src/HOL/Series.thy
changeset 56193 c726ecfb22b6
parent 56178 2a6f58938573
child 56194 9ffbb4004c81
     1.1 --- a/src/HOL/Series.thy	Tue Mar 18 14:32:23 2014 +0100
     1.2 +++ b/src/HOL/Series.thy	Tue Mar 18 15:53:48 2014 +0100
     1.3 @@ -7,122 +7,109 @@
     1.4  Additional contributions by Jeremy Avigad
     1.5  *)
     1.6  
     1.7 -header{*Finite Summation and Infinite Series*}
     1.8 +header {* Finite Summation and Infinite Series *}
     1.9  
    1.10  theory Series
    1.11  imports Limits
    1.12  begin
    1.13  
    1.14 -definition
    1.15 -   sums  :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
    1.16 -     (infixr "sums" 80) where
    1.17 -   "f sums s = (%n. setsum f {0..<n}) ----> s"
    1.18 -
    1.19 -definition
    1.20 -   summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where
    1.21 -   "summable f = (\<exists>s. f sums s)"
    1.22 -
    1.23 -definition
    1.24 -   suminf   :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a" where
    1.25 -   "suminf f = (THE s. f sums s)"
    1.26 -
    1.27 -notation suminf (binder "\<Sum>" 10)
    1.28 -
    1.29 -
    1.30 -lemma [trans]: "f=g ==> g sums z ==> f sums z"
    1.31 -  by simp
    1.32 +(* TODO: MOVE *)
    1.33 +lemma Suc_less_iff: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
    1.34 +  by (cases m) auto
    1.35  
    1.36 -lemma sumr_diff_mult_const:
    1.37 - "setsum f {0..<n} - (real n*r) = setsum (%i. f i - r) {0..<n::nat}"
    1.38 -  by (simp add: setsum_subtractf real_of_nat_def)
    1.39 -
    1.40 -lemma real_setsum_nat_ivl_bounded:
    1.41 -     "(!!p. p < n \<Longrightarrow> f(p) \<le> K)
    1.42 -      \<Longrightarrow> setsum f {0..<n::nat} \<le> real n * K"
    1.43 -using setsum_bounded[where A = "{0..<n}"]
    1.44 -by (auto simp:real_of_nat_def)
    1.45 -
    1.46 -(* Generalize from real to some algebraic structure? *)
    1.47 -lemma sumr_minus_one_realpow_zero [simp]:
    1.48 -  "(\<Sum>i=0..<2*n. (-1) ^ Suc i) = (0::real)"
    1.49 -by (induct "n", auto)
    1.50 -
    1.51 -(* FIXME this is an awful lemma! *)
    1.52 -lemma sumr_one_lb_realpow_zero [simp]:
    1.53 -  "(\<Sum>n=Suc 0..<n. f(n) * (0::real) ^ n) = 0"
    1.54 -by (rule setsum_0', simp)
    1.55 -
    1.56 -lemma sumr_group:
    1.57 -     "(\<Sum>m=0..<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {0 ..< n * k}"
    1.58 -apply (subgoal_tac "k = 0 | 0 < k", auto)
    1.59 -apply (induct "n")
    1.60 -apply (simp_all add: setsum_add_nat_ivl add_commute)
    1.61 +(* TODO: MOVE *)
    1.62 +lemma norm_ratiotest_lemma:
    1.63 +  fixes x y :: "'a::real_normed_vector"
    1.64 +  shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0"
    1.65 +apply (subgoal_tac "norm x \<le> 0", simp)
    1.66 +apply (erule order_trans)
    1.67 +apply (simp add: mult_le_0_iff)
    1.68  done
    1.69  
    1.70 -lemma sumr_offset3:
    1.71 -  "setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)) + setsum f {0..<k}"
    1.72 -apply (subst setsum_shift_bounds_nat_ivl [symmetric])
    1.73 -apply (simp add: setsum_add_nat_ivl add_commute)
    1.74 +(* TODO: MOVE *)
    1.75 +lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
    1.76 +by (erule norm_ratiotest_lemma, simp)
    1.77 +
    1.78 +(* TODO: MOVE *)
    1.79 +lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"
    1.80 +apply (drule le_imp_less_or_eq)
    1.81 +apply (auto dest: less_imp_Suc_add)
    1.82  done
    1.83  
    1.84 -lemma sumr_offset:
    1.85 -  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
    1.86 -  shows "(\<Sum>m=0..<n. f(m+k)) = setsum f {0..<n+k} - setsum f {0..<k}"
    1.87 -by (simp add: sumr_offset3)
    1.88 +(* MOVE *)
    1.89 +lemma setsum_even_minus_one [simp]: "(\<Sum>i<2 * n. (-1) ^ Suc i) = (0::'a::ring_1)"
    1.90 +  by (induct "n") auto
    1.91 +
    1.92 +(* MOVE *)
    1.93 +lemma setsum_nat_group: "(\<Sum>m<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {..< n * k}"
    1.94 +  apply (subgoal_tac "k = 0 | 0 < k", auto)
    1.95 +  apply (induct "n")
    1.96 +  apply (simp_all add: setsum_add_nat_ivl add_commute atLeast0LessThan[symmetric])
    1.97 +  done
    1.98  
    1.99 -lemma sumr_offset2:
   1.100 - "\<forall>f. (\<Sum>m=0..<n::nat. f(m+k)::real) = setsum f {0..<n+k} - setsum f {0..<k}"
   1.101 -by (simp add: sumr_offset)
   1.102 +(* MOVE *)
   1.103 +lemma norm_setsum:
   1.104 +  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   1.105 +  shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
   1.106 +  apply (case_tac "finite A")
   1.107 +  apply (erule finite_induct)
   1.108 +  apply simp
   1.109 +  apply simp
   1.110 +  apply (erule order_trans [OF norm_triangle_ineq add_left_mono])
   1.111 +  apply simp
   1.112 +  done
   1.113  
   1.114 -lemma sumr_offset4:
   1.115 -  "\<forall>n f. setsum f {0::nat..<n+k} = (\<Sum>m=0..<n. f (m+k)::real) + setsum f {0..<k}"
   1.116 -by (clarify, rule sumr_offset3)
   1.117 +(* MOVE *)
   1.118 +lemma norm_bound_subset:
   1.119 +  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   1.120 +  assumes "finite s" "t \<subseteq> s"
   1.121 +  assumes le: "(\<And>x. x \<in> s \<Longrightarrow> norm(f x) \<le> g x)"
   1.122 +  shows "norm (setsum f t) \<le> setsum g s"
   1.123 +proof -
   1.124 +  have "norm (setsum f t) \<le> (\<Sum>i\<in>t. norm (f i))"
   1.125 +    by (rule norm_setsum)
   1.126 +  also have "\<dots> \<le> (\<Sum>i\<in>t. g i)"
   1.127 +    using assms by (auto intro!: setsum_mono)
   1.128 +  also have "\<dots> \<le> setsum g s"
   1.129 +    using assms order.trans[OF norm_ge_zero le]
   1.130 +    by (auto intro!: setsum_mono3)
   1.131 +  finally show ?thesis .
