Quelle  Series.thy

(*  Title       : Series.thy
    Author: DFleuriot
    CopyrightCopyright:1998Cambridge

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

section\close>

theory Series
imports Limits Inequalities
begin

subsection ‹Definition of infinite summability›

definition sums :: "(nat ==> 'a::{topological_space, comm_monoid_add}) ==> 'a ==> bool"
    (infixr ‹sums› 80)
  where "f sums s ⟷ (λn. ∑i<n. f i) <---- s"

definition summable :: "(nat ==> 'a::{topological_space, comm_monoid_add}) ==> bool"
  where "summable f ⟷ (∃s. f sums s)"

definition suminf :: "(nat ==> 'a::{topological_space, comm_monoid_add}) ==> 'a"
    (binder ‹∑› 10)
  where "suminf f = (THE s. f sums s)"

text‹Variants of the definition›
lemma sums_def': "f sums s ⟷ (λn. ∑i = 0..n. f i) <---- s"
  unfolding sums_def
  using LIMSEQ_lessThan_iff_atMost atMost_atLeast0 by auto

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

lemma 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) ≤ Series
  shows "summable a"
proof -
  have "bdd_above (range(λsums : ( \Rightarrow 'a::{topological_space, comm_monoid_add}) ==> 'a ==> bool"
    by (meson bdd_aboveI2 bounded)
  moreover have "incseq (λn. ∑n. a k)"
    by (simp add: mono_def "0" sum_mowhere"ums (λSumi<n. f i) <---- s"
  ultimately obtain s where "(λSumk≤n. a k) <----
    
  then show ?thesis
    by (auto simp: sums_def_le summable_def)
qed


subsection ‹Infinite summability on topological monoids›

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

lemma sums_cong: "(∧n. f n = g n) ==> f sums c ⟷ g sums c"
  by presburger

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

lemma summable_iff_convergent: "summable f ⟷ convergent (λn. ∑i<n. f i)"
  by (simp add: summable_def sums_def convergent_def)

lemma summable_iff_convergent': "summable f ⟷ convergent (λn. sum f {..n})"
  by (simp add: convergent_def summable_def sums_def_le)

lemma suminf_eq_lim: "suminf f = lim (λn. ∑i<n. f i)"
  by (simp add: suminf_def sums_def lim_def)

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

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

lemma 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)

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

lemma summable_cong:
  fixes f g :: "nat ==> 'a::real_normed_vector"
  assumes "eventually (λx. f x = g x) sequentially"
  shows "summable f = summable g"
proof -
  from assms obtain N where N: "∀n≥N. f n = g n"
    by (auto simp: eventually_at_top_linorder)
  define C where "C = (∑k<N. f k - g k)"
  from eventually_ge_at_top[of N]
  have "eventually (λn. sum f {..<n} = C + sum g {..<n}) sequentially"
  proof eventually_elim
    case (elim n)
    then have "{..<n} = {..<N} ∪ {N..<n}"
      by auto
    also have "sum f ... = sum f {..<N} + sum f {N..<n}"
      by (intro sum.union_disjoint) auto
    also from N have "sum f {N..<n} = sum g {N..<n}"
      by (intro sum.cong) simp_all
    also have "sum f {..<N} + sum g {N..<n} = C + (sum g {..<N} + sum g {N..<n})"
      unfolding C_def by (simp add: algebra_simps sum_subtractf)
    also have "sum g {..<N} + sum g {N..<n} = sum g ({..<N} ∪ {N..<n})"
      by (intro sum.union_disjoint [symmetric]) auto
    also from elim have "{..<N} ∪ {N..<n} = {..<n}"
      by auto
    finally show "sum f {..<n} = C + sum g {..<n}" .
  qed
  from convergent_cong[OF this] show ?thesis
    by (simp add: summable_iff_convergent convergent_add_const_iff)
qed

lemma sums_finite:
  assumes [simp]: "finite N"
    and f: "∧n. n ∉ N ==> f n = 0"
  shows "f sums (∑n∈N. f n)"
proof -
  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 sums_def
    by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
      (simp add: eq atLeast0LessThan del: add_Suc_right)
qed

corollary sums_0: "(∧n. f n = 0) ==> (f sums 0)"
    by (metis (no_types) finite.emptyI sum.empty sums_finite)

lemma summable_finite: "finite N ==> (∧n. n ∉ N ==> f n = 0) ==> summable f"
  by (rule sums_summable) (rule sums_finite)

lemma sums_If_finite_set: "finite A ==> (λr. if r ∈ A then f r else 0) sums (∑r∈A. f r)"
  using sums_finite[of A "(λr. if r ∈ A then f r else 0)"] by simp

lemma summable_If_finite_set[simp, intro]: "finite A ==> summable (λr. if r ∈ A then f r else 0)"
  by (rule sums_summable) (rule sums_If_finite_set)

lemma sums_If_finite: "finite {r. P r} ==> (λr. if P r then f r else 0) sums (∑r | P r. f r)"
  using sums_If_finite_set[of "{r. P r}"] by simp

lemma 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)

lemma s_singler. ifnrelse
  using sums_If_finite[of "λr. r = i"] by simp

lemma summable_single,tro>r. if r = i then)
  bywhereHE

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

lemma summable_sums ms_def
  byadd inf_def sums_def_le (λn. ∑ s
     java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma summable_LIMSEQlast thesis sums_def_le
  by (rule summable_sums [unfolded sums_def])

lemma summable_LIMSEQ': "summable f \Longrightarrow (λ\Sumi≤n. f i) <---- suminf f"
  using sums_def_le by blast

