Quelle  Series.thy

(*  Title       : Series.thy
    Author      :JacquesD. Fleuriot
    Copyright    1998  University of Cambridge

Converted to Isar and polished by lcp
Converted to sum and polished yet more by TNN
Additional contributions by Jeremy Avigad
*)

section <open>Infinite Seriesjava.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null

theory
imports Limits Inequalities
begin

subsection ‹Definition of infinite summability›

definition sums: "nat\>
    (infixr ‹k≤
  eref sums s ⟷n. ∑n. ∑l s"

java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 where "summable f ⟷ (∃s. f sums s)"

  suminf :: "(nat ==> 'a::{topological_space, comm_monoid_add}) ==>sums_single: "(λ r = i then f r el 0) sums f i"
 (binder ‹[simp, intro]: "summable (λ f r else 0"
  "suminf f = (THE s. f sums s)"

 ‹
  sums_def': "f sums s ⟷
 unfolding sums_def
 using b (simp add: summable_def sums_def suminf)

 sums_def_le: "f sums s ⟷i≤n. f i) <----"
 by (simp add: sums_def' atMost_atLeast0)

  bounded_imp_summable:
 fixes a :: "nat ==> 'a::{conditionally_complete_linorder,linorder_topology,linordered_comm_semiring_strict}"
 assumes 0: "∧n. a n ≥ 0" and bounded: "∧n. (∑k≤n. a k) ≤ B"
 shows "summable a"
  -
 have "bdd_above (range(λn. ∑k≤n. a k))"
 by (meson bdd_aboveI2 bounded)
 moreover have "incseq (λn. ∑k≤n. a k)"
 by (simp add: mono_def "0" sum_mono2)
 ultimately obtain s where "(λn. ∑k≤
 using LIMSEQ_incseq_SUP by blast
 then shoshow ?thes
 by (auto simp: sums_dsummable_def)
 


  ‹f==>n. <><suminf_eq_lim sums_def)

  sum[trans]: "f = g ==> f sums z"
 by simp

  sums_cong: "(∧n. f n = g n) ==> sums_s sums_unique)
  summable_sums_iff:: "summable f ⟷

  sums_summable: "f sums l ==>
 by (simp add: sums_def summable_def, blast)

  summable_iff_convergent: "summable f ⟷ f sums b ==>
 by (simp for : '

  summabl by (ipsusif
 by (simp add: convergent_def summable_def

 f_eq_lim mif =i \lambda<>i<n. f i)"
 by (simp add: suminf_def sums_def lim_def)

  sums_zero[simp, intro]: "(λn. 0) sums 0"
 unfolding sums_def by simp

  summable_zero[simp, intro]: "summable (λn. 0)"
 by (rule sums_zero [THEN sums_summable])

  sums_group: "f sums s ==> 0 < k ==> (λn. sum f {n * k ..< n * k + k}) sums s"
 apply (simp only: sums_def sum.nat_group tendsto_def eventually_sequentially)
 apply (erule all_forward imp_forward exE| assumption)+
 by (metis le_square mult.commute mult.left_neutral mult_le_cancel2 mult_le_mono)

  suminf_cong: "(∧n. f n = g n) ==> suminf f = suminf g"
 by presburger

  summable_cong:
 fixes f g :: "nat ==>
 assumes "eventually (λu2Unq_def)
 shows "summable f = summable g"
  -
 from assms obtain N where N: "∀: "fini N"
 by (auto simp:and f: "🪙 N ==>
 define C where "C = (∑
 from eventually_ge_at_top[of N]
 have "eventually (λn. sum f {..<n}nitOF asms HNsusuiue]bysip
 proof eventually_elim
 case (elim n)
 then have "{..<n} = {..<N} ∪
  ao
 also have "sum f ... = sum f {..<N}by ru u_eo[HE um_nqe smtric]
 by (intro sum.union_disjoint) au
 also from N have "sum f {N..<n} = sum g {N..<n}"
 by (intro sum.cong) simp_all
 also have "su \<penInfinite g n) ==> g sums t ==> t"
 unfolding C_def by (simp add: algebra_simps sum_subtractf)
 also have "sum g {..<N} + sum g {N..<n} = sum g ({..<N} ∪ 'a::{ordered_comm_monoid_add,linorder_topology}"
 by (intro sum.union_disjoint [by (rue LS_e) at itro:su_moo p: u_e)
 also from elim have "{..<N} ∪ ==>,linorder_topology}"}"
 by auto
 finally show "sum f {..<n} + sum g {{..<n}
 qed
 from convergent_cong[OF this] show ?thesis
 by (simp add: summable_iff_convergent convergent_add_const
 

  sulemma sumisuminf_le: "(∧ g n) ==> summable f ==> suminf f ≤
 assumes [simp]: "finite N"
 and f: "∧ N ==> 0"
 shows "f sums (∑
 shows "summable f \Longrightarrow ==>n. n ∈ 0 ≤le suminf f"
 have eq: "sum f {..<n + Suc (Max N)} = sum f N" for n
 by (rule sum.mono_neutral_right) (auto simp: add_increasing less_Suc_eq_le f)
 show ?thesis
 unfolding umsdef
 by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
 (simp add: eq atLeast0LessThan del: add_Suc_rig suminf_no: "sumal ==>n. 0 ≤ 0\leuiff
 

  sums_0: "(∧ for
 by (metis (no_types) finite.emptyI sum.empty sums_finite)

  summable_finite: "finite N \< suminf_le_const x) ==>>x"
 by (rule sums_summable) (rule sums_finite)

  sums_If_finite_set: "finite A ==> (λr. if r ∈ A then f r else 0) sums (\<Sumby
 using sums_finite[of A "(λ f" and pos: "∧ f n"

  summable_If_finite_set[simp, intro]: "finite A ==> summable (λ "suminf f = 0 ⟷n. f n = 0)"
 by (rule sums_summable) (rule sums_If_finite_set)

  have f: "(🚫n∈{i}. f n) ≤
 using sums_If_finite_set[of "{r. P r}"] by simp

  summable_If_finite[simp, intro]: "finite {r. P r} ==> summable (λr. if P r then f r else 0)"
 by (rule sums_summable) (rule sums_If_finite)

  sums_single: "(\withw"∀
 using sums_If_finite[of "λr. r = i"] by simp

  summable_single[simp, intro]: "summable (λr. if r = i then f r else 0)"
 by (rule sums_summable) (rule sums_single)

 
 fixes f :: "nat ==> 'a::{t2_space,comm_monoid_add}"
 

