<in>t. g i)"
+    using assms by (auto intro!: setsum_mono)
+  also have "\<dots> \<le> setsum g s"
+    using assms order.trans[OF norm_ge_zero le]
+    by (auto intro!: setsum_mono3)
+  finally show ?thesis .
+qed
+
+(* MOVE *)
+lemma (in linorder) lessThan_minus_lessThan [simp]:
+  "{..< n} - {..< m} = {m ..< n}"
+  by auto
+
 definition
-   sums  :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
-     (infixr "sums" 80) where
-   "f sums s = (%n. setsum f {0..<n}) ----> s"
+  sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
+  (infixr "sums" 80)
+where
+  "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) ----> s"
+
+definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where
+   "summable f \<longleftrightarrow> (\<exists>s. f sums s)"
 
 definition
-   summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where
-   "summable f = (\<exists>s. f sums s)"
-
-definition
-   suminf   :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a" where
-   "suminf f = (THE s. f sums s)"
-
-notation suminf (binder "\<Sum>" 10)
-
-
-lemma [trans]: "f=g ==> g sums z ==> f sums z"
+  suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a"
+  (binder "\<Sum>" 10)
+where
+  "suminf f = (THE s. f sums s)"
+
+lemma sums_subst[trans]: "f = g \<Longrightarrow> g sums z \<Longrightarrow> f sums z"
   by simp
 
-lemma sumr_diff_mult_const:
- "setsum f {0..<n} - (real n*r) = setsum (%i. f i - r) {0..<n::nat}"
-  by (simp add: setsum_subtractf 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)
-
-subsection{* Infinite Sums, by the Properties of Limits*}
-
-(*----------------------
-   suminf is the sum
- ---------------------*)
-lemma sums_summable: "f sums l ==> summable f"
+lemma sums_summable: "f sums l \<Longrightarrow> summable f"
   by (simp add: sums_def summable_def, blast)
 
-lemma summable_sums:
-  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
-  assumes "summable f"
-  shows "f sums (suminf f)"
-proof -
-  from assms obtain s where s: "(\<lambda>n. setsum f {0..<n}) ----> s"
-    unfolding summable_def sums_def [abs_def] ..
-  then show ?thesis unfolding sums_def [abs_def] suminf_def
-    by (rule theI, auto intro!: tendsto_unique[OF trivial_limit_sequentially])
-qed
-
-lemma summable_sumr_LIMSEQ_suminf:
-  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
-  shows "summable f \<Longrightarrow> (\<lambda>n. setsum f {0..<n}) ----> suminf f"
-by (rule summable_sums [unfolded sums_def])
-
-lemma suminf_eq_lim: "suminf f = lim (%n. setsum f {0..<n})"
+lemma summable_iff_convergent: "summable f \<longleftrightarrow> convergent (\<lambda>n. \<Sum>i<n. f i)"
+  by (simp add: summable_def sums_def convergent_def)
+
+lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)"
   by (simp add: suminf_def sums_def lim_def)
 
-(*-------------------
-    sum is unique
- ------------------*)
-lemma sums_unique:
-  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
-  shows "f sums s \<Longrightarrow> (s = suminf f)"
-apply (frule sums_summable[THEN summable_sums])
-apply (auto intro!: tendsto_unique[OF trivial_limit_sequentially] simp add: sums_def)
-done
-
-lemma sums_iff:
-  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
-  shows "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
-  by (metis summable_sums sums_summable sums_unique)
-
 lemma sums_finite:
-  assumes [simp]: "finite N"
-  assumes f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
+  assumes [simp]: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
   shows "f sums (\<Sum>n\<in>N. f n)"
 proof -
   { fix n
     have "setsum f {..<n + Suc (Max N)} = setsum f N"
     proof cases

   show ?thesis unfolding sums_def
     by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
        (simp add: eq atLeast0LessThan tendsto_const del: add_Suc_right)
 qed
 