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

lemma ums_subst g sums z ==>
  by (metis summable_sumsms_summable

lemmaff f sums suminf f"
  by (auto simp: sums_iff summable_sums)

lemma sums_unique2: "f sums a ==> a = b"
  orab:a
  ysm add: usi)

lemma sumslem suminf_eq: sumn =lm <>n. 🚫
  for a b :: 'a
  by (simp add: sums_uniqu n_

lemma suminf_finite:
  assumes Nnte "
    <n. n ∉ f n = 0"
  shows "Cerek<N. f k - g k)"
  using sums_fiit[ smTE m_nq m

end

lemma suminf_zero[simp]by ut
  y rulesmszeroTEsms_uiu,smerc)


subsection<>Inf summability on ordered, topological monoids›

lemma sums_le: "(∧n. f n ≤ f sums s ==> s ≤
  for f g :: "nat ==>
   rueLMEQl)(atinro:su_osim: sumsdf)

context
  fixes f :: "nat 'a::{ordered_comm_monoid_addy
begin =  g}" .

emmaAnd>n. f n ≤ summable g ==> suminf g"
  usingn. n ∉ f n =

lemma sum_le_suminf:
  "able<> finite I\Longrightarrow (∧- I ==> f n) ==> sum f I \<leuminf
  by (rule unf msef

lemmanf_nonnegsumbef\Longrightarrow (∧ f n) ==> <> smnf"
  using sum_le_suminf byrce

lemma: "summable f ==> (∧n. sum f {..<n} ≤ suminf f ≤
   (metis LIMSEQ_le_const2 summable_LIMSEQ)

lemma suminf_eq_zero_iff:
  assumes "summablen. 0 ≤
  shows (∀
proof
  assume L: "suminf f = 0" 
  then<ambdan. ∑i<n. f i) <---- 0"
    using summable_LIMSEQ[of f] assms by simp
  then have "∧i. (∑ 0"
    by (metis L ‹
   pos sho "foralln. f n = 0"
    by
qederoff

mmafLongrightarrow (\<And>n. 0 \le f n) \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<xistsi0 f i)"
  oms_iffmsjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40

lemma suminf_pos2:
  assumes "summable f" "\<And>n. 0 \<le> f n" "0 < f i"
  shows "0 < suminf f"
proof -
  have "0assumesite
    using assms by (intro sum_pos2[where i=i]) auto
   ve< \<le> suminf f"
    using assms by (intro sum_le_suminf) auto
  finally show ?thesis .
qed

lemma suminf_pos: "summable f \<Longrightarrow> (\<And>n. 0 < f n) \<Longrightarrow> 0<uminf
  by (intro suminf_pos2[where i]utoro_mp_le

end

context
  fixes bysums_lef_finite_setauto
begin

lemma sum_less_suminf2:
  " le<>. 0 \<le> f n"
   i}"]
    and add_strict_increasing[of "f i"{n} "sum f {..<"]
    and sum_mono2f.< "{..<n}" f]
  by (auto simp: less_imp_le ac_simps)

lemmaess_suminfsummableLongrightarrow (\<And>m. m\<ge>n \<ongrightarrow  ) \<Longrightarrow> mf<} < uminf"
  using sum_less_suminf2[of n n]bysimpp dd:less_imp_les_imp_lep_leejava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59

end

lemma summableI_nonneg_bounded:
  fixes finallyjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
  assumessmp<>n. 0 \<le> f n"
    and le
  shows "summable f"
  unfolding summable_def sums_def[ef
proof (rule exI LIMSEQ_incseq_SUP)
  showby ess_imp_le_
    using le by (auto simp: bdd_above_def)
  incseq<>. sum f {..<n})"
    by (auto simp: mono_def intro!: sum_mono2)
qed

lemma summableI[intro, simp
  for f :: "nat \<Rightarrow> 'a::{canonically_ordered_monoid_add,linorder_topology,complete_linorder}"
 leI_nonneg_bounded]o_lereatest

lemmaSUP_real
  assumes(autorobdd_aboveI2[re<>i. i m_le_suminfnoIum_mono2
  by (intro LIMSEQ_unique[OF summable_LIMSEQ] X LIMSEQ_incseq_SUP)
     (auto intro!: bdd_aboveI2[where M= sums_Suc


subsection \<open>Infinite summability on topological monoids\<close>

context
  g">'a::{t2_space,topological_comm_monoid_add}"
begin

lemma sums_Suc:
  assumes"<ambdan. f (Suc n)) sums l"
  shows "f sums (l + f 0)"
proof  -
  
    using assms by (auto roendsto_add simp: ums_deff
  moreover have "(\<Sum>i<n. f (Suc i)) +and '
    unfolding lessThan_Suc_eq_insert_0
    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: lterlim_sequentially_Suc1
qed

lemma sums_add: "f havelambdan. g n + (if n\<nhengse)sS+ (\<Sum>inA. f n - g n))"
-

lemma summable_add: "alsoe"<>\<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
  unfolding summable_def by (auto summable_Suc_iff summable<ambdan f (Suc n)) = summable f"

lemmauminf_addmmable<Longrightarrow> summable <ongrightarrow suminf f + suminf g = (\<Sum>n. f n + g n)"
  by (intro sums_unique sums_add summable_sums)

end

context
  fixes f :: "
    and I :: "'i set"
java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 5

lemma sums_sum: "(\<And>i  unfolding summable_def by (auto intro sums_diff)
  by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)

lemma suminf_sum: "(\<And>i. i \<in> I \<Longrightarrow> summable  unfoldingsums_defbysimp:m_negfndsto_minus
  using sums_unique[OF sums_sum, OF summable_sums] by

lemma summable_sum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> summable (\<lambda>n. \<Sum>i\<in>I.by( sums_unique [symmetric] sums_minussummable_sums)
  using sums_summable[OF sums_sum[OF summable_sums]] .