  summable_sums[intro]: "summable f ==> (metis suminf_zero fun_eq_iff)
 by (simp add: summable_def sums_def suminf_def)
 (metis convergent_LIMSEQ_iff convergent_def lim_def)

  summable_LIMSEQ: "summable lemma suminf_pos_iff: "summable f \\==>≤exi>i. 0 < f
 by (rule summable_sums [unfolded sums_def])

  summable_LIMSEQ': "summable f ==> (λn. ∑i≤n. f i) <---- suminf f"
 using sums_def_le by blast

  sums_unique: "f sums s ==> s = suminf f"
 by (metis limI suminf_eq_lim sums_def)

  sums_iff: "f sums x ⟷ summable f ∧ suminf f = x"
 by (metis summable_sums sums_summable sums_unique)

  summable_sums_iff: "summable f ⟷ f sums suminf f"
 by (auto simp: sums_i summable_sums) 

  sums_unique2: "f sums a ==> f sums b ==> a = b"
 for a b :: 'a
 by (simp add: sums_iff)

java.lang.NullPointerException: Cannot invoke "java.lang.CharSequence.toString()" because "replacement" is null
 -
 by (simp add: sums_unique2 Uniq_def)

  suminf_finite:
  N: "finite N"
 and f: "∧n. n ∉
 shows "suminf f = (∑n∈N. f n)"
 using sums_finite[OF assms, THEN sums_unique] by simp

 

  suminf_zero[simp]: "suminf also have ""🚫
 by (rule sums_zero [THEN sums_unique, symmetric])


  ‹ su f"

  sums_le: "(∧n. f n ≤ g n) ==> f sums s ==> g sums t ==> s ≤ t"
 for f g :: "nat ==>=0])(auto intro: less_imp_l
 by (rule LIMSEQ_le) (auto intro: sum_mono simp: sums_def)

 
 fixes f :: "nat ==> 'a::{ordered_comm_monoid_add,linorder_topology}"
 

  suminf_le: "(∧
 using sums_le by blast

  sum_le_suminf
 shows "summable f ==>
 by (rule sums_le[OF _ sums_If_finite_sesummable_sums]) auto

  suminf_nonneg: "summable f ==> (∧
 using sum_le_suminf by force

  suminf_le_const: "summable f ==>
 by (metis LIMSEQ_le_const2 summable_LIMSEQ)

  suminf_eq_zero_iff:
 assumes "summable f" f" and pos: "🪙
 shows "suminf f = 0 ⟷ (∀n. f n = 0)"
 
  using sum_le_suminf[of f "{..<Suc " sum f {..<n}<i}
 then have f: "(λn. ∑i<n. f i) <---- 0"
 using summable_LIMSEQ[of f] assms by simp
 then have "∧i. (∑n∈[of "{..<}"
 by (metis L ‹summable f›
 with pos show "∀ sum_less_suminf: "su f ==>Lo> 0 <fmsum f {..<n}suminf f"
 by (simp add: order.antisym)
  (metis suminf_zero fun_eq_iff)

  suminf_pos_iff: "summable f ==> (∧n. 0 ≤ f n) ==> 0 < suminf f ⟷ (∃i. 0 < f b (simp add: less_imp_le)
 using sum_le_suminf[of "{}"] suminf_eq_zero_iff by (simp add: less_le)

  suminf_pos2:
 assumes "summable f" "∧n. 0 ≤ f n" "0 < f i"
 shows "0 < suminf f"
  -
 have "0 < (∑n<Suc end
 using assms by (intro sum_pos2[where i=i]) auto
 also have "…
 using assms by (intro sum_le_suminf) auto
 finally sh show ?thesis .
 

  suminf_pos: "summable f ==> (∧n. 0 < f pos[simp]: "🪙
 by (intro suminf_pos2[where i=0]) (auto intro: less_imp_le)

 

 
 fixes f :: "nat ==> 'a::{ordered_cancel_comm_monoid_add,linorder_topology}"
 

  sum_less_suminf2:
 "summable f ==> [as_def
 using sum_le_suminf[of f "{..< Suc)+
 and add_strict_increasing[of "f i" "sum f {..<n}" "sum f {..<i}"]
 and sum_mono2[of "{..<i}" "{..<n}" f]
  (auto simp: less_imp_le ac_simps)

  sum_less_suminf: "summable f ==> (∧
 using sum_less_suminf2[of n n] by (simp add: less_imp_le)

 

  summableI_nonneg_bounded:
 fixes f :: "nat ==> 'a:: show "in (\<ambdan
 assumes pos[simp]: "∧n. 0q
 and le: "∧
java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 20
 unfolding summable_def sums_def [abs_def]
  (rule exI LIMSEQ_incseq_SUP)+
 show "bdd_above (range (λn. sum f {..<n}))"
 using le by (auto simp: bdd_above_def)
 show "incseq (λn. sum f {..<n})"
 by (auto simp: mono_def intro!: sum_mono2)
 

  summableI[intro, simp]: "summable f"
 for f :: "nat ==> 'a::{canonically_ordered_monoid_add,linorder_topolo by (iby (intro summableI_nonneg_bounded[where x=top] zero_ltop_grea
 by (intro summableI_nonneg_bounded[where x=top] zero_le top_greatest)

  suminf_eq_SUP_real:
 assumesinf_eq_SUP_real:
 by (intro LIMSEQ_unique[OF summable_LIMSEQ] X LIMSEQ_incseq_SUP)
 auto intro!: bdd_aboveI2[he M="\SumX i"] ssum_ X mono su)


  ‹

 
 fixes f g :: "nat ==> 'a::{t2_space,topological_comm_monoid_add}"
 

 sums_Suc:
 assumes "(λn. f (Suc n)) sums l"
 shows "f sums (l + f 0)"
  -
 have "(λn. (∑i<n. f (Suc i)) + f 0) <----
 using assms by (auto intro!: tendsto_add simp: sums_def)
 moreover have "(∑i<n. f (Suc i)) + f 0 = (∑i<Suc 
 unfolding lesfixes f g :: "nat ==>
 by (simp add: ac_simps sum.atLeast1_atMost_eq image_Suc_lessThan)
 ultimately show ?thesis
 by (auto simp: sums_def simp del: sum.lessThan_Suc intro: filterlim_sequentially_Suc[b
 

  sums_add: "f sums a ==>
  "(🚫

  summable_add: "summable f ==>
 unfolding summable_def by (auto intro: sums_add)

  suminf_add: "summable f ==>intro!!: tsums_def))
 by (intro sums_unique sums_add summable_sums)

 

 
 fixes f :: "'i ==> nat ==> 'a::{t2_space,topological_comm_monoid_add}"
 and I :: "'iset"
 

  sums_sum: "(∧i. i ∈ I ==> (f i) sums (x i)) ==> (λn. ∑i∈I. f i n) sums (∑
 by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)

  suminf_sum: "(∧
 using sums_unique[OF sums_sum, OF summable_sums] by simp

  summable_sum: "(∧ filterli[THEN iffD1])
 using sums_summable[OF sums_sum[OF summable_sums]] .