+lemma sums_If_finite_set: "finite A \<Longrightarrow> (\<lambda>r. if r \<in> A then f r else 0) sums (\<Sum>r\<in>A. f r)"
+  using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simp
+
+lemma sums_If_finite: "finite {r. P r} \<Longrightarrow> (\<lambda>r. if P r then f r else 0) sums (\<Sum>r | P r. f r)"
+  using sums_If_finite_set[of "{r. P r}"] by simp
+
+lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i"
+  using sums_If_finite[of "\<lambda>r. r = i"] by simp
+
+lemma series_zero: (* REMOVE *)
+  "(\<And>m. n \<le> m \<Longrightarrow> f m = 0) \<Longrightarrow> f sums (\<Sum>i<n. f i)"
+  by (rule sums_finite) auto
+
+lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0"
+  unfolding sums_def by (simp add: tendsto_const)
+
+lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)"
+  by (rule sums_zero [THEN sums_summable])
+
+lemma sums_group: "f sums s \<Longrightarrow> 0 < k \<Longrightarrow> (\<lambda>n. setsum f {n * k ..< n * k + k}) sums s"
+  apply (simp only: sums_def setsum_nat_group tendsto_def eventually_sequentially)
+  apply safe
+  apply (erule_tac x=S in allE)
+  apply safe
+  apply (rule_tac x="N" in exI, safe)
+  apply (drule_tac x="n*k" in spec)
+  apply (erule mp)
+  apply (erule order_trans)
+  apply simp
+  done
+
+context
+  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
+begin
+
+lemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)"
+  by (simp add: summable_def sums_def suminf_def)
+     (metis convergent_LIMSEQ_iff convergent_def lim_def)
+
+lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) ----> suminf f"
+  by (rule summable_sums [unfolded sums_def])
+
+lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f"
+  by (metis limI suminf_eq_lim sums_def)
+
+lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
+  by (metis summable_sums sums_summable sums_unique)
+
 lemma suminf_finite:
-  fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add,t2_space}"
   assumes N: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
   shows "suminf f = (\<Sum>n\<in>N. f n)"
   using sums_finite[OF assms, THEN sums_unique] by simp
 
-lemma sums_If_finite_set:
-  "finite A \<Longrightarrow> (\<lambda>r. if r \<in> A then f r else 0 :: 'a::{comm_monoid_add,t2_space}) sums (\<Sum>r\<in>A. f r)"
-  using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simp
-
-lemma sums_If_finite:
-  "finite {r. P r} \<Longrightarrow> (\<lambda>r. if P r then f r else 0 :: 'a::{comm_monoid_add,t2_space}) sums (\<Sum>r | P r. f r)"
-  using sums_If_finite_set[of "{r. P r}" f] by simp
-
-lemma sums_single:
-  "(\<lambda>r. if r = i then f r else 0::'a::{comm_monoid_add,t2_space}) sums f i"
-  using sums_If_finite[of "\<lambda>r. r = i" f] by simp
-
-lemma sums_split_initial_segment:
-  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
-  shows "f sums s ==> (\<lambda>n. f(n + k)) sums (s - (SUM i = 0..< k. f i))"
-  apply (unfold sums_def)
-  apply (simp add: sumr_offset)
-  apply (rule tendsto_diff [OF _ tendsto_const])
-  apply (rule LIMSEQ_ignore_initial_segment)
-  apply assumption
-done
-
-lemma summable_ignore_initial_segment:
-  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
-  shows "summable f ==> summable (%n. f(n + k))"
-  apply (unfold summable_def)
-  apply (auto intro: sums_split_initial_segment)
-done
-
-lemma suminf_minus_initial_segment:
-  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
-  shows "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:
-  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
-  shows "summable f ==>
-    suminf f = (SUM i = 0..< k. f i) + (\<Sum>n. f(n + k))"
-by (auto simp add: suminf_minus_initial_segment)
-
-lemma suminf_exist_split: fixes r :: real assumes "0 < r" and "summable a"
-  shows "\<exists> N. \<forall> n \<ge> N. \<bar> \<Sum> i. a (i + n) \<bar> < r"
-proof -
-  from LIMSEQ_D[OF summable_sumr_LIMSEQ_suminf[OF `summable a`] `0 < r`]
-  obtain N :: nat where "\<forall> n \<ge> N. norm (setsum a {0..<n} - suminf a) < r" by auto
-  thus ?thesis unfolding suminf_minus_initial_segment[OF `summable a` refl] abs_minus_commute real_norm_def
-    by auto
-qed
-
-lemma sums_Suc:
-  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
-  assumes sumSuc: "(\<lambda> n. f (Suc n)) sums l" shows "f sums (l + f 0)"
-proof -
-  from sumSuc[unfolded sums_def]
-  have "(\<lambda>i. \<Sum>n = Suc 0..<Suc i. f n) ----> l" unfolding setsum_reindex[OF inj_Suc] image_Suc_atLeastLessThan[symmetric] comp_def .
-  from tendsto_add[OF this tendsto_const, where b="f 0"]
-  have "(\<lambda>i. \<Sum>n = 0..<Suc i. f n) ----> l + f 0" unfolding add_commute setsum_head_upt_Suc[OF zero_less_Suc] .
-  thus ?thesis unfolding sums_def by (rule LIMSEQ_imp_Suc)
-qed
-
-lemma series_zero:
-  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
-  assumes "\<forall>m. n \<le> m \<longrightarrow> f m = 0"
-  shows "f sums (setsum f {0..<n})"
-proof -
-  { fix k :: nat have "setsum f {0..<k + n} = setsum f {0..<n}"
-      using assms by (induct k) auto }
-  note setsum_const = this
-  show ?thesis
-    unfolding sums_def
-    apply (rule LIMSEQ_offset[of _ n])
-    unfolding setsum_const
-    apply (rule tendsto_const)
-    done
-qed
-
-lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0"
-  unfolding sums_def by (simp add: tendsto_const)
-
-lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)"
-by (rule sums_zero [THEN sums_summable])
+end
 
 lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 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 tendsto, 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)
-
-lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
-lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
-lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
-
-lemma sums_mult:
-  fixes c :: "'a::real_normed_algebra"
-  shows "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
-  by (rule bounded_linear.sums [OF bounded_linear_mult_right])
-
-lemma summable_mult:
-  fixes c :: "'a::real_normed_algebra"
-  shows "summable f \<Longrightarrow> summable (%n. c * f n)"
-  by (rule bounded_linear.summable [OF bounded_linear_mult_right])
-
-lemma suminf_mult:
-  fixes c :: "'a::real_normed_algebra"
-  shows "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
-  by (rule bounded_linear.suminf [OF bounded_linear_mult_right, 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 bounded_linear.sums [OF bounded_linear_mult_left])
-
-lemma summable_mult2:
-  fixes c :: "'a::real_normed_algebra"
-  shows "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
-  by (rule bounded_linear.summable [OF bounded_linear_mult_left])
-
-lemma suminf_mult2:
-  fixes c :: "'a::real_normed_algebra"
-  shows "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
-  by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
-
-lemma sums_divide:
-  fixes c :: "'a::real_normed_field"
-  shows "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
-  by (rule bounded_linear.sums [OF bounded_linear_divide])
-
-lemma summable_divide:
-  fixes c :: "'a::real_normed_field"
-  shows "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
-  by (rule bounded_linear.summable [OF bounded_linear_divide])
-
-lemma suminf_divide:
-  fixes c :: "'a::real_normed_field"
-  shows "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
-  by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
-
-lemma sums_add:
-  fixes a b :: "'a::real_normed_field"
-  shows "\<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 tendsto_add)
-
-lemma summable_add:
-  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
-  shows "\<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:
-  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
-  shows "\<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:
-  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
-  shows "\<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 tendsto_diff)
-
-lemma summable_diff:
-  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
-  shows "\<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:
-  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
-  shows "\<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:
-  fixes X :: "nat \<Rightarrow> 'a::real_normed_field"
-  shows "X sums a ==> (\<lambda>n. - X n) sums (- a)"
-  unfolding sums_def by (simp add: setsum_negf tendsto_minus)
-
-lemma summable_minus:
-  fixes X :: "nat \<Rightarrow> 'a::real_normed_field"
-  shows "summable X \<Longrightarrow> summable (\<lambda>n. - X n)"
-unfolding summable_def by (auto intro: sums_minus)
-
-lemma suminf_minus:
-  fixes X :: "nat \<Rightarrow> 'a::real_normed_field"
-  shows "summable X \<Longrightarrow> (\<Sum>n. - X n) = - (\<Sum>n. X n)"
-by (intro sums_unique [symmetric] sums_minus summable_sums)
-
-lemma sums_group:
-  fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
-  shows "\<lbrakk>f sums s; 0 < k\<rbrakk> \<Longrightarrow> (\<lambda>n. setsum f {n*k..<n*k+k}) sums s"
-apply (simp only: sums_def sumr_group)
-apply (unfold LIMSEQ_iff, 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 pos_summable:
-  fixes f:: "nat \<Rightarrow> real"
-  assumes pos: "\<And>n. 0 \<le> f n" and le: "\<And>n. setsum f {0..<n} \<le> x"
-  shows "summable f"
-proof -
-  have "convergent (\<lambda>n. setsum f {0..<n})"
-    proof (rule Bseq_mono_convergent)
-      show "Bseq (\<lambda>n. setsum f {0..<n})"
-        by (intro BseqI'[of _ x]) (auto simp add: setsum_nonneg pos intro: le)
-    next
-      show "\<forall>m n. m \<le> n \<longrightarrow> setsum f {0..<m} \<le> setsum f {0..<n}"
-        by (auto intro: setsum_mono2 pos)
-    qed
-  thus ?thesis
-    by (force simp add: summable_def sums_def convergent_def)
-qed
-
-lemma series_pos_le:
+  by (rule sums_zero [THEN sums_unique, symmetric])
+
+context
   fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
-  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 (rule LIMSEQ_le_const)
+begin
+
+lemma series_pos_le: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 \<le> f m \<Longrightarrow> setsum f {..<n} \<le> suminf f"
+  apply (rule LIMSEQ_le_const[OF summable_LIMSEQ])
   apply assumption
   apply (intro exI[of _ n])
-  apply (auto intro!: setsum_mono2)
-  done
-
-lemma series_pos_less:
-  fixes f :: "nat \<Rightarrow> 'a::{ordered_ab_semigroup_add_imp_le, ordered_comm_monoid_add, linorder_topology}"
-  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)
-  using add_less_cancel_left [of "setsum f {0..<n}" 0 "f n"]
-  apply simp
-  apply (erule series_pos_le)
-  apply (simp add: order_less_imp_le)
-  done
-
-lemma suminf_eq_zero_iff:
-  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
-  shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
+  apply (auto intro!: setsum_mono2 simp: not_le[symmetric])
+  done
+
+lemma suminf_eq_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
 proof
   assume "summable f" "suminf f = 0" and pos: "\<forall>n. 0 \<le> f n"
   then have "f sums 0"
     by (simp add: sums_iff)
   then have f: "(\<lambda>n. \<Sum>i<n. f i) ----> 0"