   1.132 +qed
   1.133  
   1.134 -subsection{* Infinite Sums, by the Properties of Limits*}
   1.135 +(* MOVE *)
   1.136 +lemma (in linorder) lessThan_minus_lessThan [simp]:
   1.137 +  "{..< n} - {..< m} = {m ..< n}"
   1.138 +  by auto
   1.139 +
   1.140 +definition
   1.141 +  sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
   1.142 +  (infixr "sums" 80)
   1.143 +where
   1.144 +  "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) ----> s"
   1.145  
   1.146 -(*----------------------
   1.147 -   suminf is the sum
   1.148 - ---------------------*)
   1.149 -lemma sums_summable: "f sums l ==> summable f"
   1.150 +definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool" where
   1.151 +   "summable f \<longleftrightarrow> (\<exists>s. f sums s)"
   1.152 +
   1.153 +definition
   1.154 +  suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a"
   1.155 +  (binder "\<Sum>" 10)
   1.156 +where
   1.157 +  "suminf f = (THE s. f sums s)"
   1.158 +
   1.159 +lemma sums_subst[trans]: "f = g \<Longrightarrow> g sums z \<Longrightarrow> f sums z"
   1.160 +  by simp
   1.161 +
   1.162 +lemma sums_summable: "f sums l \<Longrightarrow> summable f"
   1.163    by (simp add: sums_def summable_def, blast)
   1.164  
   1.165 -lemma summable_sums:
   1.166 -  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
   1.167 -  assumes "summable f"
   1.168 -  shows "f sums (suminf f)"
   1.169 -proof -
   1.170 -  from assms obtain s where s: "(\<lambda>n. setsum f {0..<n}) ----> s"
   1.171 -    unfolding summable_def sums_def [abs_def] ..
   1.172 -  then show ?thesis unfolding sums_def [abs_def] suminf_def
   1.173 -    by (rule theI, auto intro!: tendsto_unique[OF trivial_limit_sequentially])
   1.174 -qed
   1.175 +lemma summable_iff_convergent: "summable f \<longleftrightarrow> convergent (\<lambda>n. \<Sum>i<n. f i)"
   1.176 +  by (simp add: summable_def sums_def convergent_def)
   1.177  
   1.178 -lemma summable_sumr_LIMSEQ_suminf:
   1.179 -  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
   1.180 -  shows "summable f \<Longrightarrow> (\<lambda>n. setsum f {0..<n}) ----> suminf f"
   1.181 -by (rule summable_sums [unfolded sums_def])
   1.182 -
   1.183 -lemma suminf_eq_lim: "suminf f = lim (%n. setsum f {0..<n})"
   1.184 +lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)"
   1.185    by (simp add: suminf_def sums_def lim_def)
   1.186  
   1.187 -(*-------------------
   1.188 -    sum is unique
   1.189 - ------------------*)
   1.190 -lemma sums_unique:
   1.191 -  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
   1.192 -  shows "f sums s \<Longrightarrow> (s = suminf f)"
   1.193 -apply (frule sums_summable[THEN summable_sums])
   1.194 -apply (auto intro!: tendsto_unique[OF trivial_limit_sequentially] simp add: sums_def)
   1.195 -done
   1.196 -
   1.197 -lemma sums_iff:
   1.198 -  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
   1.199 -  shows "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
   1.200 -  by (metis summable_sums sums_summable sums_unique)
   1.201 -
   1.202  lemma sums_finite:
   1.203 -  assumes [simp]: "finite N"
   1.204 -  assumes f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
   1.205 +  assumes [simp]: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
   1.206    shows "f sums (\<Sum>n\<in>N. f n)"
   1.207  proof -
   1.208    { fix n
   1.209 @@ -146,266 +133,76 @@
   1.210         (simp add: eq atLeast0LessThan tendsto_const del: add_Suc_right)
   1.211  qed
   1.212  
   1.213 -lemma suminf_finite:
   1.214 -  fixes f :: "nat \<Rightarrow> 'a::{comm_monoid_add,t2_space}"
   1.215 -  assumes N: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
   1.216 -  shows "suminf f = (\<Sum>n\<in>N. f n)"
   1.217 -  using sums_finite[OF assms, THEN sums_unique] by simp
   1.218 -
   1.219 -lemma sums_If_finite_set:
   1.220 -  "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)"
   1.221 +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)"
   1.222    using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simp
   1.223  
   1.224 -lemma sums_If_finite:
   1.225 -  "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)"
   1.226 -  using sums_If_finite_set[of "{r. P r}" f] by simp
   1.227 -
   1.228 -lemma sums_single:
   1.229 -  "(\<lambda>r. if r = i then f r else 0::'a::{comm_monoid_add,t2_space}) sums f i"
   1.230 -  using sums_If_finite[of "\<lambda>r. r = i" f] by simp
   1.231 -
   1.232 -lemma sums_split_initial_segment:
   1.233 -  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.234 -  shows "f sums s ==> (\<lambda>n. f(n + k)) sums (s - (SUM i = 0..< k. f i))"
   1.235 -  apply (unfold sums_def)
   1.236 -  apply (simp add: sumr_offset)
   1.237 -  apply (rule tendsto_diff [OF _ tendsto_const])
   1.238 -  apply (rule LIMSEQ_ignore_initial_segment)
   1.239 -  apply assumption
   1.240 -done
   1.241 -
   1.242 -lemma summable_ignore_initial_segment:
   1.243 -  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.244 -  shows "summable f ==> summable (%n. f(n + k))"
   1.245 -  apply (unfold summable_def)
   1.246 -  apply (auto intro: sums_split_initial_segment)
   1.247 -done
   1.248 -
   1.249 -lemma suminf_minus_initial_segment:
   1.250 -  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.251 -  shows "summable f ==>
   1.252 -    suminf f = s ==> suminf (%n. f(n + k)) = s - (SUM i = 0..< k. f i)"
   1.253 -  apply (frule summable_ignore_initial_segment)
   1.254 -  apply (rule sums_unique [THEN sym])
   1.255 -  apply (frule summable_sums)
   1.256 -  apply (rule sums_split_initial_segment)
   1.257 -  apply auto
   1.258 -done
   1.259 +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)"
   1.260 +  using sums_If_finite_set[of "{r. P r}"] by simp
   1.261  
   1.262 -lemma suminf_split_initial_segment:
   1.263 -  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.264 -  shows "summable f ==>
   1.265 -    suminf f = (SUM i = 0..< k. f i) + (\<Sum>n. f(n + k))"
   1.266 -by (auto simp add: suminf_minus_initial_segment)
   1.267 -
   1.268 -lemma suminf_exist_split: fixes r :: real assumes "0 < r" and "summable a"
   1.269 -  shows "\<exists> N. \<forall> n \<ge> N. \<bar> \<Sum> i. a (i + n) \<bar> < r"
   1.270 -proof -
   1.271 -  from LIMSEQ_D[OF summable_sumr_LIMSEQ_suminf[OF `summable a`] `0 < r`]
   1.272 -  obtain N :: nat where "\<forall> n \<ge> N. norm (setsum a {0..<n} - suminf a) < r" by auto
   1.273 -  thus ?thesis unfolding suminf_minus_initial_segment[OF `summable a` refl] abs_minus_commute real_norm_def
   1.274 -    by auto
   1.275 -qed
   1.276 +lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i"
   1.277 +  using sums_If_finite[of "\<lambda>r. r = i"] by simp
   1.278  
   1.279 -lemma sums_Suc:
   1.280 -  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.281 -  assumes sumSuc: "(\<lambda> n. f (Suc n)) sums l" shows "f sums (l + f 0)"
   1.282 -proof -
   1.283 -  from sumSuc[unfolded sums_def]
   1.284 -  have "(\<lambda>i. \<Sum>n = Suc 0..<Suc i. f n) ----> l" unfolding setsum_reindex[OF inj_Suc] image_Suc_atLeastLessThan[symmetric] comp_def .
   1.285 -  from tendsto_add[OF this tendsto_const, where b="f 0"]
   1.286 -  have "(\<lambda>i. \<Sum>n = 0..<Suc i. f n) ----> l + f 0" unfolding add_commute setsum_head_upt_Suc[OF zero_less_Suc] .