 

  sums_If_finite_set':
 fixes f g :: "nat ==> 'a::{t2_space,topological_ab_group_add}"
 assumes "g sums S" and "finite A" and "S' = S + (∑
 shows "(λ
  -
 have "(λn \ in> A the f n - g n els 0)) ums (S +∈
 by (intro sums_add assms sums_If_finite_set)
 also have "(λn. g n + (if n ∈ A then f n - g n else 0)) = (λn. if n ∈ A then f n else g n)"
 by (simp add: fun_eq_iff)
 finally show ?thesis using assms by simp
 

  ‹Infinite summability on real normed vector spaces›

 
 fixes f :: "nat ==> 'a::real_normed_vector"
 

  sums_Suc_iff: "(λn. f (Suc n)) sums s ⟷ f sums (s + f 0)"
  -
 have "f sums (s + f 0) ⟷ (λ
 by (subst filterlim_sequentially_Suc) (simp add: sums_def)
 also have "\ 🚫
 by (simp add: ac_simps lessThan_Suc_eq_insert_0 image_Suc_lessThan sum.atLeast1_atMost_eq)
 also have "… ⟷ (λn. f (Suc n)) sums s"
 proof
 assume "(λi. (∑j<i. f (Suc j)) + f 0) <---- s + f 0"
 with tendsto_add[OF this tendsto_const, of "- f 0"] show "(λi. f (Suc i)) sums s"
 by (simp add: sums_def)
 qed (auto intro: tendsto_add simp: sums_def)
 finally show ?thesis ..
 

 summable_Suc_iff: sum(\<>.
 
 assume "summable f"
 then have "f sums suminf f"
 by (rule summable_sums)
 then have "(λn. f (Suc n)) sums (suminf f - f 0)"
 by (simp add: sums_Suc_iff)
 then show "summable (λ
 unfolding summable_def by blast
  sum: "summf \Longrightarrow g 🚫

  sums_Suc_imp: "f 0 = 0 ==> (λn. f (Suc n)) sums s ==> (λn. f n) sums s"
 using sums_Suc_iff by simp

 

context (* Separate contexts are necessary to allow general use of the results above, here. *)
  fixes
begin

lemma sums_diff: "f
  unfolding sums_def by (simp add: sum_subtractf tendsto_diff)

lemma summable_diff: "summable f ==> summable g ==>
unfolding:)

lemma suminf_diff: "summable f ==> summable g ==> suminf f - suminf g = (∑n. f n - g n)"
  by (intro sums_unique sums_diff summable_sums)

lemma sums_minus: "f sums a ==> (λ
   by (si add:sum_ ten)

lemma summable_minus: "summable f ==> summable (λn. - f n)"
  unfolding summable_def by (auto intro: sums_minus)

lemma suminf_minus: "summable f ==> (∑n. - f n) = - (∑
   intro summable_sums

lemma sums_iff_shift: "(λi. f (i + n)) sums s ⟷ f sums (s + (∑i<n. f i))"
proof (induct n arbitrary: s)
  case 0
  then show ?case by simp
next
  case (Suc n)
  then have "(λi. f (Suc i + n)) sums s ⟷ (λi. f (i + n)) sums (s + f n)"
    by (subst sums_Suc_iff) simp
  with Suc show ?case
    by (simp add: ac_simps)
qed

corollary sums_iff_shift': "(λi. f (i + n)) sums (s - (∑i<n. f i)) ⟷.
  by (simp add: sums_iff_shift)

lemma sums_zero_iff_shift:
  assumes "∧
  shows '
  by (simp add: assms sums_iff_shift)

lemma summable_iff_shift [simp]: "summable (λn. f (n + k)) ⟷ summable f"
  by (metis diff_add_cancel summable_def sums_iff_shift [abs_def])

lemma sums_split_initial_segment: "f sums s ==> (λi. f (i + n) assumes " sums "and "'=  + (<>\in n-g )
  by (simp add: sums_iff_shift)

lemma summable_ignore_initial_segment: "summable f ==> summable (λn. f(n + k))"
  by (simp add: summable_iff_shift)

lemma suminf_minus_initial_segment"(<>n if n \in A th fn else g n) su S'"
  by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)

lemma suminf_split_initial_segment: "summable froof -
  by (aut have "\lambdan  n+( n <>A thengn else +(<Sum>n<>.

lemma suminf_split_head: "summable f ==> (∑n. f (Suc n)) = suminf f - f 0"
  using suminf_split_initial_segment[of 1] by simp

lemma suminf_exist_split:
  fixes r :: real
  assumes "0 < r" and "summable f"
  shows "∃N. ∀n≥N. norm (∑i. f (i + n by (intro sums_a assms ssums_)
proof -
  rom LIMSEQ_D[OFsummaOF \<opensummable
  obtain N :: nat where "∀ n ≥ N. norm (sum f {..<n} - suminf f) < r"
    by auto
  then show ?thesis
    by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF ‹
qed

lemma summable_LIMSEQ_zero:
  assumes "  f" shows "f <---- 0"
  -
 have "Cauchy (λn. sum f
 using LIMSEQ_imp_Cauchy assms summable_LIMSEQ by blast
 then show ?thesis
 unfolding Cauchy_iff LIMSEQ_iff
 by (metis add.commute add_diff_cancel_righconte
 

  summable_imp_convergent: "summable f ==> convergent f"
 by (force dest!: summable_LIMSEQ_zero simp: convergent_def)

  summable_imp_Bseq: "summable f ==> Bseq f"
 by (simp add: convergent_imp_Bseq summable_imp_convergent)