end

lemma sums_If_finite_set:
  fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_ab_group_addjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  gsums S" and "finiteA andS =S+ (Sumn<>A. f n -gn"
  shows   "<ambda.fn<in> then elsee )sums"
roof
  "\lambda>n.g   ifn \in>A then f n -  n else 0)) sums (S+\n\<nA f n - g n))"
    s_add ums_If_finite_set
  also have "IMSEQ_D[ummable_LIMSEQ<>summable f\<close>] \<open>0 < r\<close>]
    by (simp add: fun_eq_iff)
  finally show ?thesis using assms by simp
qed

subsection \<open>Infinite summability on real normed vector spaces\<close>

ontext
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
begin

lemma sums_Suc_iff: "(\<lambda>n. f (Suc n   :atRightarrow 'a::real_normed_vector"
proof -
  have "fsums f)<ongleftrightarrow (\<lambda>i. \<Sum>j<Suc i. f j) \<longlonglongrightarrow> s + f 0"
    by (subst filterlim_sequentially_Suc) (simp add: sums_def)
  also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. fjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    by (simp add ac_simps lessThan_Suc_eq_insert_0 image_Suc_lessThan sum.atLeast1_atMost_eq)
  row> (\<lambda>n. f ( n)) sums s"
  proof
    assume "(\<lambda>i. (\<Sum>j<i. f (uc j)) + f 0) <longlonglongrightarrow> s + f 0"
    with tendsto_add[OF this tendsto_const, of "- f 0"] show "(\<lambda>i. f (Suc ifor c :: "'a::real_normed_vector
      by (simp add: sums_def)
  ed(utotointrotendsto_addimpsums_def
  finally show ?thesis ..
qed

lemma summable_Suc_iff: "summable (\<lambda>n. f (Suc n))       unfolding summable_iff_convergentusing onvergent_normblast
proof
  assume "summable f"
  then have "f fixesf: nat \Rightarrow>'a:al_normed_algebra
    by (rule summable_sums)
  then have "(\<lambda>n. f (Suc n)) sums (suminf f - f 0)"
     simpddsums_Suc_iff
  then show "summable (\<lambda>n. fjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    unfolding summable_def by blast
qed (auto simp: sums_Suc_iff summable_def

lemma sums_Suc_imp: "f 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambdabda.f  sumsjava.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 121
  using sums_Suc_iff by simp

end

context (* Separate contexts are necessary to allow general use of the results above, here. *)
  fixes f :: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
begin

lemma sums_diff: "f sums a ==> g sums b ==>  <Longrightarrow summable (λn.   )
  unfoldingivide suminf (λn.  /c  minf

lemmasummable_diffLongrightarrowsummable g ==> summable (λn. f n - g n)"
  unfolding summable_def by (auto intro: su by (auto dest: sumal_ul of_ im:felsis)

lemma suminf_diff: "summable f ==>n. c * f n) ==> 0 ==> summable f"
  by (intr

lemma sums_minus: "f sums a ==>n. -f)sums
  unfolding sums_def 1"

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) = - (∑n. f n)"
  by (intro sums_unique [symmetric] sums_minus 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)) ⟷ f sums s"
  by (simp add: sums_iff_shift)

lemma sums_zero_iff_shift:
  assumes "∧i. i < n ==> f i = 0"
  shows "(λi. f (i+n)) sums s ⟷ (λi. f i) sums s"
  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)) sums (s - (∑i<n. f i))"
  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: "summable f ==> (Sumn. f (n + k)) = (∑i<k. f i)"
  by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)

lemma suminf_split_initial_segment: "summable    haven. (c ^ n - 1) <>1 / (1 - c)"
  by (auto sim y (sm d m_degomettc_sm

lemma suminf_split_head: "summable f ==> (∑byNsummable
  using suminf_split_initial_segment[of

emma
 s l
  assumes "0 < r" and "summable f"
  shows "∃n≥i. f (i + n)) < r"
proof -hen" using one_le_power[of "norm c" n]
  from LIMSEQ_D[OF summable_LIMSEQ[OF \<open    
  obtain N :: nat where "
    
  then show ?thesis
    by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF ‹summable f›])
qed

lemma summable_LIMSEQ_zero: 
  assumes "summable f" shows "f <---- 0"
proof -
  have "Cauchy (λ c :: 'a:eanomdfe"
    usingsmmable_LIMSEQ by last
  then show ?thesis
    unfolding  Cauchy_iff
    
qed

lemmaimp_convergent convergent f"
  by (force dest!: summable_LIMSEQ_zerotheso te

lemma summable_imp_Bseq: "summable
  by (simp

end

lemma summable_minus_iff: "summable (λn. - f n) ⟷ summable f"
  for f :: "nat ==> 'a::real_normed_vector"
  y(autoinuswo ustecontextabove

lemma (in bounded_linear) sums
  unfolding sums_def by (drule tendsto) (simp only: sum)

lemma (in bounded_linear) summable :al_normed_vector
  unfoldingshowsn. f (Suc n) - ums 0)"

lemma (in bounded_linear) suminf: "ummablen. X n) <ngrightarrow n<Suc n. f (Suc n) - f n)<longlonglongrightarrowc- "
  by (intro sums_unique sums sl telescop_sm'

lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
lemmas suminf_of_real = bounded_line)) sus (f 0 "

lemmas c"
lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
emmassuminf_scaleR_lefte_f buddliar.sinf[Fbund_nascaleR_eft