   1.287 -  thus ?thesis unfolding sums_def by (rule LIMSEQ_imp_Suc)
   1.288 -qed
   1.289 -
   1.290 -lemma series_zero:
   1.291 -  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
   1.292 -  assumes "\<forall>m. n \<le> m \<longrightarrow> f m = 0"
   1.293 -  shows "f sums (setsum f {0..<n})"
   1.294 -proof -
   1.295 -  { fix k :: nat have "setsum f {0..<k + n} = setsum f {0..<n}"
   1.296 -      using assms by (induct k) auto }
   1.297 -  note setsum_const = this
   1.298 -  show ?thesis
   1.299 -    unfolding sums_def
   1.300 -    apply (rule LIMSEQ_offset[of _ n])
   1.301 -    unfolding setsum_const
   1.302 -    apply (rule tendsto_const)
   1.303 -    done
   1.304 -qed
   1.305 +lemma series_zero: (* REMOVE *)
   1.306 +  "(\<And>m. n \<le> m \<Longrightarrow> f m = 0) \<Longrightarrow> f sums (\<Sum>i<n. f i)"
   1.307 +  by (rule sums_finite) auto
   1.308  
   1.309  lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0"
   1.310    unfolding sums_def by (simp add: tendsto_const)
   1.311  
   1.312  lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)"
   1.313 -by (rule sums_zero [THEN sums_summable])
   1.314 +  by (rule sums_zero [THEN sums_summable])
   1.315 +
   1.316 +lemma sums_group: "f sums s \<Longrightarrow> 0 < k \<Longrightarrow> (\<lambda>n. setsum f {n * k ..< n * k + k}) sums s"
   1.317 +  apply (simp only: sums_def setsum_nat_group tendsto_def eventually_sequentially)
   1.318 +  apply safe
   1.319 +  apply (erule_tac x=S in allE)
   1.320 +  apply safe
   1.321 +  apply (rule_tac x="N" in exI, safe)
   1.322 +  apply (drule_tac x="n*k" in spec)
   1.323 +  apply (erule mp)
   1.324 +  apply (erule order_trans)
   1.325 +  apply simp
   1.326 +  done
   1.327 +
   1.328 +context
   1.329 +  fixes f :: "nat \<Rightarrow> 'a::{t2_space, comm_monoid_add}"
   1.330 +begin
   1.331 +
   1.332 +lemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)"
   1.333 +  by (simp add: summable_def sums_def suminf_def)
   1.334 +     (metis convergent_LIMSEQ_iff convergent_def lim_def)
   1.335 +
   1.336 +lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) ----> suminf f"
   1.337 +  by (rule summable_sums [unfolded sums_def])
   1.338 +
   1.339 +lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f"
   1.340 +  by (metis limI suminf_eq_lim sums_def)
   1.341 +
   1.342 +lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> (suminf f = x)"
   1.343 +  by (metis summable_sums sums_summable sums_unique)
   1.344 +
   1.345 +lemma suminf_finite:
   1.346 +  assumes N: "finite N" and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
   1.347 +  shows "suminf f = (\<Sum>n\<in>N. f n)"
   1.348 +  using sums_finite[OF assms, THEN sums_unique] by simp
   1.349 +
   1.350 +end
   1.351  
   1.352  lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0"
   1.353 -by (rule sums_zero [THEN sums_unique, symmetric])
   1.354 -
   1.355 -lemma (in bounded_linear) sums:
   1.356 -  "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
   1.357 -  unfolding sums_def by (drule tendsto, simp only: setsum)
   1.358 -
   1.359 -lemma (in bounded_linear) summable:
   1.360 -  "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
   1.361 -unfolding summable_def by (auto intro: sums)
   1.362 -
   1.363 -lemma (in bounded_linear) suminf:
   1.364 -  "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
   1.365 -by (intro sums_unique sums summable_sums)
   1.366 -
   1.367 -lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
   1.368 -lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
   1.369 -lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
   1.370 -
   1.371 -lemma sums_mult:
   1.372 -  fixes c :: "'a::real_normed_algebra"
   1.373 -  shows "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
   1.374 -  by (rule bounded_linear.sums [OF bounded_linear_mult_right])
   1.375 -
   1.376 -lemma summable_mult:
   1.377 -  fixes c :: "'a::real_normed_algebra"
   1.378 -  shows "summable f \<Longrightarrow> summable (%n. c * f n)"
   1.379 -  by (rule bounded_linear.summable [OF bounded_linear_mult_right])
   1.380 -
   1.381 -lemma suminf_mult:
   1.382 -  fixes c :: "'a::real_normed_algebra"
   1.383 -  shows "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
   1.384 -  by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
   1.385 -
   1.386 -lemma sums_mult2:
   1.387 -  fixes c :: "'a::real_normed_algebra"
   1.388 -  shows "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
   1.389 -  by (rule bounded_linear.sums [OF bounded_linear_mult_left])
   1.390 -
   1.391 -lemma summable_mult2:
   1.392 -  fixes c :: "'a::real_normed_algebra"
   1.393 -  shows "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
   1.394 -  by (rule bounded_linear.summable [OF bounded_linear_mult_left])
   1.395 -
   1.396 -lemma suminf_mult2:
   1.397 -  fixes c :: "'a::real_normed_algebra"
   1.398 -  shows "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
   1.399 -  by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
   1.400 -
   1.401 -lemma sums_divide:
   1.402 -  fixes c :: "'a::real_normed_field"
   1.403 -  shows "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
   1.404 -  by (rule bounded_linear.sums [OF bounded_linear_divide])
   1.405 -
   1.406 -lemma summable_divide:
   1.407 -  fixes c :: "'a::real_normed_field"
   1.408 -  shows "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
   1.409 -  by (rule bounded_linear.summable [OF bounded_linear_divide])
   1.410 -
   1.411 -lemma suminf_divide:
   1.412 -  fixes c :: "'a::real_normed_field"
   1.413 -  shows "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
   1.414 -  by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
   1.415 -
   1.416 -lemma sums_add:
   1.417 -  fixes a b :: "'a::real_normed_field"
   1.418 -  shows "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) sums (a + b)"
   1.419 -  unfolding sums_def by (simp add: setsum_addf tendsto_add)
   1.420 -
   1.421 -lemma summable_add:
   1.422 -  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
   1.423 -  shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n + Y n)"
   1.424 -unfolding summable_def by (auto intro: sums_add)
   1.425 +  by (rule sums_zero [THEN sums_unique, symmetric])
   1.426  
   1.427 -lemma suminf_add:
   1.428 -  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
   1.429 -  shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X + suminf Y = (\<Sum>n. X n + Y n)"
   1.430 -by (intro sums_unique sums_add summable_sums)
   1.431 -
   1.432 -lemma sums_diff:
   1.433 -  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
   1.434 -  shows "\<lbrakk>X sums a; Y sums b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) sums (a - b)"
   1.435 -  unfolding sums_def by (simp add: setsum_subtractf tendsto_diff)
   1.436 -
   1.437 -lemma summable_diff:
   1.438 -  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
   1.439 -  shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> summable (\<lambda>n. X n - Y n)"
   1.440 -unfolding summable_def by (auto intro: sums_diff)