     fix i show "\<exists>N. \<forall>n\<ge>N. (\<Sum>n\<in>{i}. f n) \<le> setsum f {..<n}"
       using pos by (intro exI[of _ "Suc i"] allI impI setsum_mono2) auto
   qed
   with pos show "\<forall>n. f n = 0"
     by (auto intro!: antisym)
-next
-  assume "\<forall>n. f n = 0"
-  then have "f = (\<lambda>n. 0)"
-    by auto
-  then show "suminf f = 0"
-    by simp
-qed
-
-lemma suminf_gt_zero_iff:
-  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
-  shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)"
-  using series_pos_le[of f 0] suminf_eq_zero_iff[of f]
-  by (simp add: less_le)
-
-lemma suminf_gt_zero:
-  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
-  shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f"
-  using suminf_gt_zero_iff[of f] by (simp add: less_imp_le)
-
-lemma suminf_ge_zero:
-  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
-  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;
-        \<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
-      \<Longrightarrow> setsum f {0..<k} < suminf f"
-unfolding One_nat_def
-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 [OF summable_sums], simp)
-apply simp
-apply (frule sums_unique)
-apply (drule sums_summable)
-apply simp
-apply (erule suminf_gt_zero)
-apply (simp add: add_ac)
-done
+qed (metis suminf_zero fun_eq_iff)
+
+lemma suminf_gt_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)"
+  using series_pos_le[of 0] suminf_eq_zero_iff by (simp add: less_le)
+
+lemma suminf_gt_zero: "summable f \<Longrightarrow> \<forall>n. 0 < f n \<Longrightarrow> 0 < suminf f"
+  using suminf_gt_zero_iff by (simp add: less_imp_le)
+
+lemma suminf_ge_zero: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 \<le> suminf f"
+  by (drule_tac n="0" in series_pos_le) simp_all
+
+lemma suminf_le: "summable f \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
+  by (metis LIMSEQ_le_const2 summable_LIMSEQ)
+
+lemma summable_le: "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
+  by (rule LIMSEQ_le) (auto intro: setsum_mono summable_LIMSEQ)
+
+end
+
+lemma series_pos_less:
+  fixes f :: "nat \<Rightarrow> 'a::{ordered_ab_semigroup_add_imp_le, ordered_comm_monoid_add, linorder_topology}"
+  shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {..<n} < suminf f"
+  apply simp
+  apply (rule_tac y="setsum f {..<Suc n}" in order_less_le_trans)
+  using add_less_cancel_left [of "setsum f {..<n}" 0 "f n"]
+  apply simp
+  apply (erule series_pos_le)
+  apply (simp add: order_less_imp_le)
+  done
+
+lemma sums_Suc_iff:
+  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+  shows "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
+proof -
+  have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) ----> s + f 0"
+    by (subst LIMSEQ_Suc_iff) (simp add: sums_def)
+  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
+    by (simp add: ac_simps setsum_reindex image_iff lessThan_Suc_eq_insert_0)
+  also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
+  proof
+    assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
+    with tendsto_add[OF this tendsto_const, of "- f 0"]
+    show "(\<lambda>i. f (Suc i)) sums s"
+      by (simp add: sums_def)
+  qed (auto intro: tendsto_add tendsto_const simp: sums_def)
+  finally show ?thesis ..
+qed
+
+context
+  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+begin
+
+lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
+  unfolding sums_def by (simp add: setsum_addf tendsto_add)
+
+lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)"
+  unfolding summable_def by (auto intro: sums_add)
+
+lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)"
+  by (intro sums_unique sums_add summable_sums)
+
+lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)"