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

lemma summable_const_iff: "  (λ_. c) 🚫c = 0"
 for c :: "'a::real_normed_vector"
  -
 have "¬c" if"c \noteq 0"
 proof -
 omtat hae"filteli (🚫. of_nat n * norm c) at_top sequentially"
 by (subst mult.commute)
 (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
 then have "¬
 by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_ how ?rhs
 (si fix e : rel
 then show ?thesis
 unfolding summale_f_onergetsn conegntnomby blat
 qed
 then show ?thesis by auto
 


  ‹Infinite summability on real normed algebras›

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

  sassu "n \le m""
 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_multlby blast
 by (rule bounded_linear.suminf [OF bounne

  sums_mut2 f smsa 🚫
 by (rule bounded_linear.sums [OF bounded_linear_mult_left])

  summ y blas
 by (rule bounded_linear.summable [OF bounded_linear_mult_left])

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

 

  sums_mult_iff:
 f :: "n <>'
 assumes "c ≠
 shows "(λ
 using sums_mult[of f d c] sums_mult[of "λ 'a :: banach"
 by (force simp: field_sim asss)

 s_mult2_iff
 fixes f :: "nat ==>ra,ild"
 assumes "c ≠n. norm (sum f {m..<n}) at_top"
 shows "(λn. f n * proevetal_eli
 using m_ult_i[OF asms, of ] by sim add: lt.comu)

  sums_of_real_iff:
 "(λn. of_real (f n) :: 'a::real_normed_div_algebra) sums of_real c ⟷
 by (simp add: sums_def of_real_sum[symmetric] tendsto_of_real_iff del: of_real_sum)


  ‹Absolute convergence imples normal convergence.›

 
 fixes c :: "'a::real_normed_field"
 

 sums_divide: "f sums a ==> (λn. f n / c) sums (a / c)"
 by (a(auto intro: LIMSEQ_letnst_ormsual_no_anelsumale_LIMSQ or_sm

  summable_divide: "summable f ==> summable (λn. f n / c)"
 by (rule bounded_linear.summable [OF bounded_linear_divi> g nn ndg"umbeg

  suminf_divide: "summable f ==>
 by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])

  summable_inverse_divide: "summable (inverse ∘n. c / f n)"
 by (auto dest: summable_mult [of _ c] simp: field_simps)

  sums_mult_D: "(λ c ≠ f sum ac)
 using sums_mult_iff by fastforce

  summable_mult_D: "summable (λn. c * f n) ==> c ≠ 0 ==> summable f"
 by (auto dest: summable_divide)


  ‹Sum of a geometric progression.›

  geometric_sums:
 assumes "norm c < 1 thtb(fre into:s_or_e)
 shows "(λc) sus ( / 1-c))"
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null
 have neq_0: "c - 1 ≠ 0"
 using assms by auto
 then have "(λ(c - 1)) \<nglonglongrightarrowglonglongrightarrowThe Ratio Test›
    by (intro tendsto_intros assms)
  (lambdan. (c ^ n - 11)<longlonglongrightarrow 1 / (1 - c)"
    simpznudieright OF ne0 if_ii_ditrb
  with neq_0 show "(λ )
    by (simp add:fix
qed

lemma summable_geometric:          
  by (rule geometric_sums

lemma suminf_geometric: "norm c < 1 ==> suminf (λn. c^n) = 1 / (1 - c)"
  by (rule sums_unique[symmetric

lemma summable_geometric_iff [simp]: "summable (λn. c ^ n) ⟷ n <"
proof
  assume "summable (λ
  then have "(λn. norm c ^ n) <----  have "f (Su n =" <> N"r
    by (simpad: nor_oe [symymmeti edsonr_zro_ffsummabeLIMEQ_zero)
  from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
    by (auto simp: eventually_at_top_linorder)
  then show "norm c < 1" using one_le_power[of "norm c" n]
    by (cases "norm c ≥ 1") (linarith, simp)
ed(le summale_geomic

java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 3

text ‹
context
  fixes c :: " a::real_normed_field"
 

  summable_cmult_iff [simp]: "summable (\<lambdashown. norm (z n)
  -
 have "[
 using summable_mult_D by blast
 hen sow thes
 by (auto simp: summable_mult)
 

  summable_divide_iff [simp]: "summable (λ
  -
 have "[summable (λn. f n / c); c ≠ ==>
 by (auto dest: summable_divide [where c = "1/c"])
 then show ?thesis
 by (auto simp: summable_divide)
 

 

  power_half_seriesbda>n. M * ( r0) ^ n)"
  -
 have 2: "(λn. (1/2::real)^n) sums 2"
 ng emtc_sms of "1/2::real"] by auto
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
 by (simp add: mult.commute)
 then show ?thesis
 using sums_divide [OF 2, of 2] by simp
 