   1.441 -
   1.442 -lemma suminf_diff:
   1.443 -  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
   1.444 -  shows "\<lbrakk>summable X; summable Y\<rbrakk> \<Longrightarrow> suminf X - suminf Y = (\<Sum>n. X n - Y n)"
   1.445 -by (intro sums_unique sums_diff summable_sums)
   1.446 -
   1.447 -lemma sums_minus:
   1.448 -  fixes X :: "nat \<Rightarrow> 'a::real_normed_field"
   1.449 -  shows "X sums a ==> (\<lambda>n. - X n) sums (- a)"
   1.450 -  unfolding sums_def by (simp add: setsum_negf tendsto_minus)
   1.451 -
   1.452 -lemma summable_minus:
   1.453 -  fixes X :: "nat \<Rightarrow> 'a::real_normed_field"
   1.454 -  shows "summable X \<Longrightarrow> summable (\<lambda>n. - X n)"
   1.455 -unfolding summable_def by (auto intro: sums_minus)
   1.456 -
   1.457 -lemma suminf_minus:
   1.458 -  fixes X :: "nat \<Rightarrow> 'a::real_normed_field"
   1.459 -  shows "summable X \<Longrightarrow> (\<Sum>n. - X n) = - (\<Sum>n. X n)"
   1.460 -by (intro sums_unique [symmetric] sums_minus summable_sums)
   1.461 +context
   1.462 +  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
   1.463 +begin
   1.464  
   1.465 -lemma sums_group:
   1.466 -  fixes f :: "nat \<Rightarrow> 'a::real_normed_field"
   1.467 -  shows "\<lbrakk>f sums s; 0 < k\<rbrakk> \<Longrightarrow> (\<lambda>n. setsum f {n*k..<n*k+k}) sums s"
   1.468 -apply (simp only: sums_def sumr_group)
   1.469 -apply (unfold LIMSEQ_iff, safe)
   1.470 -apply (drule_tac x="r" in spec, safe)
   1.471 -apply (rule_tac x="no" in exI, safe)
   1.472 -apply (drule_tac x="n*k" in spec)
   1.473 -apply (erule mp)
   1.474 -apply (erule order_trans)
   1.475 -apply simp
   1.476 -done
   1.477 -
   1.478 -text{*A summable series of positive terms has limit that is at least as
   1.479 -great as any partial sum.*}
   1.480 -
   1.481 -lemma pos_summable:
   1.482 -  fixes f:: "nat \<Rightarrow> real"
   1.483 -  assumes pos: "\<And>n. 0 \<le> f n" and le: "\<And>n. setsum f {0..<n} \<le> x"
   1.484 -  shows "summable f"
   1.485 -proof -
   1.486 -  have "convergent (\<lambda>n. setsum f {0..<n})"
   1.487 -    proof (rule Bseq_mono_convergent)
   1.488 -      show "Bseq (\<lambda>n. setsum f {0..<n})"
   1.489 -        by (intro BseqI'[of _ x]) (auto simp add: setsum_nonneg pos intro: le)
   1.490 -    next
   1.491 -      show "\<forall>m n. m \<le> n \<longrightarrow> setsum f {0..<m} \<le> setsum f {0..<n}"
   1.492 -        by (auto intro: setsum_mono2 pos)
   1.493 -    qed
   1.494 -  thus ?thesis
   1.495 -    by (force simp add: summable_def sums_def convergent_def)
   1.496 -qed
   1.497 -
   1.498 -lemma series_pos_le:
   1.499 -  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
   1.500 -  shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 \<le> f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} \<le> suminf f"
   1.501 -  apply (drule summable_sums)
   1.502 -  apply (simp add: sums_def)
   1.503 -  apply (rule LIMSEQ_le_const)
   1.504 +lemma series_pos_le: "summable f \<Longrightarrow> \<forall>m\<ge>n. 0 \<le> f m \<Longrightarrow> setsum f {..<n} \<le> suminf f"
   1.505 +  apply (rule LIMSEQ_le_const[OF summable_LIMSEQ])
   1.506    apply assumption
   1.507    apply (intro exI[of _ n])
   1.508 -  apply (auto intro!: setsum_mono2)
   1.509 +  apply (auto intro!: setsum_mono2 simp: not_le[symmetric])
   1.510    done
   1.511  
   1.512 -lemma series_pos_less:
   1.513 -  fixes f :: "nat \<Rightarrow> 'a::{ordered_ab_semigroup_add_imp_le, ordered_comm_monoid_add, linorder_topology}"
   1.514 -  shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {0..<n} < suminf f"
   1.515 -  apply (rule_tac y="setsum f {0..<Suc n}" in order_less_le_trans)
   1.516 -  using add_less_cancel_left [of "setsum f {0..<n}" 0 "f n"]
   1.517 -  apply simp
   1.518 -  apply (erule series_pos_le)
   1.519 -  apply (simp add: order_less_imp_le)
   1.520 -  done
   1.521 -
   1.522 -lemma suminf_eq_zero_iff:
   1.523 -  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
   1.524 -  shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
   1.525 +lemma suminf_eq_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
   1.526  proof
   1.527    assume "summable f" "suminf f = 0" and pos: "\<forall>n. 0 \<le> f n"
   1.528    then have "f sums 0"
   1.529 @@ -419,77 +216,208 @@
   1.530    qed
   1.531    with pos show "\<forall>n. f n = 0"
   1.532      by (auto intro!: antisym)
   1.533 -next
   1.534 -  assume "\<forall>n. f n = 0"
   1.535 -  then have "f = (\<lambda>n. 0)"
   1.536 -    by auto
   1.537 -  then show "suminf f = 0"
   1.538 -    by simp
   1.539 +qed (metis suminf_zero fun_eq_iff)
   1.540 +
   1.541 +lemma suminf_gt_zero_iff: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)"
   1.542 +  using series_pos_le[of 0] suminf_eq_zero_iff by (simp add: less_le)
   1.543 +
   1.544 +lemma suminf_gt_zero: "summable f \<Longrightarrow> \<forall>n. 0 < f n \<Longrightarrow> 0 < suminf f"
   1.545 +  using suminf_gt_zero_iff by (simp add: less_imp_le)
   1.546 +
   1.547 +lemma suminf_ge_zero: "summable f \<Longrightarrow> \<forall>n. 0 \<le> f n \<Longrightarrow> 0 \<le> suminf f"
   1.548 +  by (drule_tac n="0" in series_pos_le) simp_all
   1.549 +
   1.550 +lemma suminf_le: "summable f \<Longrightarrow> (\<And>n. setsum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
   1.551 +  by (metis LIMSEQ_le_const2 summable_LIMSEQ)
   1.552 +
   1.553 +lemma summable_le: "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
   1.554 +  by (rule LIMSEQ_le) (auto intro: setsum_mono summable_LIMSEQ)
   1.555 +
   1.556 +end
   1.557 +
   1.558 +lemma series_pos_less:
   1.559 +  fixes f :: "nat \<Rightarrow> 'a::{ordered_ab_semigroup_add_imp_le, ordered_comm_monoid_add, linorder_topology}"
   1.560 +  shows "\<lbrakk>summable f; \<forall>m\<ge>n. 0 < f m\<rbrakk> \<Longrightarrow> setsum f {..<n} < suminf f"
   1.561 +  apply simp
   1.562 +  apply (rule_tac y="setsum f {..<Suc n}" in order_less_le_trans)
   1.563 +  using add_less_cancel_left [of "setsum f {..<n}" 0 "f n"]
   1.564 +  apply simp
   1.565 +  apply (erule series_pos_le)
   1.566 +  apply (simp add: order_less_imp_le)
   1.567 +  done
   1.568 +
   1.569 +lemma sums_Suc_iff:
   1.570 +  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.571 +  shows "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
   1.572 +proof -
   1.573 +  have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) ----> s + f 0"
   1.574 +    by (subst LIMSEQ_Suc_iff) (simp add: sums_def)
   1.575 +  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
   1.576 +    by (simp add: ac_simps setsum_reindex image_iff lessThan_Suc_eq_insert_0)
   1.577 +  also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
   1.578 +  proof
   1.579 +    assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) ----> s + f 0"
   1.580 +    with tendsto_add[OF this tendsto_const, of "- f 0"]
   1.581 +    show "(\<lambda>i. f (Suc i)) sums s"
   1.582 +      by (simp add: sums_def)
   1.583 +  qed (auto intro: tendsto_add tendsto_const simp: sums_def)
   1.584 +  finally show ?thesis ..