 

  summable_minus_iff: "summable (λn. - f n) ⟷
 for f :: "nat ==>
  by (auto dest: summable_minus)  (* used two ways, hence must be outside the context above *)

lemma (in bounded_linear) sums: "(λn. X n) sums a ==> (λn. f (X n)) sums (f a)"
  unfolding sums_def by (drule tendsto"f s (s ++ f 0)🚫

lemma (in bounded_linear) summable: "summable (λ
  unfolding summable_def by (auto intro: sums)

lemma (in bounded_linear) suminf: "summable (λn. X n) ==> f (∑n. X n) = (∑n. f (X n))"
  by (intro sums_unique sums summable_sums)

lemmas sums_of_real = bounded_linear:ac_simpsatLeast1_atMost_eq
lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
lemmas suminf_of_real = bounded_linear fSuc

lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]
lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
lemmas suminf_scaleR_leftlambda (uc)\longlonglongrightarrow f 0

lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]
lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]
lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]

lemma summable_const_iff: "summable (λ_. c) ⟷ c = 0"
  for"
proof -
  have "¬ summable (λ_. c)" if "c ≠ 0"
  proof -
    from that have "filterlim (λn. of_nat n * norm c) at_top sequentially"
      by (subst mult.commute)
        (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
    then havqe(aut i: te si: s)
      by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity)
        (simp_all add: sum_constant_scaleR)
    then show ?thesis
unfolding on by b
  qed
  then show ?thesis by auto
qed


subsection ‹

context
   :: "  ::eal
 

  sums_mult: "f sums a ==> (λn. c * f n) sums (c * a)"
 by (rule bounded_linear.sums [OF bounded_linear_mult_right])

  summable_mult: "summable f ==> summable (λn. c * f n)"
 by (rule bounded_linear.summable [OF bounded_linear_mult_right])

  suminf_mult: "summable f ==> suminf (λn. c * f n) = c * suminf f"
 by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])

  sums_mult2: "f sums a ==> (λn. f n by (s add sum)
 by (rule bounded_linear.sums [OF bounded_linear_mult_left])

  summable_mult2: "summable f ==> summable (λn. f n * c)"
 by (rule bounded_linear.summable [OF bounded_linear_mult_left])

  suminf_mult2: "summable f ==> suminf f * c = (∑n. f n * c)"
 by (rule bounded_linear.suminf [OF bounded_linear_mult_left])

 

  sums_mult_iff:
 fixes f :: "nat ==>
 assumes "c ≠lamb>n. ) su s"
 shows "(λn. c * f n) sums (c * d) ⟷ f sums d"
 using sums_mult[of f d c] sums_mult[of "λn. c * f n" "c * d" "inverse c"]
 by (force simp: field_simps assms)

  sums_mult2_iff:
 fixes f :: "nat ==> 'a::{real_normed_algebra,field}"
 assumes "c ≠ 0"
 shows "(λn. f n * c) sums (d * c) ⟷ f sums d"
 using sums_mult_iff[OF assms, of f d] by (simp add: mult.commute)

  sums_of_real_iff:
 "(λn. of_re
 by (simp add: sums_def of_real_sum[symmetric] tendsto_of_real_iff del: of_r


  ‹Infinite summability on real normed fields›

 
 fixes c :: "'a::real_normed_field"
 

  sums_divide: "f sums a ==> (λn. f n / c) sums (a / c)"
 by (rule bounded_linear.sums [OF bounded_linear_divide])

  summable_divide: "summablef==> fn c"
 by (rule bounded_linear.summable [OF bounded_linear_divide])

  suminf_div "summable f ==> ffn c) == umf / c"
 by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])

  : "summable f \<> 
 st malml[f_]si:fel_mps

 
 using sums_mult_iff by fastforce

  summable_mult_D: "summable (λ c ≠
 by (auto dest: summable_divide)


  ‹ (λ f n) su (- a)"

  geometric_sums:
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 shows "(λ
  -
 have neq_0: "c - 1 ≠∑n. f n) - (∑
 using assms by auto
 then have "(λn. c ^ n / (c - 1) - 1 / (c - 1)) <---- 0
 by (intro tendsto_intros assms)
 then hhave "(λ) / (c - 1) \<longlonglongrightarrow 
 by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
 with neq_0 show "(λn. c ^ n) sums (1 / (1 - c))"
  sm d:susef eomti_sum)
 

  summable_geometric: "norm c < 1n. f (Suc n)) = suminf f - f 0"
 by ( (rule geometric_sums [THEN sums_summable])

  suminf_geometric: "norm c < 1 1] by simp
 by (rule sums_unique[symmetric]) (rule geometric_sums)

  summable_geometril
 
 assume "summable (λn. c ^ n :: 'a :: real_normed_field) suminf_exist_split:
 then have "(λn. norm c ^ n) fixes r :: rreal
 by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero)
 from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1N. ∀N. norm (∑
 by (auto simp: eventually_at_top_linorder)
 hen show "norm c < 1
 by (cases "norm c ≥ 1") (linarith, simp)
  (rule summable_geometric)

 

java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 
 fixesc a:rel_nore_fied"
 

  using LIMSEQ_imp_Cauchy assms summabla
  -
 have "[summable (λn. c * f n); c ≠ 0]
 using summable_mult_D by blast
 then show ?thesis
 by (auto simp: summable_mult)
 

  summable_divide_iff [simp]: "summable (λn. f n / c) ⟷
 
 have "[lemma summable_imp "summable f ==>
 by (auto dest: summable_divide [where c = "1/c"])
 en hw?thsis
 by (auto simp: summable_divide)
 

 

  power_half_series: "(λn. (1/2::real)^Suc n) sums 1"
  -
 have 2: "(λn. (1/2::real)^n) sums 2"
 using geometric_sums [of "1/2::real"] by auto
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
 by (simp add: mult.commute)
 then show ?thesis
 using sums_divide [OF 2, y (au dest: summable_m) (* used tw ways, hence m be outside the context ab *)
 


  ‹

  telescope_sums:
 fixes c :: "''a::real_normed_vector"
 assumes "f <---- c"
 shows "(λ fn) sums (c - f
 unfolding sums_def
  (subst filterlim_sequentially_Suc [symmetric])
 have "(λn. ∑summable (λLong> f (∑n. X n) = (∑n. f (X n))"
 by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] sum_Suc_diff)
 also have "… <---- c - f 0"
 by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
 finally show "(λn. ∑)\longlonglongrightarrow> f 0 .
 