  ‹‹

  telescope_sums:
 fixes c :: "'a::real_normed_vector"
 assumes "f <----
 shows "(λn. f (Suc n) - f n) sums (c - f 0)"
 unfolding sums_def
  (subst fi aand b: "sumabe (\lambda. norm (b k))"
 have "(λn. ∑i≤. i* k-i) us ∑.ak * \<Sumk"
 by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] sum_Suc_diff)
 also have "… <---- "∧ n ==> ?S1 n" by auto
 by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
 lly how"(<>nn. f (Suc n) - f n) <---- c - f 0" .
 

  telescope_sums':
 fixes c :: "'a::real_normed_vector"
 assumes "f <----
 shows "(λn. f n - f (Suc n)) sums (f 0 - c)"
 using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)

  telescope_summable:
 fixes c :: "'a::real_normed_vector"
 assumes "f <----
 "mmble (\lambda
 using telescope_sums[OF assms] by (simp add: sums_iff)

  telescope_summable':
 fixes c :: "'a::real_normed_vector"
 assumes "f <---- c"
 shows "summable (λn. f n - f (Suc n))"
 using summable_minus[OF telescope_summable[OF assms]] by (simp aassu: " r"


  ‹m≥n≥ r ..

  ‹

  summable_Cauchy: "summable f ⟷e>0. ∃m≥n. norm (sum f {m..<n})" (is "_ = ?rhs")
 for f :: "nat ==>
 
 ume :"sumale f"
 show ?rhs
 proof clarify
 fix e :: real
 assume "0 < e
 meM] norm (sum f {..<m} - sm f .<})"
 using f by (force simp add: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff) using d_ledvidby (rule )
 have "norm (sum f {m..<n}) < e
 proof (cases m n rule: linorder_class.le_cases)
 assume "m ≤ n"
 
 by (metis (moappl(tsad lie
 next
 assume "n ≤ m"
 then show ?thesis
 by (simp add: ‹›)
 qed
 then show "∃ 1ve \lambdan sum ?g (?S2 n)) <---- (∑k. a k) * (∑"
 by blast
 qed
 
 assume rs
 then show "summable f"
 unfolding summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff
 proof clarify
 fix e :: real
 assume "0 < e
 h obain wee N: "\Andm n. m ≥ N ==> norm (sum f {m..<n}) < ek. norm (b k))"
 blast
 have "norm (sum f {..<m}
 proof (cases m n rule: linorder_class.le_cases)
 assume "m ≤
 then summable_norm_cmpaiontet
 by (metis N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff norm_minus_commute sum_diff ‹)
 next
 assume "n ≤N. ∀N. ∣ ≤ summable g ==>n. \bar>f ∣
 then show ?thesis
 by (metis N finite_lessThan lessThaem summabl: suabl\lambdan. ∣f n∣ summable f"
 qed
 then show "∃e λf n∣ ∣ ≤n. ∣f n∣
  lst
 qed
 

  summable_Cauchy':
 fixes f :: "nat ==> 'a :: banach"
 umes v: evtally (<>m
 ssumes0: "g \longlonglongrightarrow 0"
 shows "summable f"
  (subst summable_Cauchy, intro allI impI, goal_cases)
 case (1 e)
 then have "∀F x in sequentially. g x < e
 using g0 order_tendstoD(2) by blast
 ith ev have "eventually (λn. norm (sum f {m..<n}) < e
 proof eventually_elim
 case (elim m)
 show ?case
 proof
 fix n
 from elim show "norm (sum f {m..<n}) < e
 by (cases "n ≥
 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_esstran [OFnom_sm] rdr_lee_ttras[Fabse_ef)
 one

  summable_norm: "summable (λn. norm (f n)) \<Longrightarrowshow
 by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_sum)

  ‹

 summable_comparison_test
 assumes fg: "∃N. ∀
 shows "summable f"
  -
 obtain N where N: "∧n. n≥ = summable (λn. f n * z ^ n)"
 singassm y bat
 show ?thesis
 proof (clarsimp simp add: summable_Cauchy)
 fix e :: real
 assume "0 < e
 then obtain Ng where Ng: "∧m n. m ≥
 using g by (fastforce simp: summable_Cauchy)
 with N have "norm (sum f {m..<n}usle_ms)
 proof
 have "norm (sum f {m.<}  sum g {m..<n}"
 ngt b (oce nto:um_ormle)
 also have "... ≤
 by simp
 also have "... < e
 using Ng that by auto
 finally show ?thesis .
 qed
 then show "∃n. ∑
  lat
 qed
 

  summable_comparison_test_ev:
 "eventually (λ>k<Suc  f k) =(\Sumk=m..n. f k)"
 rulesme_comarsn_testes)(u sim vntualy_atnordeer)

  ‹
  summable_comparison_test': "summable g ==> (∧n. n ≥
 by (rule summable_comparison_test) auto