+  unfolding sums_def by (simp add: setsum_subtractf tendsto_diff)
+
+lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)"
+  unfolding summable_def by (auto intro: sums_diff)
+
+lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)"
+  by (intro sums_unique sums_diff summable_sums)
+
+lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)"
+  unfolding sums_def by (simp add: setsum_negf tendsto_minus)
+
+lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)"
+  unfolding summable_def by (auto intro: sums_minus)
+
+lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)"
+  by (intro sums_unique [symmetric] sums_minus summable_sums)
+
+lemma sums_Suc: "(\<lambda> n. f (Suc n)) sums l \<Longrightarrow> f sums (l + f 0)"
+  by (simp add: sums_Suc_iff)
+
+lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"
+proof (induct n arbitrary: s)
+  case (Suc n)
+  moreover have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)"
+    by (subst sums_Suc_iff) simp
+  ultimately show ?case
+    by (simp add: ac_simps)
+qed simp
+
+lemma summable_iff_shift: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f"
+  by (metis diff_add_cancel summable_def sums_iff_shift[abs_def])
+
+lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))"
+  by (simp add: sums_iff_shift)
+
+lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))"
+  by (simp add: summable_iff_shift)
+
+lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)"
+  by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
+
+lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)"
+  by (auto simp add: suminf_minus_initial_segment)
+
+lemma suminf_exist_split: 
+  fixes r :: real assumes "0 < r" and "summable f"
+  shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
+proof -
+  from LIMSEQ_D[OF summable_LIMSEQ[OF `summable f`] `0 < r`]
+  obtain N :: nat where "\<forall> n \<ge> N. norm (setsum f {..<n} - suminf f) < r" by auto
+  thus ?thesis
+    by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF `summable f`])
+qed
+
+lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
+  apply (drule summable_iff_convergent [THEN iffD1])
+  apply (drule convergent_Cauchy)
+  apply (simp only: Cauchy_iff LIMSEQ_iff, 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
+
+end
+
+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 tendsto, 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)
+
+lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
+lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
+lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
+
+context
+  fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"
+begin
+
+lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
+  by (rule bounded_linear.sums [OF bounded_linear_mult_right])
+
+lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)"
+  by (rule bounded_linear.summable [OF bounded_linear_mult_right])
+
+lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
+  by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
+
+lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
+  by (rule bounded_linear.sums [OF bounded_linear_mult_left])
+
+lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
+  by (rule bounded_linear.summable [OF bounded_linear_mult_left])
+
+lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
+  by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
+
+end
+
+context
+  fixes c :: "'a::real_normed_field"
+begin
+
+lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
+  by (rule bounded_linear.sums [OF bounded_linear_divide])
+
+lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
+  by (rule bounded_linear.summable [OF bounded_linear_divide])
+
+lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
+  by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
 