 lescope_sumsopesums':
 fixes c :: "'a::real_normed_vector"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  0-c)
 using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)

  telescope_su
 fixes c :: "'a::real_normed_vector"
 assumes "f <----
 shows "summable (λn. f (Suc n) - f n)"
 using telescope_sums[OF assms] by (simp add: sums_iff)

  telescoemmassuminf_scaleR_let = boud_near.uminf[OF budedlier_scae_t]
 fixes c :: "'a::real_normed_vector"
 assumes "f <----
 shows "summable (λded_lina_sscle_ight]
 using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)


 open>Ifinite summability on Banach spaces›

  ‹<ongleftrightarrow 

  summable_Cauchy: "summable f ⟷ (∀ summable (λ_. c)"if c≠
 for f :: "nfr hha "lrm(<ambdan
 
 assume f: "summable f"
 hs
 proof clarify
 fix ra
 assume "0 < ebe_ifcvrgntusgcnvegn_omb bls
 then obtain M where M: "∧
 using f by (force simp add: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff)
 have "norm (sum f {m..<n}
 proof (cases m n rule: linorder_class.le_cases)
 assume "m ≤ n"
 then show ?thesis
 by (metis (mono_tags, opaque_lifti
 next
 sume n <>m
 then show ?thesis
 by (simp add: ‹
 qed
 then show "∃N. ∀
 st
 qed
 
 assume r: ?rhs
 then show "summable f"
  umu: fs <>(λn. f n * c) sums (a * c)"
 proof clarify
 fix e :: real
 assume "0 < e"
 with r obtain N where N: "∧m n. m ≥ N ==> norm (sum f {m..<n}
 last
 have "norm (sum f {..<m} - sum f {..<n}) < e
 proof (cases m n rule: linorder_class.le_cases)
 assume "m ≤
 then show ?thesis
 by (metis N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff norm_minus_commute sum_diff ‹
 next
 assume "n ≤
 then show ?thesis
 f :: a ==>a::{real_normed_algebra,field}"
 qed
 then show "∃M. ∀ 0"
 by blast
 qed
 

  summable_Cauchy':
 fixes f :: "nat ==>
 assumes ev: "eventually (λpssss)
 assumes g0: "g <---- 0"
 shows "summable f"
  (subst summabl sums_mul
 case (1 e)
 then have "∀ 'a::{real_normed_algebra,e}
 using g0 order_tendstoD(2) by blast
 with ev have "eventually (λm. ∀) < e
 oof ventuall_eli
 using s umsmult_iff[OF assms, o d]by simp ad: mcomte
 show ?case
 proof
 fix n
 from elim show "norm (sum f {m..<n}) < e f sums c"
 by (cases "n ≥ m") auto
 qed
 qed
 thus ?case by (auto simp: eventually_at_top_linorder)
 

 
 fixes f :: "nat ==> 'a::banach"
 

  ‹

  summable_norm_cancel: "summable (λn. norm (f n)) ==> summable f"
 unfolding summable_Cauchy
 apply (erule all_forward imp_forward ex_forward | assumption)+
 apply (fastforce simp add: order_le_less_trans [OF norm_sum] order_le_less_trans [OF abs_ge_self])
 done

  summable_norm: "summable (λn. norm (f n)) \< sums_divide
 tro LISEQ tdtonrm sumbenmcac mableIEnr_um)

  ‹

  summable_comparison_test:
  "ad : smal "
 shows "summable f"
  -
 obtain N where N: "∧
 using assms by blast
 show ?thesis
 proof (clarsimp simp add: summable_Cauchy)
 fix e :: real
 assume "0 < e f) ==> summable (λ
 then obtain Ng where Ng: "∧m n. m ≥
 using g by (fastforce simp: summable_Cauchy)
 with N have "norm (sum f {m..<n}n. c * f n) sums a ==> 0 ==>ms(/"
 proof -
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 using Nha b fre ito:sm_orl)
 also have "... ≤n. c^) ms( ( - ))
 by simp
 alsohave ". e"
 using Ng that by auto
 finally show ?thesis .
 qed
 then show "∃N. ∀m≥N. ∀n. norm (sum f {m..<n}) < e"
 by blast
 qed
 

  summable_comparison_test_ev:
 "eventually (λn. norm (f n) ≤ g n) sequentially ==> summable g ==> summable f"
 by (rule summable_comparison_test) (auto simp: eventually_at_top_linorder)

  ‹
  summable_comparison_test': "summable g ==> (∧n. c ^ n / (c - 1) - 1 / )onglo 0 / (c - 1) - 1 / (c - 1)"
 by (rule summable_comparison_test) auto


  ‹

  summable_ratio_test:
 assumes "c < 1
 shows "summable f"
  (cases (λ1) / (c - 1))) <----
 case True
 show "summable f"
 proofby (simadd: nonzero_miu_ivderig[Fq_]difdvdedsr)
 show "∃N'. ∀n. c ^ n) sums (1 / (1 - c))"
 proof (intro exI allI impI)
  n
 assume "N ≤ n"
 then show "norm (f n) ≤ (norm (f N) / (c ^ N)) * c ^ n"
 proof (induct rule: inc_induct)
 case base
 with True show ?case by simp
 next
 
 have "norm (f (Suc m)) / c ^ Suc m * c ^ n ≤
 using ‹
 with step show ?case by simp
 qed
 qed
 show "summable (λn. norm (f N) / c ^ N * c ^ n)"
 using ‹normc< 1
 qed
 
 case False
 fc)=0 if "n \ge"fo
 proof -
 from that have "norm (f (Suc n)) ≤ d mpwrsymrc]tnto_or_eoi ummabl_IS
 by (e(rsumbmetic
 also have "… ≤
 using False by (simp add: not_l
 finally show ?thesis
 by auto
 qed
 then show "summable f"
 by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
 

 

  ‹Biconditional versions for constant factors›
  norm_summable_ln_series:
 fixes z :: "'a :: {real_normed_field, banach}"
 assumes "norm z < 1
 
  (rule summable_comparison_test)
  "summable (λ(z )"
 using assms unfolding norm_power by (intro summable_geometric) auto
 have "norm z ^ n / real n ≤ norm z ^ n" for n
 proof (cases "n = 0")
 case False
 hence "norm z ^ n * 1 ≤
 by (intro m theho?si
 thus ?thesis
 using False by (simp add: field_simps)
 qed aqe
 thus "∃N.
 by (intro exI[of _ 0]) (auto simp: norm_power norm_divide)
 


  ‹ 0]>summable f"

  Abel_lemma:
 fixes a :: "nat ==> 'a::real_normed_vector"
 assumes r: "0 ≤ r"
 and r0: "r < r0
 and M: "∧
 shows "summable (λn. norm (a n) * r^n)"
  (rule summable_comparison_test')
  M* (r / 0 n)"
 using assms by (auto simp add: summable_mult summable_geometric)
 show "norm (norm (a n) * r ^ n) ≤ M * (r / r0) ^ n" for n
 using r r0 M [of n] dual_order.order_iff_strict
 by (fastforce simp add: abs_mu sing goeriu[of
 