   1.585  qed
   1.586  
   1.587 -lemma suminf_gt_zero_iff:
   1.588 -  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
   1.589 -  shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)"
   1.590 -  using series_pos_le[of f 0] suminf_eq_zero_iff[of f]
   1.591 -  by (simp add: less_le)
   1.592 +context
   1.593 +  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.594 +begin
   1.595 +
   1.596 +lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
   1.597 +  unfolding sums_def by (simp add: setsum_addf tendsto_add)
   1.598 +
   1.599 +lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)"
   1.600 +  unfolding summable_def by (auto intro: sums_add)
   1.601 +
   1.602 +lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)"
   1.603 +  by (intro sums_unique sums_add summable_sums)
   1.604 +
   1.605 +lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)"
   1.606 +  unfolding sums_def by (simp add: setsum_subtractf tendsto_diff)
   1.607 +
   1.608 +lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)"
   1.609 +  unfolding summable_def by (auto intro: sums_diff)
   1.610 +
   1.611 +lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)"
   1.612 +  by (intro sums_unique sums_diff summable_sums)
   1.613 +
   1.614 +lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)"
   1.615 +  unfolding sums_def by (simp add: setsum_negf tendsto_minus)
   1.616 +
   1.617 +lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)"
   1.618 +  unfolding summable_def by (auto intro: sums_minus)
   1.619  
   1.620 -lemma suminf_gt_zero:
   1.621 -  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
   1.622 -  shows "\<lbrakk>summable f; \<forall>n. 0 < f n\<rbrakk> \<Longrightarrow> 0 < suminf f"
   1.623 -  using suminf_gt_zero_iff[of f] by (simp add: less_imp_le)
   1.624 +lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)"
   1.625 +  by (intro sums_unique [symmetric] sums_minus summable_sums)
   1.626 +
   1.627 +lemma sums_Suc: "(\<lambda> n. f (Suc n)) sums l \<Longrightarrow> f sums (l + f 0)"
   1.628 +  by (simp add: sums_Suc_iff)
   1.629 +
   1.630 +lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"
   1.631 +proof (induct n arbitrary: s)
   1.632 +  case (Suc n)
   1.633 +  moreover have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)"
   1.634 +    by (subst sums_Suc_iff) simp
   1.635 +  ultimately show ?case
   1.636 +    by (simp add: ac_simps)
   1.637 +qed simp
   1.638  
   1.639 -lemma suminf_ge_zero:
   1.640 -  fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
   1.641 -  shows "\<lbrakk>summable f; \<forall>n. 0 \<le> f n\<rbrakk> \<Longrightarrow> 0 \<le> suminf f"
   1.642 -  by (drule_tac n="0" in series_pos_le, simp_all)
   1.643 +lemma summable_iff_shift: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f"
   1.644 +  by (metis diff_add_cancel summable_def sums_iff_shift[abs_def])
   1.645 +
   1.646 +lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))"
   1.647 +  by (simp add: sums_iff_shift)
   1.648 +
   1.649 +lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))"
   1.650 +  by (simp add: summable_iff_shift)
   1.651 +
   1.652 +lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)"
   1.653 +  by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
   1.654 +
   1.655 +lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)"
   1.656 +  by (auto simp add: suminf_minus_initial_segment)
   1.657  
   1.658 -lemma sumr_pos_lt_pair:
   1.659 -  fixes f :: "nat \<Rightarrow> real"
   1.660 -  shows "\<lbrakk>summable f;
   1.661 -        \<And>d. 0 < f (k + (Suc(Suc 0) * d)) + f (k + ((Suc(Suc 0) * d) + 1))\<rbrakk>
   1.662 -      \<Longrightarrow> setsum f {0..<k} < suminf f"
   1.663 -unfolding One_nat_def
   1.664 -apply (subst suminf_split_initial_segment [where k="k"])
   1.665 -apply assumption
   1.666 -apply simp
   1.667 -apply (drule_tac k="k" in summable_ignore_initial_segment)
   1.668 -apply (drule_tac k="Suc (Suc 0)" in sums_group [OF summable_sums], simp)
   1.669 -apply simp
   1.670 -apply (frule sums_unique)
   1.671 -apply (drule sums_summable)
   1.672 -apply simp
   1.673 -apply (erule suminf_gt_zero)
   1.674 -apply (simp add: add_ac)
   1.675 -done
   1.676 +lemma suminf_exist_split: 
   1.677 +  fixes r :: real assumes "0 < r" and "summable f"
   1.678 +  shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
   1.679 +proof -
   1.680 +  from LIMSEQ_D[OF summable_LIMSEQ[OF `summable f`] `0 < r`]
   1.681 +  obtain N :: nat where "\<forall> n \<ge> N. norm (setsum f {..<n} - suminf f) < r" by auto
   1.682 +  thus ?thesis
   1.683 +    by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF `summable f`])
   1.684 +qed
   1.685 +
   1.686 +lemma summable_LIMSEQ_zero: "summable f \<Longrightarrow> f ----> 0"
   1.687 +  apply (drule summable_iff_convergent [THEN iffD1])
   1.688 +  apply (drule convergent_Cauchy)
   1.689 +  apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
   1.690 +  apply (drule_tac x="r" in spec, safe)
   1.691 +  apply (rule_tac x="M" in exI, safe)
   1.692 +  apply (drule_tac x="Suc n" in spec, simp)
   1.693 +  apply (drule_tac x="n" in spec, simp)
   1.694 +  done
   1.695 +
   1.696 +end
   1.697 +
   1.698 +lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
   1.699 +  unfolding sums_def by (drule tendsto, simp only: setsum)
   1.700 +
   1.701 +lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
   1.702 +  unfolding summable_def by (auto intro: sums)
   1.703 +
   1.704 +lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
   1.705 +  by (intro sums_unique sums summable_sums)
   1.706 +
   1.707 +lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
   1.708 +lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
   1.709 +lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
   1.710 +
   1.711 +context
   1.712 +  fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"
   1.713 +begin
   1.714 +
   1.715 +lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
   1.716 +  by (rule bounded_linear.sums [OF bounded_linear_mult_right])
   1.717 +
   1.718 +lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)"
   1.719 +  by (rule bounded_linear.summable [OF bounded_linear_mult_right])
   1.720 +
   1.721 +lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
   1.722 +  by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
   1.723 +
   1.724 +lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
   1.725 +  by (rule bounded_linear.sums [OF bounded_linear_mult_left])
   1.726 +
   1.727 +lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
   1.728 +  by (rule bounded_linear.summable [OF bounded_linear_mult_left])
   1.729 +
   1.730 +lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
   1.731 +  by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
   1.732 +
   1.733 +end
   1.734 +
   1.735 +context
   1.736 +  fixes c :: "'a::real_normed_field"
   1.737 +begin
   1.738 +
   1.739 +lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
   1.740 +  by (rule bounded_linear.sums [OF bounded_linear_divide])
   1.741 +
   1.742 +lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
   1.743 +  by (rule bounded_linear.summable [OF bounded_linear_divide])
   1.744 +
   1.745 +lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
   1.746 +  by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
   1.747  
   1.748  text{*Sum of a geometric progression.*}
   1.749  
   1.750 -lemmas sumr_geometric = geometric_sum [where 'a = real]
   1.751 -
   1.752 -lemma geometric_sums:
   1.753 -  fixes x :: "'a::{real_normed_field}"
   1.754 -  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) sums (1 / (1 - x))"
   1.755 +lemma geometric_sums: "norm c < 1 \<Longrightarrow> (\<lambda>n. c^n) sums (1 / (1 - c))"