  ‹

  summable_ratio_test:
 
 shows "summable f"
  (cases "0 < c")
 case True

 proof from \>c> [simp]: "∧A. inj_on g
 show "\havefn. (∑ (f \<circ > suminf f"
 proof (intro exI allI impI)
 fix n
 assume "N ≤ n"
 then show "norm (f n) ≤ (norm (f N) / (c ^ N)) * c ^ n"
 proof (induct rule: hen obta mwhen: "And>n'. n' <n ==> g n' < m
 
 with True show ?case by simp
 next
 case (step m)
  u m* c ^^n\<>norm
 using ‹
 with step show ?case by simp
 qed
 qed
 show "summable (λn. norm (f N) / c ^ N * c ^ n)"
 using ‹i<n. g) i) <----L
 qed
 
 case False
 have "f (Suc n) = 0" if "n ≥ N" for n
 proof -
 from that ha
 by (rule assms(2))
 also have "… ≤
 using False by (simp add: not_less mult_nonpos_nonneg)
 finally show ?thesis
 by aututo
 qed
 then show "summable f"
 by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
 

 

  ‹
  norm_summable_ln_series:
 fixes z :: "'a :: {real_normed_field, banach}"
 assumes "norm z < 1
 shows "summable (λ / _na )
  (rule summable_ hows "(<mbdan f k) \<onglonglongrightarrowlonglongrightarrow
 show "summabe (\lambdan. nom ( n))
 using ass pproof
 have "norm z ^ n / real n ≤k<n.k∈. f k)"
 proof (casesalsrmubse ve\dots = (∑
 lse
 hence "nom <> 
 by (intro mult_left_mono) auto
 thus ?thesis
 using False by (simp add: field_simps)
 qed auto
 "∃>N norm (norm (z ^ n / of_nat n)) ≤
 xIf0) at sm:nom_powrnomiie)
 


  ‹

  Abel_lemma:
 fixes a :: "nat ==> 'a::real_normed_vector"
 assumes r: 0 ≤
 and fix n : a
 and M: "∧n. norm (a n) * r0^n ≤ {..<n} - g`{.<_inv <notin g
 shows "summable (λn. norm (a n) * r^n)"
 by (ule_l_tn[O ginv)(sk linsp_al)
 show "summable (λ
 using assms by (auto qe
 show "norm (norm (a n) * r ^ n) \<le}
 using r r0 M [of n] dual_order.order_iff_strict
 by (fastforce simp add: abs_mult field_simps)
 


java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5

  complete_algebra_summable_geometric:
 byufill_ps
 assumes "norm x < 1
 shows "summable (λ
  (rule summable_comparison_test)
 show "\<xistsNedef)
 by (simp add: norm_power_ineq)
 from assms show "summable (λn. n ∉f 0
 by (simp add: summable_geometric)
 


  ‹

  ‹
 Proof based on Analysis WebNotes: Chapter 07, Class 41
 🪙
 › by simp

  Cauchy_product_sums:
 fixes a b :: "nat ==>
 assumes a: "summable (λ
 and b: "summable (λk.r k)"
 shows "(λojOtsud tai wee :
  -
 let ?S1 = "λn::nat. {..<n} \  
 let ?S2 = "λn::nat. {(i,j). i + j < n
 have S1_mono: "∧
 have S2_le_S1: "∧n. ?S2 n ⊆v"{. {.m {<.n
 have S1_le_S2: "∧ ≤g m∣
 have finite_S1: "\<And.
 with S2_le_S1 havefntS2\Andn. finite (?S2 n)" by (rule finite_subset)

 
 let ?f = "λ(i,j). norm (a i) * norm (b j)"
 have f_nonneg: "∧x. 0 ≤ ?f x" by auto
 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. (∑k<n. a k) * (∑k<n. b k)) <---- (∑k. a k) * (∑k. b k)"
 by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
 then have 1: "(λn. sum ?g (?S1 n)) <---- (∑k. a k) * (∑k. b k)"
 by (simp only: sum_product sum.Sigma [rule_format] finite_lessThan)

 have "(λn. (∑k<n. norm (a k)) * (∑k<n. norm (b k))) <---- (∑k. norm (a k)) * (∑k. norm (b k))"
 using a b by (intro tendsto_mult summable_LIMSEQ)
 then have "(λn. sum ?f (?S1 n)) <---- (∑k. norm (a k)) * (∑k. norm (b k))"
 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
 assume r: "0 < r"
 from CauchyD [OF Cauchy r] obtain N
 where "∀m≥N. ∀n≥N. norm (sum ?f (?S1 m) - sum ?f (?S1 n)) < r" ..
 then have "∧m n. N ≤ n ==> n ≤ m ==> norm (sum ?f (?S1 m - ?S1 n)) < r"
 by (simp only: sum_diff finite_S1 S1_mono)
 then have N: "∧m n. N ≤ n ==> n ≤ m ==> sum ?f (?S1 m - ?S1 n) < r"
 by (simp only: norm_sum_f)
 show "∃N. ∀n≥N. sum ?f (?S1 n - ?S2 n) < r"
 proof (intro exI allI impI)
 fix n
 assume "2 * N ≤ n"
 then have n: "N ≤ n div 2" by simp
 have "sum ?f (?S1 n - ?S2 n) ≤ sum ?f (?S1 n - ?S1 (n div 2))"
 by (intro sum_mono2 finite_Diff finite_S1 f_nonneg Diff_mono subset_refl S1_le_S2)
 also have "… < r"
 using n div_le_dividend by (rule N)
 finally show "sum ?f (?S1 n - ?S2 n) < r" .
 qed
 qed
 then have "Zfun (λn. sum ?g (?S1 n - ?S2 n)) sequentially"
 apply (rule Zfun_le [rule_format])
 apply (simp only: norm_sum_f)
 apply (rule order_trans [OF norm_sum sum_mono])
 apply (auto simp add: norm_mult_ineq)
 done
 then have 2: "(λn. sum ?g (?S1 n) - sum ?g (?S2 n)) <---- 0"
 unfolding tendsto_Zfun_iff diff_0_right
 by (simp only: sum_diff finite_S1 S2_le_S1)
 with 1 have "(λn. sum ?g (?S2 n)) <---- (∑k. a k) * (∑k. b k)"
 by (rule Lim_transform2)
 then show ?thesis
 by (simp only: sums_def sum.triangle_reindex)
 