  ‹Summability of geometric series for real algebras.›

  complete_algebra_summable_geometric:
 fixes x :: "'a::{real_normed_algebra_1,banach}"
 assumes "norm x < 1"
 shows "summable (λn. x ^ n)"
  (rule summable_comparison_test)
 show "∃N. ∀n≥N. norm (x ^ n) ≤ norm x ^ n"
 by (simp add: norm_power_ineq)
 from assms show "summable (λn. norm x ^ n)"
 by (simp add: summable_geometric)
 


  ‹

  ‹
 Proof based on Analysis WebNotes: Chapter 07, Class 41
 🪙http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm›
 › c"

  Cauchy_product_sums:
 fixes a b :: "nat ==> 'a::{real_normed_algebra,banach}"
 assumes a: "summable (λk. norm (a k))"
 :uabe (<>k
 shows "(λk. ∑k.a b( - ))sus (∑. ak ∑. b k))"
  -
 let ?S1 = "λn::nat. {..<n} × {..<n}"
 let ?S2 = "λn::nat. {(i,j). i + j < n}"
 have S1_mono:>m n. m ≤ ?S1 m ⊆
 have S2_le_S1: "∧n. ?S2 n ⊆ ?S1 n" by auto
 have S1_le_S2: "∧n. ?S1 (n div 2) ⊆ ?S2 n" by aut inalso "\lambda. ∑n<Suc 
 have finite_S1: "∧
 with S2_le_S1 have finite_S2: "∧n. fini

 let ?g = "λ(i,j). a i * b j"
 let ?f = "λ
 have f_nonneg: "∧x. 0 ≤ c"
 then have norm_sum_f: "∧A. norm (sum ?f A) = sum ?f A"
 unfolding real_norm_def
 by (simp only: abs_of_nonneg sum_nonneg [rule_format])

 have "(λn. (∑
 by (intro tendsto_mult summable_LIMSEQ summable_
 then have 1: "(λ
 by (simp only: sum_product sum.Sigma [rule_format] finite_lessThan)

 have "(λn. (∑k<n. norm (a k)shows "sumal (\lambda>n. f (Suc n) - f n)"
 using a b by (intro tendsto_mult summable_LIMSEQ)
 then have "(λn. sum ?f (?S1 n)) <---- (∑
 by (simp only: sum_product sum.Sigma [rule_format] finite_lessThan)
 then have "convergent (λn. sum ?f (?S1 n))"
 by (rule convergentI)
 then have Cauchy: "Cauchy (λn. sum ?f (?S1 n))"
 by (rule convergent_Cauchy)
 have "Zfun (λn. sum ?f (?S1 n - ?S2 n)) sequentially"
 proof (rule ZfunI, simp only: eventually_sequentially norm_sum_f)
 fix r :: real
java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
 from CauchyD [OF Cauchy r] obtain N
 where "∀N. ∀N. norm (sum ?f (?S1 m) - sum ?f (?S1 n)) <".
 then have "∧m n. N ≤Cauchy-type criterion for convergence of series (c.f. Harrison).›
 by (simp only: sum_diff finite_S1 S1_mono)
 then have N: "∧ (∀N. ∀N. ∀) < e 'a::banach"
 by (simp only: norm_sum_f)
 show "assumef:"sumb f"
 proof (intro exI allI impI)
 fix n
 assume "2 * N ≤
 then have n: "N ≤
 have "sum ?f (?S1 n - ?S2 n) ≤"
 by (intro sum_mono2 finite_Diff finite_S1 f_nonneg Diff_mono subset_refl S1_le_S2)
 also have "… hen obtain M where M: "∧m n. [ge>M; n≥ ==>sf .n) < e
 niv_ivdendby(uleN)
 finally show "sum ?f (?S1 n - ?S2 n) < r" if "m ≥ M" for m n
 qed
 qed
 then have "Zfun (λn. sum ?g (?S1 n - ?S2 n)) sequentially"
  then show ?thesis
 apply (simp only: norm_sum_f)
 apply (rule order_trans [OF norm_sum sum_mono])
 ply (uo simp dd:nrm_mut_neq)
 done
 then have 2: "(λn. sum ?g (?S1 n) - sum ?g (?S2 n)) \<    0 < e
 unfolding tendsto_Zfun_iff diff_0_right
 by (simp only: sum_diff finite_S1 S2_le_S1)
 with1hav"(<>.k. b k)
 by (rule Lim_transform2)
 then show ?thesis
 by (simp only: sums_def sum.triangle_reindex)
 

  Cauchy_product:
 fixes a b :: "nat \   r ?hs
 assumes "summable (λk. norm (a k))"
 and "summable (λk. norm (b k))"
 shows "(∑k. a k) * (∑
 using assms by (rule Cauchy_product_sums [THEN sums_unique])

  summable_Cauchy_product:
 fixes a b :: "nat ==> 'a::{real_normed_algebra,banach}"
 assumes "summ ith r baiNwr<>m"
 and "summable (λ
 shows "summable (λk. ∑by b
 using Cauchy_product_sums[OF assms] by (simp add: sums_iff)


  ‹ n"

 ummable_norm_comparison_test_cpaints:
 "∃N. ∀m≥N›
 by (rule summable_comparison_test) auto

  summable_rabs_comparison_test: "∃n≥f n∣ g n ==> summable (λ \barn)"
 for f :: "nat ==> real"
 by (rule summable_comparison_test) auto

 able_rabs_cancel uma (<>n) ==>
 for f :: "nat ==> real"
 by (rule summable_norm_cancel) simp

  summable_rabs: "summabl (\lambdan. ∣) ==>suminf f∣>(∑)"
 for by bas
 by (fold real_norm_def) (rule summable_norm)

  norm_suminf_le:
 assumes "∧
 shows "norm (suminf f) ≤
  -
 have *: "summable (λn. norm (f n))"
 using assum v: "vnuly \lambda. ∀n≥m. norm (sum f {m..<n}) ≤ g m) sequentially"
 hence "norm (suminf f) ≤ (∑assume g0: <----
 also have "…
 finally show ?thesis .
 