 text{*Sum of a geometric progression.*}
 
-lemmas sumr_geometric = geometric_sum [where 'a = real]
-
-lemma geometric_sums:
-  fixes x :: "'a::{real_normed_field}"
-  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"
+lemma geometric_sums: "norm c < 1 \<Longrightarrow> (\<lambda>n. c^n) sums (1 / (1 - c))"
+proof -
+  assume less_1: "norm c < 1"
+  hence neq_1: "c \<noteq> 1" by auto
+  hence neq_0: "c - 1 \<noteq> 0" by simp
+  from less_1 have lim_0: "(\<lambda>n. c^n) ----> 0"
     by (rule LIMSEQ_power_zero)
-  hence "(\<lambda>n. x ^ n / (x - 1) - 1 / (x - 1)) ----> 0 / (x - 1) - 1 / (x - 1)"
+  hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) ----> 0 / (c - 1) - 1 / (c - 1)"
     using neq_0 by (intro tendsto_intros)
-  hence "(\<lambda>n. (x ^ n - 1) / (x - 1)) ----> 1 / (1 - x)"
+  hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) ----> 1 / (1 - c)"
     by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
-  thus "(\<lambda>n. x ^ n) sums (1 / (1 - x))"
+  thus "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
     by (simp add: sums_def geometric_sum neq_1)
 qed
 
-lemma summable_geometric:
-  fixes x :: "'a::{real_normed_field}"
-  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
-by (rule geometric_sums [THEN sums_summable])
-
-lemma half: "0 < 1 / (2::'a::linordered_field)"
-  by simp
+lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
+  by (rule geometric_sums [THEN sums_summable])
+
+lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)"
+  by (rule sums_unique[symmetric]) (rule geometric_sums)
+
+end
 
 lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
 proof -
   have 2: "(\<lambda>n. (1/2::real)^n) sums 2" using geometric_sums [of "1/2::real"]
     by auto

     by simp
 qed
 
 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:
-  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
-  shows "summable f \<Longrightarrow> f ----> 0"
-apply (drule summable_convergent_sumr_iff [THEN iffD1])
-apply (drule convergent_Cauchy)
-apply (simp only: Cauchy_iff LIMSEQ_iff, 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 suminf_le:
-  fixes x :: "'a :: {ordered_comm_monoid_add, linorder_topology}"
-  shows "summable f \<Longrightarrow> (!!n. setsum f {0..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
-  apply (drule summable_sums)
-  apply (simp add: sums_def)
-  apply (rule LIMSEQ_le_const2)
-  apply assumption
-  apply auto
-  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_iff, 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 norm_bound_subset:
-  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
-  assumes "finite s" "t \<subseteq> s"
-  assumes le: "(\<And>x. x \<in> s \<Longrightarrow> norm(f x) \<le> g x)"
-  shows "norm (setsum f t) \<le> setsum g s"
-proof -
-  have "norm (setsum f t) \<le> (\<Sum>i\<in>t. norm (f i))"
-    by (rule norm_setsum)
-  also have "\<dots> \<le> (\<Sum>i\<in>t. g i)"
-    using assms by (auto intro!: setsum_mono)
-  also have "\<dots> \<le> setsum g s"
-    using assms order.trans[OF norm_ge_zero le]
-    by (auto intro!: setsum_mono3)
-  finally show ?thesis .
-qed
-
-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
+  shows "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (setsum f {m..<n}) < e)"
+  apply (simp only: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff, 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_all add: setsum_diff [symmetric])
+  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_all add: setsum_diff [symmetric])
+  done
+
+context
+  fixes f :: "nat \<Rightarrow> 'a::banach"
+begin  
+
+text{*Absolute convergence imples normal convergence*}
+
+lemma summable_norm_cancel:
+  "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_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
+  by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_setsum)
+
+text {* Comparison tests *}
+
+lemma summable_comparison_test: "\<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
+
+subsection {* The Ratio Test*}
+
+lemma summable_ratio_test: 
+  assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
+  shows "summable f"
+proof cases
+  assume "0 < c"
+  show "summable f"
+  proof (rule summable_comparison_test)
+    show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
+    proof (intro exI allI impI)
+      fix n assume "N \<le> n" then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
+      proof (induct rule: inc_induct)
+        case (step m)
+        moreover have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
+          using `0 < c` `c < 1` assms(2)[OF `N \<le> m`] by (simp add: field_simps)
+        ultimately show ?case by simp
+      qed (insert `0 < c`, simp)
+    qed
+    show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
+      using `0 < c` `c < 1` by (intro summable_mult summable_geometric) simp
+  qed
+next
+  assume c: "\<not> 0 < c"
+  { fix n assume "n \<ge> N"
+    then have "norm (f (Suc n)) \<le> c * norm (f n)"
+      by fact
+    also have "\<dots> \<le> 0"
+      using c by (simp add: not_less mult_nonpos_nonneg)
+    finally have "f (Suc n) = 0"
+      by auto }
+  then show "summable f"
+    by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_iff)
+qed
+
+end
 