   1.756  proof -
   1.757 -  assume less_1: "norm x < 1"
   1.758 -  hence neq_1: "x \<noteq> 1" by auto
   1.759 -  hence neq_0: "x - 1 \<noteq> 0" by simp
   1.760 -  from less_1 have lim_0: "(\<lambda>n. x ^ n) ----> 0"
   1.761 +  assume less_1: "norm c < 1"
   1.762 +  hence neq_1: "c \<noteq> 1" by auto
   1.763 +  hence neq_0: "c - 1 \<noteq> 0" by simp
   1.764 +  from less_1 have lim_0: "(\<lambda>n. c^n) ----> 0"
   1.765      by (rule LIMSEQ_power_zero)
   1.766 -  hence "(\<lambda>n. x ^ n / (x - 1) - 1 / (x - 1)) ----> 0 / (x - 1) - 1 / (x - 1)"
   1.767 +  hence "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) ----> 0 / (c - 1) - 1 / (c - 1)"
   1.768      using neq_0 by (intro tendsto_intros)
   1.769 -  hence "(\<lambda>n. (x ^ n - 1) / (x - 1)) ----> 1 / (1 - x)"
   1.770 +  hence "(\<lambda>n. (c ^ n - 1) / (c - 1)) ----> 1 / (1 - c)"
   1.771      by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
   1.772 -  thus "(\<lambda>n. x ^ n) sums (1 / (1 - x))"
   1.773 +  thus "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
   1.774      by (simp add: sums_def geometric_sum neq_1)
   1.775  qed
   1.776  
   1.777 -lemma summable_geometric:
   1.778 -  fixes x :: "'a::{real_normed_field}"
   1.779 -  shows "norm x < 1 \<Longrightarrow> summable (\<lambda>n. x ^ n)"
   1.780 -by (rule geometric_sums [THEN sums_summable])
   1.781 +lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
   1.782 +  by (rule geometric_sums [THEN sums_summable])
   1.783  
   1.784 -lemma half: "0 < 1 / (2::'a::linordered_field)"
   1.785 -  by simp
   1.786 +lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)"
   1.787 +  by (rule sums_unique[symmetric]) (rule geometric_sums)
   1.788 +
   1.789 +end
   1.790  
   1.791  lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
   1.792  proof -
   1.793 @@ -503,110 +431,104 @@
   1.794  
   1.795  text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
   1.796  
   1.797 -lemma summable_convergent_sumr_iff:
   1.798 - "summable f = convergent (%n. setsum f {0..<n})"
   1.799 -by (simp add: summable_def sums_def convergent_def)
   1.800 +lemma summable_Cauchy:
   1.801 +  fixes f :: "nat \<Rightarrow> 'a::banach"
   1.802 +  shows "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (setsum f {m..<n}) < e)"
   1.803 +  apply (simp only: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff, safe)
   1.804 +  apply (drule spec, drule (1) mp)
   1.805 +  apply (erule exE, rule_tac x="M" in exI, clarify)
   1.806 +  apply (rule_tac x="m" and y="n" in linorder_le_cases)
   1.807 +  apply (frule (1) order_trans)
   1.808 +  apply (drule_tac x="n" in spec, drule (1) mp)
   1.809 +  apply (drule_tac x="m" in spec, drule (1) mp)
   1.810 +  apply (simp_all add: setsum_diff [symmetric])
   1.811 +  apply (drule spec, drule (1) mp)
   1.812 +  apply (erule exE, rule_tac x="N" in exI, clarify)
   1.813 +  apply (rule_tac x="m" and y="n" in linorder_le_cases)
   1.814 +  apply (subst norm_minus_commute)
   1.815 +  apply (simp_all add: setsum_diff [symmetric])
   1.816 +  done
   1.817  
   1.818 -lemma summable_LIMSEQ_zero:
   1.819 -  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.820 -  shows "summable f \<Longrightarrow> f ----> 0"
   1.821 -apply (drule summable_convergent_sumr_iff [THEN iffD1])
   1.822 -apply (drule convergent_Cauchy)
   1.823 -apply (simp only: Cauchy_iff LIMSEQ_iff, safe)
   1.824 -apply (drule_tac x="r" in spec, safe)
   1.825 -apply (rule_tac x="M" in exI, safe)
   1.826 -apply (drule_tac x="Suc n" in spec, simp)
   1.827 -apply (drule_tac x="n" in spec, simp)
   1.828 -done
   1.829 +context
   1.830 +  fixes f :: "nat \<Rightarrow> 'a::banach"
   1.831 +begin  
   1.832 +
   1.833 +text{*Absolute convergence imples normal convergence*}
   1.834  
   1.835 -lemma suminf_le:
   1.836 -  fixes x :: "'a :: {ordered_comm_monoid_add, linorder_topology}"
   1.837 -  shows "summable f \<Longrightarrow> (!!n. setsum f {0..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
   1.838 -  apply (drule summable_sums)
   1.839 -  apply (simp add: sums_def)
   1.840 -  apply (rule LIMSEQ_le_const2)
   1.841 -  apply assumption
   1.842 -  apply auto
   1.843 +lemma summable_norm_cancel:
   1.844 +  "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
   1.845 +  apply (simp only: summable_Cauchy, safe)
   1.846 +  apply (drule_tac x="e" in spec, safe)
   1.847 +  apply (rule_tac x="N" in exI, safe)
   1.848 +  apply (drule_tac x="m" in spec, safe)
   1.849 +  apply (rule order_le_less_trans [OF norm_setsum])
   1.850 +  apply (rule order_le_less_trans [OF abs_ge_self])
   1.851 +  apply simp
   1.852    done
   1.853  
   1.854 -lemma summable_Cauchy:
   1.855 -     "summable (f::nat \<Rightarrow> 'a::banach) =
   1.856 -      (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. norm (setsum f {m..<n}) < e)"
   1.857 -apply (simp only: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_iff, safe)
   1.858 -apply (drule spec, drule (1) mp)
   1.859 -apply (erule exE, rule_tac x="M" in exI, clarify)
   1.860 -apply (rule_tac x="m" and y="n" in linorder_le_cases)
   1.861 -apply (frule (1) order_trans)
   1.862 -apply (drule_tac x="n" in spec, drule (1) mp)
   1.863 -apply (drule_tac x="m" in spec, drule (1) mp)
   1.864 -apply (simp add: setsum_diff [symmetric])
   1.865 -apply simp
   1.866 -apply (drule spec, drule (1) mp)
   1.867 -apply (erule exE, rule_tac x="N" in exI, clarify)
   1.868 -apply (rule_tac x="m" and y="n" in linorder_le_cases)
   1.869 -apply (subst norm_minus_commute)
   1.870 -apply (simp add: setsum_diff [symmetric])
   1.871 -apply (simp add: setsum_diff [symmetric])
   1.872 -done
   1.873 +lemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
   1.874 +  by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_setsum)
   1.875 +
   1.876 +text {* Comparison tests *}
   1.877  
   1.878 -text{*Comparison test*}
   1.879 +lemma summable_comparison_test: "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f"
   1.880 +  apply (simp add: summable_Cauchy, safe)
   1.881 +  apply (drule_tac x="e" in spec, safe)
   1.882 +  apply (rule_tac x = "N + Na" in exI, safe)
   1.883 +  apply (rotate_tac 2)
   1.884 +  apply (drule_tac x = m in spec)
   1.885 +  apply (auto, rotate_tac 2, drule_tac x = n in spec)
   1.886 +  apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
   1.887 +  apply (rule norm_setsum)
   1.888 +  apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
   1.889 +  apply (auto intro: setsum_mono simp add: abs_less_iff)
   1.890 +  done
   1.891 +
   1.892 +subsection {* The Ratio Test*}
   1.893  
   1.894 -lemma norm_setsum:
   1.895 -  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   1.896 -  shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
   1.897 -apply (case_tac "finite A")
   1.898 -apply (erule finite_induct)
   1.899 -apply simp
   1.900 -apply simp
   1.901 -apply (erule order_trans [OF norm_triangle_ineq add_left_mono])
   1.902 -apply simp
   1.903 -done
   1.904 -
   1.905 -lemma norm_bound_subset:
   1.906 -  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   1.907 -  assumes "finite s" "t \<subseteq> s"
   1.908 -  assumes le: "(\<And>x. x \<in> s \<Longrightarrow> norm(f x) \<le> g x)"
   1.909 -  shows "norm (setsum f t) \<le> setsum g s"
   1.910 -proof -
   1.911 -  have "norm (setsum f t) \<le> (\<Sum>i\<in>t. norm (f i))"
   1.912 -    by (rule norm_setsum)
   1.913 -  also have "\<dots> \<le> (\<Sum>i\<in>t. g i)"
   1.914 -    using assms by (auto intro!: setsum_mono)
   1.915 -  also have "\<dots> \<le> setsum g s"
   1.916 -    using assms order.trans[OF norm_ge_zero le]
   1.917 -    by (auto intro!: setsum_mono3)
   1.918 -  finally show ?thesis .