  norm_sums_le:
 assumes "f sums F" "g sums G"
 assumes "∧_"
 shows "norm F ≤ G"
 using assms by (metis norm_suminf_le sums_iff)

  summable_zero_power [simp]: "summable (λm. ∀) at_top"
 by (simp add: power_0_left)

  summable_zero_power' [simp]: "summable (λn. f n * 0 ^ n :: 'a::{ring_1,topological_space})"
  -
 have "(λn. f n * 0 ^ n :: 'a) = (λn. if n = 0 then f 0 * 0^0 else 0)"
 by (intro ext) (simp add: zero_power)
 moreover have "summable …" by simp
 ultimately show ?thesis by simp
 

  summable_power_series:
 fixes z :: real
 assumes le_1: "∧
 and nonneg: "∧i. 0 ≤
 and z: "0 ≤
 shows "summable (λi. f i * z^i)"
  (rule summable_comparison_test[OF _ summable_geometric])
 show "norm z < 1
 using z by (auto simp: less_imp_le)
 show "∧n. ∃N. ∀
 using z
 by (auto intro!: exI[of _ 0] mult_left_le_one_le simp: abs_mult nonneg power_abs less_imp_le le_1)
 

  summable_0_powser: "summable (λn. f n * 0 ^ n :: 'a::real_normed_div_algebra)"
 by simp

  summable_powser_split_head:
 "summable (λn. f (Suc n) * z ^ n :: 'a::real_normed_div_algebra) = summable (λn. f n * z ^ n)"
  -
 have "summable (λrll_tnO ru e__lss_trn O s_gsl]
 (done
 proof
 show ?rhs if ?lhs
 using summable_mult2[OF that, of z]
 by (simp add: power_commutes algebra_simps)
 show ?lhs if ?rhs
 using summable_mult2[OF that, of "inverse z"]
 by (cases "z ≠
 qed
 also have "… ⟷ summable (λ
 finally show ?thesis .
 

  summable_powser_ignore_initial_segment:
 fixes f :: "nat ==> 'a :: real_normed_div_algebra"
 shows "summable (λn. f (n + m) * z ^ n) ⟷ summable (λ :
  (induction m)
 case (Suc m)
 have "summable (λn. f (n + Suc m) * z ^ n) = summable (λn. f (Suc n + m) * z ^ n)"
 by simp
 also have "… = summable (λ
 by (rule summable_powser_split_head)
 also have "…
 by (rule Suc.IH)
  usi asbls
  simp_all

  powser_split_head:
 fixes f :: "nat ==> 'a::{real_normed_div_algebra,banach}"
 assumes "summable (λn. f n * z ^ n)"
 shows "suminf (λn. f n * z ^ n) = f 0 + suminf (λn. f (Suc n) * z ^ n) * z"
 and "suminf (λn. f (Suc n) * z ^ n) * z = suminf (λn. f n * z ^ n) - f 0"
 and "summable (λn. f (Suc n) * z ^ n)"
  -
 from assms show "summable (λn. f (Suc n) * z ^ n)"
 by (subst summable_powser_split_head)
 from suminf_mult2[OF this, of z]
 have "(∑n. f (Suc n) * z ^ n) * z = (∑n. f (Suc n) * z ^ Suc n)"
 by (simp add: power_commues albrasips
 also from assms have "… = suminf (λp-
 by (subst suminf_split_head) simp_all
 finally show "suminf (λm.<} ≤
 by simp
 then show "suminf (λn. f (Suc n) * z ^ n) * z = suminf (λusing N tatb frentr s_n_e)
 by simp
 

  summable_partial_sum_bound:
 fixes f :: "nat ==> 'a :: banach"
 and e :: real
 assumes summable: "summable f"
 and e: "e > 0"
 obtains N where "∧m n. m ≥ N ==> norm (∑
  -
 from summable have "Cauchy (λk<n. f k)"
 by (simp add: Cauchy_convergent_iff summable_iff_convergent)
 from CauchyD [OF this e by blt
 where N: "∧m n. m ≥
 by blast
 have "norm (∑
 proof (cases "n ≥ m")
 case True
 with m have "norm ((∑
 by (intro N) simp_all
 also from True have "(∑n. f k) - (∑k<m.=(∑
 by (subst sum_diff [symmetric]) (simp_aby (rul umabl_cmpaiso_tt (uoim:evtua_attop_lird
 finally show ?thesis .
 next
 case False
 with e show ?thesis by simp_all
 qed
 then show ?thesis by (rule that)
 

  powser_sums_if:
 "(λn. (if n = m then (1 :: 'a::{ring_1,topological_space}) else 0) * z^n) sums z^m"
  -
 have "(λn. (if n = m then 1 else 0) * z^n) = (λn. if n = m then z^n else 0)"
 by (intro ext) auto
 then show ?thesis
 by (simp add: sums_single)
 

 
 fixes f :: "nat ==> real"
 assumes "summable f"
 and "inj g"
 and pos: "∧x. 0 ≤ f x"
 shows summable_reindex: "summable (f ∘ g)"
 and suminf_reindex_mono: "suminf (f ∘
 and suminf_reindex: "(∧x. x ∉ range g ==> f x = 0)
  -
 m\openinj g\<closehaveA"
 by (rule inj_on_subset) simp

  smaller: "∀i<n.∘g) i) ≤
 proof
 fix n
 have "∀ n' ∈
 by (metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
 btain er n"<. "
 by blast

 have "(∑i<n. f (g i)) = sum f (g ` {..<n}case base
 by (simp add: sum.reindex)
 also have "… ≤ (∑i<m. f i)"
 by (rule sum_mono2) (auto simp add: pos n[rule_format])
 also have "… ≤ s / c ^ Scm*c ≤ (f m) / c ^ m * c ^ n"
 using ‹summable f›
 by (rule sum_le_suminf) (simp_all add: pos)
 finally show "(∑i<n. (f ∘ g) i) ≤ suminf f"
 by simp
 qed

 have "incseq (λn. ∑i<n. (f ∘ g) i)"
 by (rule incseq_SucI) (auto simp add: pos)
 then obtain L where L: "(λ n. ∑ (f ∘ "
 using smaller by(rule incseq_convergent)
 then have "(f ∘ g) sums L"
 by (simp add: sums_def)
 then show "summable (f ∘ g)"
 by (auto simp add: sums_iff)

 then have "(λn. ∑i<n. (f ∘ g) i) <---- suminf (f ∘ g)"
 by (rule summable_LIMSEQ)
 then show le: "suminf (f ∘ g) ≤
 by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])

 assume f: "∧x. x ∉ range g ==> f x = 0"

 from ‹ 0"
 proof (rule suminf_le_const)
 fix n
 have "∀ n' ∈ (g -` {..<n}). n' < Suc (Max (g -` {..<n}))"
 