 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:
+  "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable (\<lambda>n. norm (f n))"
+  by (rule summable_comparison_test) auto
+
+lemma summable_rabs_cancel:
   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
+  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f"
+  by (rule summable_norm_cancel) simp
 
 text{*Summability of geometric series for real algebras*}
 
 lemma complete_algebra_summable_geometric:
   fixes x :: "'a::{real_normed_algebra_1,banach}"

     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> 'a::{ordered_comm_monoid_add, linorder_topology}"
-  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"
-  using suminf_ge_zero[OF sm gt0] by simp
-
-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:
+
+text{*A summable series of positive terms has limit that is at least as
+great as any partial sum.*}
+
+lemma pos_summable:
+  fixes f:: "nat \<Rightarrow> real"
+  assumes pos: "\<And>n. 0 \<le> f n" and le: "\<And>n. setsum f {..<n} \<le> x"
+  shows "summable f"
+proof -
+  have "convergent (\<lambda>n. setsum f {..<n})"
+  proof (rule Bseq_mono_convergent)
+    show "Bseq (\<lambda>n. setsum f {..<n})"
+      by (intro BseqI'[of _ x]) (auto simp add: setsum_nonneg pos intro: le)
+  qed (auto intro: setsum_mono2 pos)
+  thus ?thesis
+    by (force simp add: summable_def sums_def convergent_def)
+qed
+
+lemma summable_rabs_comparison_test:
   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 tendsto_norm summable_norm_cancel
-                summable_sumr_LIMSEQ_suminf norm_setsum)
+  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>)"
+  by (rule summable_comparison_test) auto
 
 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)
-
-(* TODO: MOVE *)
-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 - Suc 0" 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 del: setsum_cong 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 (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 *}
 
 text {*
   Proof based on Analysis WebNotes: Chapter 07, Class 41
   @{url "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))"
+  shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k<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))"
+    (\<Sum>(k, i)\<in>(SIGMA k:{..<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})"
+    show "inj_on (\<lambda>(k,i). (i, k - i)) (SIGMA k:{..<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})"
+    show "{(i,j). i + j < n} = (\<lambda>(k,i). (i, k - i)) ` (SIGMA k:{..<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)

   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 ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<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

     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 tendsto_mult summable_sumr_LIMSEQ_suminf
-        summable_norm_cancel [OF a] summable_norm_cancel [OF b])
+  have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
+    by (intro tendsto_mult summable_LIMSEQ 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 tendsto_mult summable_sumr_LIMSEQ_suminf)
+    by (simp only: setsum_product setsum_Sigma [rule_format] finite_lessThan)
+
+  have "(\<lambda>n. (\<Sum>k<n. norm (a k)) * (\<Sum>k<n. norm (b k))) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
+    using a b by (intro tendsto_mult summable_LIMSEQ)
   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)
+    by (simp only: setsum_product setsum_Sigma [rule_format] finite_lessThan)
   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 "Zfun (\<lambda>n. setsum ?f (?S1 n - ?S2 n)) sequentially"