   1.919 +lemma summable_ratio_test: 
   1.920 +  assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
   1.921 +  shows "summable f"
   1.922 +proof cases
   1.923 +  assume "0 < c"
   1.924 +  show "summable f"
   1.925 +  proof (rule summable_comparison_test)
   1.926 +    show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
   1.927 +    proof (intro exI allI impI)
   1.928 +      fix n assume "N \<le> n" then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
   1.929 +      proof (induct rule: inc_induct)
   1.930 +        case (step m)
   1.931 +        moreover have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
   1.932 +          using `0 < c` `c < 1` assms(2)[OF `N \<le> m`] by (simp add: field_simps)
   1.933 +        ultimately show ?case by simp
   1.934 +      qed (insert `0 < c`, simp)
   1.935 +    qed
   1.936 +    show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
   1.937 +      using `0 < c` `c < 1` by (intro summable_mult summable_geometric) simp
   1.938 +  qed
   1.939 +next
   1.940 +  assume c: "\<not> 0 < c"
   1.941 +  { fix n assume "n \<ge> N"
   1.942 +    then have "norm (f (Suc n)) \<le> c * norm (f n)"
   1.943 +      by fact
   1.944 +    also have "\<dots> \<le> 0"
   1.945 +      using c by (simp add: not_less mult_nonpos_nonneg)
   1.946 +    finally have "f (Suc n) = 0"
   1.947 +      by auto }
   1.948 +  then show "summable f"
   1.949 +    by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_iff)
   1.950  qed
   1.951  
   1.952 -lemma summable_comparison_test:
   1.953 -  fixes f :: "nat \<Rightarrow> 'a::banach"
   1.954 -  shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f"
   1.955 -apply (simp add: summable_Cauchy, safe)
   1.956 -apply (drule_tac x="e" in spec, safe)
   1.957 -apply (rule_tac x = "N + Na" in exI, safe)
   1.958 -apply (rotate_tac 2)
   1.959 -apply (drule_tac x = m in spec)
   1.960 -apply (auto, rotate_tac 2, drule_tac x = n in spec)
   1.961 -apply (rule_tac y = "\<Sum>k=m..<n. norm (f k)" in order_le_less_trans)
   1.962 -apply (rule norm_setsum)
   1.963 -apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
   1.964 -apply (auto intro: setsum_mono simp add: abs_less_iff)
   1.965 -done
   1.966 +end
   1.967  
   1.968  lemma summable_norm_comparison_test:
   1.969 -  fixes f :: "nat \<Rightarrow> 'a::banach"
   1.970 -  shows "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk>
   1.971 -         \<Longrightarrow> summable (\<lambda>n. norm (f n))"
   1.972 -apply (rule summable_comparison_test)
   1.973 -apply (auto)
   1.974 -done
   1.975 +  "\<lbrakk>\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable (\<lambda>n. norm (f n))"
   1.976 +  by (rule summable_comparison_test) auto
   1.977  
   1.978 -lemma summable_rabs_comparison_test:
   1.979 +lemma summable_rabs_cancel:
   1.980    fixes f :: "nat \<Rightarrow> real"
   1.981 -  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>)"
   1.982 -apply (rule summable_comparison_test)
   1.983 -apply (auto)
   1.984 -done
   1.985 +  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f"
   1.986 +  by (rule summable_norm_cancel) simp
   1.987  
   1.988  text{*Summability of geometric series for real algebras*}
   1.989  
   1.990 @@ -620,119 +542,34 @@
   1.991      by (simp add: summable_geometric)
   1.992  qed
   1.993  
   1.994 -text{*Limit comparison property for series (c.f. jrh)*}
   1.995  
   1.996 -lemma summable_le:
   1.997 -  fixes f g :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add, linorder_topology}"
   1.998 -  shows "\<lbrakk>\<forall>n. f n \<le> g n; summable f; summable g\<rbrakk> \<Longrightarrow> suminf f \<le> suminf g"
   1.999 -apply (drule summable_sums)+
  1.1000 -apply (simp only: sums_def, erule (1) LIMSEQ_le)
  1.1001 -apply (rule exI)
  1.1002 -apply (auto intro!: setsum_mono)
  1.1003 -done
  1.1004 -
  1.1005 -lemma summable_le2:
  1.1006 -  fixes f g :: "nat \<Rightarrow> real"
  1.1007 -  shows "\<lbrakk>\<forall>n. \<bar>f n\<bar> \<le> g n; summable g\<rbrakk> \<Longrightarrow> summable f \<and> suminf f \<le> suminf g"
  1.1008 -apply (subgoal_tac "summable f")
  1.1009 -apply (auto intro!: summable_le)
  1.1010 -apply (simp add: abs_le_iff)
  1.1011 -apply (rule_tac g="g" in summable_comparison_test, simp_all)
  1.1012 -done
  1.1013 +text{*A summable series of positive terms has limit that is at least as
  1.1014 +great as any partial sum.*}
  1.1015  
  1.1016 -(* specialisation for the common 0 case *)
  1.1017 -lemma suminf_0_le:
  1.1018 -  fixes f::"nat\<Rightarrow>real"
  1.1019 -  assumes gt0: "\<forall>n. 0 \<le> f n" and sm: "summable f"
  1.1020 -  shows "0 \<le> suminf f"
  1.1021 -  using suminf_ge_zero[OF sm gt0] by simp
  1.1022 +lemma pos_summable:
  1.1023 +  fixes f:: "nat \<Rightarrow> real"
  1.1024 +  assumes pos: "\<And>n. 0 \<le> f n" and le: "\<And>n. setsum f {..<n} \<le> x"
  1.1025 +  shows "summable f"
  1.1026 +proof -
  1.1027 +  have "convergent (\<lambda>n. setsum f {..<n})"
  1.1028 +  proof (rule Bseq_mono_convergent)
  1.1029 +    show "Bseq (\<lambda>n. setsum f {..<n})"
  1.1030 +      by (intro BseqI'[of _ x]) (auto simp add: setsum_nonneg pos intro: le)
  1.1031 +  qed (auto intro: setsum_mono2 pos)
  1.1032 +  thus ?thesis
  1.1033 +    by (force simp add: summable_def sums_def convergent_def)
  1.1034 +qed
  1.1035  
  1.1036 -text{*Absolute convergence imples normal convergence*}
  1.1037 -lemma summable_norm_cancel:
  1.1038 -  fixes f :: "nat \<Rightarrow> 'a::banach"
  1.1039 -  shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
  1.1040 -apply (simp only: summable_Cauchy, safe)
  1.1041 -apply (drule_tac x="e" in spec, safe)
  1.1042 -apply (rule_tac x="N" in exI, safe)
  1.1043 -apply (drule_tac x="m" in spec, safe)
  1.1044 -apply (rule order_le_less_trans [OF norm_setsum])
  1.1045 -apply (rule order_le_less_trans [OF abs_ge_self])
  1.1046 -apply simp
  1.1047 -done
  1.1048 -
  1.1049 -lemma summable_rabs_cancel:
  1.1050 +lemma summable_rabs_comparison_test:
  1.1051    fixes f :: "nat \<Rightarrow> real"
  1.1052 -  shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f"
  1.1053 -by (rule summable_norm_cancel, simp)
  1.1054 -
  1.1055 -text{*Absolute convergence of series*}
  1.1056 -lemma summable_norm:
  1.1057 -  fixes f :: "nat \<Rightarrow> 'a::banach"
  1.1058 -  shows "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
  1.1059 -  by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel
  1.1060 -                summable_sumr_LIMSEQ_suminf norm_setsum)
  1.1061 +  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>)"
  1.1062 +  by (rule summable_comparison_test) auto
  1.1063  
  1.1064  lemma summable_rabs:
  1.1065    fixes f :: "nat \<Rightarrow> real"
  1.1066    shows "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
  1.1067  by (fold real_norm_def, rule summable_norm)
  1.1068  
  1.1069 -subsection{* The Ratio Test*}
  1.1070 -
  1.1071 -lemma norm_ratiotest_lemma:
  1.1072 -  fixes x y :: "'a::real_normed_vector"
  1.1073 -  shows "\<lbrakk>c \<le> 0; norm x \<le> c * norm y\<rbrakk> \<Longrightarrow> x = 0"
  1.1074 -apply (subgoal_tac "norm x \<le> 0", simp)
  1.1075 -apply (erule order_trans)
  1.1076 -apply (simp add: mult_le_0_iff)
  1.1077 -done
  1.1078 -
  1.1079 -lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
  1.1080 -by (erule norm_ratiotest_lemma, simp)
  1.1081 -
  1.1082 -(* TODO: MOVE *)
  1.1083 -lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"
  1.1084 -apply (drule le_imp_less_or_eq)
  1.1085 -apply (auto dest: less_imp_Suc_add)
  1.1086 -done
  1.1087 -
  1.1088 -lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"
  1.1089 -by (auto simp add: le_Suc_ex)
  1.1090 -
  1.1091 -(*All this trouble just to get 0<c *)
  1.1092 -lemma ratio_test_lemma2:
  1.1093 -  fixes f :: "nat \<Rightarrow> 'a::banach"
  1.1094 -  shows "\<lbrakk>\<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> 0 < c \<or> summable f"
  1.1095 -apply (simp (no_asm) add: linorder_not_le [symmetric])
  1.1096 -apply (simp add: summable_Cauchy)
  1.1097 -apply (safe, subgoal_tac "\<forall>n. N < n --> f (n) = 0")
  1.1098 - prefer 2
  1.1099 - apply clarify
  1.1100 - apply(erule_tac x = "n - Suc 0" in allE)
  1.1101 - apply (simp add:diff_Suc split:nat.splits)
  1.1102 - apply (blast intro: norm_ratiotest_lemma)
  1.1103 -apply (rule_tac x = "Suc N" in exI, clarify)
  1.1104 -apply(simp cong del: setsum_cong cong: setsum_ivl_cong)
  1.1105 -done
  1.1106 -
  1.1107 -lemma ratio_test:
  1.1108 -  fixes f :: "nat \<Rightarrow> 'a::banach"
  1.1109 -  shows "\<lbrakk>c < 1; \<forall>n\<ge>N. norm (f (Suc n)) \<le> c * norm (f n)\<rbrakk> \<Longrightarrow> summable f"
  1.1110 -apply (frule ratio_test_lemma2, auto)
  1.1111 -apply (rule_tac g = "%n. (norm (f N) / (c ^ N))*c ^ n"
  1.1112 -       in summable_comparison_test)
  1.1113 -apply (rule_tac x = N in exI, safe)
  1.1114 -apply (drule le_Suc_ex_iff [THEN iffD1])
  1.1115 -apply (auto simp add: power_add field_power_not_zero)
  1.1116 -apply (induct_tac "na", auto)
  1.1117 -apply (rule_tac y = "c * norm (f (N + n))" in order_trans)
  1.1118 -apply (auto intro: mult_right_mono simp add: summable_def)
  1.1119 -apply (rule_tac x = "norm (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
  1.1120 -apply (rule sums_divide)
  1.1121 -apply (rule sums_mult)
  1.1122 -apply (auto intro!: geometric_sums)
  1.1123 -done
  1.1124 -
  1.1125  subsection {* Cauchy Product Formula *}
  1.1126  
  1.1127  text {*
  1.1128 @@ -742,14 +579,14 @@
  1.1129  
  1.1130  lemma setsum_triangle_reindex:
  1.1131    fixes n :: nat
  1.1132 -  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))"
  1.1133 +  shows "(\<Sum>(i,j)\<in>{(i,j). i+j < n}. f i j) = (\<Sum>k<n. \<Sum>i=0..k. f i (k - i))"
  1.1134  proof -
  1.1135    have "(\<Sum>(i, j)\<in>{(i, j). i + j < n}. f i j) =
  1.1136 -    (\<Sum>(k, i)\<in>(SIGMA k:{0..<n}. {0..k}). f i (k - i))"
  1.1137 +    (\<Sum>(k, i)\<in>(SIGMA k:{..<n}. {0..k}). f i (k - i))"
  1.1138    proof (rule setsum_reindex_cong)
  1.1139 -    show "inj_on (\<lambda>(k,i). (i, k - i)) (SIGMA k:{0..<n}. {0..k})"
  1.1140 +    show "inj_on (\<lambda>(k,i). (i, k - i)) (SIGMA k:{..<n}. {0..k})"
  1.1141        by (rule inj_on_inverseI [where g="\<lambda>(i,j). (i+j, i)"], auto)
  1.1142 -    show "{(i,j). i + j < n} = (\<lambda>(k,i). (i, k - i)) ` (SIGMA k:{0..<n}. {0..k})"
  1.1143 +    show "{(i,j). i + j < n} = (\<lambda>(k,i). (i, k - i)) ` (SIGMA k:{..<n}. {0..k})"
  1.1144        by (safe, rule_tac x="(a+b,a)" in image_eqI, auto)
  1.1145      show "\<And>a. (\<lambda>(k, i). f i (k - i)) a = split f ((\<lambda>(k, i). (i, k - i)) a)"
  1.1146        by clarify
  1.1147 @@ -763,7 +600,7 @@
  1.1148    assumes b: "summable (\<lambda>k. norm (b k))"
  1.1149    shows "(\<lambda>k. \<Sum>i=0..k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
  1.1150  proof -
  1.1151 -  let ?S1 = "\<lambda>n::nat. {0..<n} \<times> {0..<n}"
  1.1152 +  let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}"
  1.1153    let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
  1.1154    have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
  1.1155    have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
  1.1156 @@ -779,20 +616,15 @@
  1.1157      unfolding real_norm_def
  1.1158      by (simp only: abs_of_nonneg setsum_nonneg [rule_format])
  1.1159  
  1.1160 -  have "(\<lambda>n. (\<Sum>k=0..<n. a k) * (\<Sum>k=0..<n. b k))
  1.1161 -           ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
  1.1162 -    by (intro tendsto_mult summable_sumr_LIMSEQ_suminf
  1.1163 -        summable_norm_cancel [OF a] summable_norm_cancel [OF b])
  1.1164 +  have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
  1.1165 +    by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
  1.1166    hence 1: "(\<lambda>n. setsum ?g (?S1 n)) ----> (\<Sum>k. a k) * (\<Sum>k. b k)"
  1.1167 -    by (simp only: setsum_product setsum_Sigma [rule_format]
  1.1168 -                   finite_atLeastLessThan)
  1.1169 +    by (simp only: setsum_product setsum_Sigma [rule_format] finite_lessThan)
  1.1170  
  1.1171 -  have "(\<lambda>n. (\<Sum>k=0..<n. norm (a k)) * (\<Sum>k=0..<n. norm (b k)))
  1.1172 -       ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
  1.1173 -    using a b by (intro tendsto_mult summable_sumr_LIMSEQ_suminf)
  1.1174 +  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))"
  1.1175 +    using a b by (intro tendsto_mult summable_LIMSEQ)
  1.1176    hence "(\<lambda>n. setsum ?f (?S1 n)) ----> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
  1.1177 -    by (simp only: setsum_product setsum_Sigma [rule_format]
  1.1178 -                   finite_atLeastLessThan)
  1.1179 +    by (simp only: setsum_product setsum_Sigma [rule_format] finite_lessThan)
  1.1180    hence "convergent (\<lambda>n. setsum ?f (?S1 n))"
  1.1181      by (rule convergentI)
  1.1182    hence Cauchy: "Cauchy (\<lambda>n. setsum ?f (?S1 n))"