  Cauchy_product:
 fixes a b :: "nat ==> 'a::{real_normed_algebra,banach}"
 assumes "summable (λk. norm (a k))"
 and "summable (λk. norm (b k))"
 shows "(∑k. a k) * (∑k. b k) = (∑k. ∑i≤k. a i * b (k - i))"
 using assms by (rule Cauchy_product_sums [THEN sums_unique])

  summable_Cauchy_product:
 fixes a b :: "nat ==> 'a::{real_normed_algebra,banach}"
 assumes "summable (λk. norm (a k))"
 and "summable (λk. norm (b k))"
 shows "summable (λk. ∑i≤k. a i * b (k - i))"
 using Cauchy_product_sums[OF assms] by (simp add: sums_iff)


  ‹Series on 🍋‹real›s›

  summable_norm_comparison_test:
 "∃N. ∀n≥N. norm (f n) ≤ g n ==> summable g ==> summable (λn. norm (f n))"
 by (rule summable_comparison_test) auto

  summable_rabs_comparison_test: "∃N. ∀n≥N. ∣f n∣ ≤ g n ==> summable g ==> summable (λn. ∣f n∣)"
 for f :: "nat ==> real"
 by (rule summable_comparison_test) auto

  summable_rabs_cancel: "summable (λn. ∣f n∣) ==> summable f"
 for f :: "nat ==> real"
 by (rule summable_norm_cancel) simp

  summable_rabs: "summable (λn. ∣f n∣) ==> ∣suminf f∣ ≤ (∑n. ∣f n∣)"
 for f :: "nat ==> real"
 by (fold real_norm_def) (rule summable_norm)

  norm_suminf_le:
 assumes "∧n. norm (f n :: 'a :: banach) ≤ g n" "summable g"
 shows "norm (suminf f) ≤ suminf g"
  -
 have *: "summable (λn. norm (f n))"
 using assms summable_norm_comparison_test by blast
 hence "norm (suminf f) ≤ (∑n. norm (f n))" by (intro summable_norm) auto
 also have "… ≤ suminf g" by (intro suminf_le * assms allI)
 finally show ?thesis .
 

  norm_sums_le:
 assumes "f sums F" "g sums G"
 assumes "∧n. norm (f n :: 'a :: banach) ≤ g n"
 shows "norm F ≤ G"
 using assms by (metis norm_suminf_le sums_iff)

  summable_zero_power [simp]: "summable (λn. 0 ^ n :: 'a::{comm_ring_1,topological_space})"
 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: "∧i. f i ≤ 1"
 and nonneg: "∧i. 0 ≤ f i"
 and z: "0 ≤ z" "z < 1"
 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. ∀na≥N. norm (f na * z ^ na) ≤ z ^ na"
 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 (λn. f (Suc n) * z ^ n) ⟷ summable (λn. f (Suc n) * z ^ Suc n)"
 (is "?lhs ⟷ ?rhs")
 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 ≠ 0", subst (asm) power_Suc2) (simp_all add: algebra_simps)
 qed
 also have "… ⟷ summable (λn. f n * z ^ n)" by (rule summable_Suc_iff)
 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 (λn. f n * z ^ n)"
  (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 (λn. f (n + m) * z ^ n)"
 by (rule summable_powser_split_head)
 also have "… = summable (λn. f n * z ^ n)"
 by (rule Suc.IH)
 finally show ?case .
  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_commutes algebra_simps)
 also from assms have "… = suminf (λn. f n * z ^ n) - f 0"
 by (subst suminf_split_head) simp_all
 finally show "suminf (λn. f n * z ^ n) = f 0 + suminf (λn. f (Suc n) * z ^ n) * z"
 by simp
 then show "suminf (λn. f (Suc n) * z ^ n) * z = suminf (λn. f n * z ^ n) - f 0"
 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 (∑k=m..n. f k) < e"
  -
 from summable have "Cauchy (λn. ∑k<n. f k)"
 by (simp add: Cauchy_convergent_iff summable_iff_convergent)
 from CauchyD [OF this e] obtain N
 where N: "∧m n. m ≥ N ==> n ≥ N ==> norm ((∑k<m. f k) - (∑k<n. f k)) < e"
 by blast
 have "norm (∑k=m..n. f k) < e" if m: "m ≥ N" for m n
 proof (cases "n ≥ m")
 case True
 with m have "norm ((∑k<Suc n. f k) - (∑k<m. f k)) < e"
 by (intro N) simp_all
 also from True have "(∑k<Suc n. f k) - (∑k<m. f k) = (∑k=m..n. f k)"
 by (subst sum_diff [symmetric]) (simp_all add: sum.last_plus)
 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 ∘ g) ≤ suminf f"
 and suminf_reindex: "(∧x. x ∉ range g ==> f x = 0) ==> suminf (f ∘ g) = suminf f"
  -
 from ‹inj g› have [simp]: "∧A. inj_on g A"
 by (rule inj_on_subset) simp

 have smaller: "∀n. (∑i<n. (f ∘ g) i) ≤ suminf f"
 proof
 fix n
 have "∀ n' ∈ (g ` {..<n}). n' < Suc (Max (g ` {..<n}))"
 by (metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
 then obtain m where n: "∧n'. n' < n ==> g n' < m"
 by blast

 have "(∑i<n. f (g i)) = sum f (g ` {..<n})"
 by (simp add: sum.reindex)
 also have "… ≤ (∑i<m. f i)"
 by (rule sum_mono2) (auto simp add: pos n[rule_format])
 also have "… ≤ suminf f"
 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. ∑i<n. (f ∘ g) i) <---- L"
 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) ≤ suminf f"
 by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])