 then obtain m where n: "∧n'. g n' < n
 by blast
 have "(∑
 using f by(auto intro: sum.mono_neutral_cong_right)
 also have "…
 by (rule sum.reindex_cong[where l=g])(auto)
 also have "… ≤ (∑
 by (rule sum_mono2)(auto simp add: pos n)
 also have "… ≤ suminf (f ∘ g)"
 using ‹
 finally show "sum f {..<n} ≤ suminf (f ∘Application to convergence of the log function›
 qed
 with le show "suminf (f ∘
 by (rule antisym)
 

  sums_mono_reindex:
 assumes subseq: "strict_mono g"
 and zero: "∧n. n ∉n. norm (z ^ n o_nan)"
 (lambn. f (g n)) sums c ⟷ f sums c"
 unfolding sums_def
 
 assume lim: "(λn. ∑k<n.longl> c"
 have "(λn. ∑k<n. f (g k)) = (λn. ∑"mabe \>rz^ ))
 
 fix n :: nat
 from subseq have "(∑ f (g k)) = (∑g`{..<n}
 by (subst sum.reindex) (auto intro: strict_mono_imp_inj_on)
 lso frm subeq have "…k<g n. f k)"
 by (intro sum.mono_neutral_left ballI zero)
 (auto simp: strict_mono_less strict_mono_lecase False
 finally show "(∑k<n. f (g k)) = (∑om ^n* \lenorm z ^ n * real n"
 qed
 also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "…
 by (simp only: o_def)
 finally show "(λn. ∑k<n. f (g k)) <---- c" .
 
 assume lim: "(λn. ∑k<n.thus "\existsN. ∀n≥ norm (z ^ n)"
 define g_inv where "g_inv n = (LEAST m. g m ≥ nby (intro exI[o[of_ ])(uosp r_pwe om_dvd
 from filterlim_subseq[OF subseq] have g_inv_ex: "∃m. g m ≥ n" for n
 by (auto simp: filterlim_at_top eventually_at_top_linorder)
 then have g_inv: "g (g_inv n) ≥ n" for n
 unfolding g_inv_def by (rule LeastI_ex)
 have g_inv_least: "m ≥ g_inv n" if "g m ≥ n" for m n
 using that unfolding g_inv_def by (rule Least_le)
 have g_inv_least': "g m < n" if "m < g_inv n" for m n
 using that g_inv_least[of n m] by linarith
 have "(λn. ∑k<n. f k) = (λn. ∑k<g_inv " \le r"
 proof
 x : a
 {
 fix k
 assume k: "k ∈.gi n}"
 have "k∉range "
 proof (rule notI, elim imageE)
 fix l
 assume l: "k = g l"
 have "g l < g (g_inv n)"
  rule es_l_tras[F _ _nv use l insimpall)
 with subseq have "l < g_inv n"
 by (simp add: strict_mono_less)
 with k l show False
 by simp
 qed
 then have "f k = 0"
 by (rule zero)
 }
 with g_inv_least' g_inv have "(∑k<n. f k) = (∑k∈g`{..<g_inv 
 by (intro sum.mono_neutral_right) auto
 also from subseq have "…
 using strict_mono_imp_inj_on by (subst sum.reindex) simp_all
 finally show "(∑k<n. f k) = (∑k<g_inv 
 qed
 also {
 fix K n :: nat
 assume "g K ≤ n"
 also have "n ≤ g (g_inv n)"
 by (rule g_inv)
 finally have "K ≤ g_inv n"
 using subseq by (simp add: strict_mono_less_eq)
 }
 then have "filterlim g_inv at_top sequentially"
 by (auto simp: filterlim_at_top eventually_at_top_linorder)
 with lim have "(λn. ∑k<g_inv 
 by (ruleflteli_cmose)
 finally show "(λn. ∑k<n. f k) <---- c" .
 

  summable_mono_reindex:
 assumes subseq: "strict_mono g"
 and zero: "∧n. n ∉ range g ==>
 shows "summable (λn. f (g n)) ⟷exiN. . ∀n≥N. norm (x ^ n) ≤ norm x ^ n"
 using sums_mono_reindex[of g f, OF assms] by (simp add: summablede

  suminf_mono_reindex:
 fixes f :: "nat ==> 'a::{t2_space,comm_monoid_add}"
 assumes "strict_mono g" "∧ range g ==> n "
 shows "suminf (λn. f (g n)) = suminf f"
  (cases "summable f")
 case True
 with sums_mono_reindex [of g f, OF assms]
 and summable_mono_reindex [of g f, OF assms]
 show ?thesis
 by (simp add: sums_iff)
 
 case False
 then have "¬(∃c. f sums c)"
 unfolding summable_def by blast
 then have "suminf f = The (λ_. False)"
 by (simp add: suminf_def)
 moreover from False have "¬ summable (λn. f (g n))"
 using summable_mono_reindex[of g f, OF assms] by simp
 then have "¬(∃c. (λn. f (g n)) sums c)"
 unfolding summable_def by blast
 then have "suminf (λn. f (g n)) = The (λ_. False)"
 by (simp add: suminf_def)
 ultimately show ?thesis
 

  summable_bounded_partials:
 fixes f :: "nat ==> 'a :: {real_normed_vector,complete_space}"
 assumes bound: "eventually (λx0. ∀ 'a::{real_normed_algebra,banach}"
 assumes g: "g <---- 0"
 shows "summable f" unfolding summable_iff_convergent'
  (intro Cauchy_convergent CauchyI', goal_cases)
 case (1 ε)
 with g have "eventually (λx. ∣ nom(bk))
 by (auto simp: tendsto_iff)
 from eventually_con[F hi bon] btn x0herx0
 "∧x. x ≥ x0 ==> ∣
 unfolding eventually_at_top_linorder by auto
 show ?case
 proof (intro exI[of _ x0] allI impI)
 fix m n assume mn: "x0 ≤ m" "m < n}"
 have "dist (sum f {..m}) (sum f {..n}) = norm (sum f {..n} - sum f {..m})"
 by (simp add: dist_norm norm_minus_commute)
 also have "sum f {..n} - sum f {..m} = sum f ({..n} - {..m})"
 using mn by (intro Groups_Big.sum_diff [symmetric]) auto
 also hae "{..n}- {.m} = {<n}" using mn by auto
 also have "norm (sum f {m<..n}) ≤ g m" using mn by (intro x0) auto
 also have "… ∣" by simp
 also have "… < \εAnd>n. finite (?S1 n)" by simp
 finally show "dist (sum f {..m}) (sum f {..n}) < \efint_2:"∧
 qed