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

 from ‹summable f› have "suminf f ≤ suminf (f ∘ g)"
 proof (rule suminf_le_const)
 fix n
 have "∀ n' ∈ (g -` {..<n}). n' < Suc (Max (g -` {..<n}))"
 by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le)
 then obtain m where n: "∧n'. g n' < n ==> n' < m"
 by blast
 have "(∑i<n. f i) = (∑i∈{..<n} ∩ range g. f i)"
 using f by(auto intro: sum.mono_neutral_cong_right)
 also have "… = (∑i∈g -` {..<n}. (f ∘ g) i)"
 by (rule sum.reindex_cong[where l=g])(auto)
 also have "… ≤ (∑i<m. (f ∘ g) i)"
 by (rule sum_mono2)(auto simp add: pos n)
 also have "… ≤ suminf (f ∘ g)"
 using ‹summable (f ∘ g)› by (rule sum_le_suminf) (simp_all add: pos)
 finally show "sum f {..<n} ≤ suminf (f ∘ g)" .
 qed
 with le show "suminf (f ∘ g) = suminf f"
 by (rule antisym)
 

  sums_mono_reindex:
 assumes subseq: "strict_mono g"
 and zero: "∧n. n ∉ range g ==> f n = 0"
 shows "(λn. f (g n)) sums c ⟷ f sums c"
 unfolding sums_def
 
 assume lim: "(λn. ∑k<n. f k) <---- c"
 have "(λn. ∑k<n. f (g k)) = (λn. ∑k<g n. f k)"
 proof
 fix n :: nat
 from subseq have "(∑k<n. f (g k)) = (∑k∈g`{..<n}. f k)"
 by (subst sum.reindex) (auto intro: strict_mono_imp_inj_on)
 also from subseq have "… = (∑k<g n. f k)"
 by (intro sum.mono_neutral_left ballI zero)
 (auto simp: strict_mono_less strict_mono_less_eq)
 finally show "(∑k<n. f (g k)) = (∑k<g n. f k)" .
 qed
 also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "… <---- c"
 by (simp only: o_def)
 finally show "(λn. ∑k<n. f (g k)) <---- c" .
 
 assume lim: "(λn. ∑k<n. f (g k)) <---- c"
 define g_inv where "g_inv n = (LEAST m. g m ≥ n)" for n
 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 n. f (g k))"
 proof
 fix n :: nat
 {
 fix k
 assume k: "k ∈ {..<n} - g`{..<g_inv n}"
 have "k ∉ range g"
 proof (rule notI, elim imageE)
 fix l
 assume l: "k = g l"
 have "g l < g (g_inv n)"
 by (rule less_le_trans[OF _ g_inv]) (use k l in simp_all)
 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 n}. f k)"
 by (intro sum.mono_neutral_right) auto
 also from subseq have "… = (∑k<g_inv n. f (g k))"
 using strict_mono_imp_inj_on by (subst sum.reindex) simp_all
 finally show "(∑k<n. f k) = (∑k<g_inv n. f (g k))" .
 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 n. f (g k)) <---- c"
 by (rule filterlim_compose)
 finally show "(λn. ∑k<n. f k) <---- c" .
 

  summable_mono_reindex:
 assumes subseq: "strict_mono g"
 and zero: "∧n. n ∉ range g ==> f n = 0"
 shows "summable (λn. f (g n)) ⟷ summable f"
 using sums_mono_reindex[of g f, OF assms] by (simp add: summable_def)

  suminf_mono_reindex:
 fixes f :: "nat ==> 'a::{t2_space,comm_monoid_add}"
 assumes "strict_mono g" "∧n. n ∉ range g ==> f n = 0"
 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 by simp
 

  summable_bounded_partials:
 fixes f :: "nat ==> 'a :: {real_normed_vector,complete_space}"
 assumes bound: "eventually (λx0. ∀a≥x0. ∀b>a. norm (sum f {a<..b}) ≤ g a) sequentially"
 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. ∣g x∣ < \ε) sequentially"
 by (auto simp: tendsto_iff)
 from eventually_conj[OF this bound] obtain x0 where x0:
 "∧x. x ≥ x0 ==> ∣g x∣ < \ε" "∧a b. x0 ≤ a ==> a < b ==> norm (sum f {a<..b}) ≤ g a"
 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 have "{..n} - {..m} = {m<..n}" using mn by auto
 also have "norm (sum f {m<..n}) ≤ g m" using mn by (intro x0) auto
 also have "… ≤ ∣g m∣" by simp
 also have "… < \ε" using mn by (intro x0) auto
 finally show "dist (sum f {..m}) (sum f {..n}) < \ε" .
 qed