(*  Title       : Series.thy

    Author      : Jacques D. Fleuriot

    Copyright   : 1998  University of Cambridge

Converted to Isar and polished by lcp

Converted to setsum and polished yet more by TNN

*) 

header{*Finite Summation and Infinite Series*}

theory Series

imports SEQ Lim

begin

(* FIXME why not globally? *)

declare atLeastLessThan_empty[simp];

declare atLeastLessThan_iff[iff]

constdefs

   sums  :: "[nat=>real,real] => bool"     (infixr "sums" 80)

   "f sums s  == (%n. setsum f {0..<n}) ----> s"

   summable :: "(nat=>real) => bool"

   "summable f == (\<exists>s. f sums s)"

   suminf   :: "(nat=>real) => real"

   "suminf f == SOME s. f sums s"

lemma setsum_Suc[simp]:

  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"

by (simp add: atLeastLessThanSuc add_commute)

(*

lemma sumr_add: "sumr m n f + sumr m n g = sumr m n (%n. f n + g n)"

by (simp add: setsum_addf)

lemma sumr_mult: "r * sumr m n (f::nat=>real) = sumr m n (%n. r * f n)"

by (simp add: setsum_mult)

lemma sumr_split_add [rule_format]:

     "n < p --> sumr 0 n f + sumr n p f = sumr 0 p (f::nat=>real)"

apply (induct "p", auto)

apply (rename_tac k) 

apply (subgoal_tac "n=k", auto) 

done

lemma sumr_split_add: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"

by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)

lemma sumr_split_add_minus:

fixes f :: "nat \<Rightarrow> 'a::ab_group_add"

shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>

  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"

using sumr_split_add [of m n p f,symmetric]

apply (simp add: add_ac)

done

*)

lemma sumr_diff_mult_const:

 "setsum f {0..<n} - (real n*r) = setsum (%i. f i - r) {0..<n::nat}"

by (simp add: diff_minus setsum_addf real_of_nat_def)

(*

lemma sumr_rabs: "abs(sumr m n  (f::nat=>real)) \<le> sumr m n (%i. abs(f i))"

by (simp add: setsum_abs)

lemma sumr_rabs_ge_zero [iff]: "0 \<le> sumr m n (%n. abs (f n))"

by (simp add: setsum_abs_ge_zero)

text{*Just a congruence rule*}

lemma sumr_fun_eq:

     "(\<forall>r. m \<le> r & r < n --> f r = g r) ==> sumr m n f = sumr m n g"

by (auto intro: setsum_cong) 

lemma sumr_less_bounds_zero [simp]: "n < m ==> sumr m n f = 0"

by (simp add: atLeastLessThan_empty)

lemma sumr_minus: "sumr m n (%i. - f i) = - sumr m n f"

by (simp add: Finite_Set.setsum_negf)

lemma sumr_shift_bounds:

  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"

by (induct "n", auto)

*)

(* Generalize from real to some algebraic structure? *)

lemma sumr_minus_one_realpow_zero [simp]:

  "setsum (%i. (-1) ^ Suc i) {0..<2*n} = (0::real)"

by (induct "n", auto)

(*

lemma sumr_interval_const2:

     "[|\<forall>n\<ge>m. f n = r; m \<le> k|]

      ==> sumr m k f = (real (k - m) * r)"

apply (induct "k", auto)

apply (drule_tac x = k in spec)

apply (auto dest!: le_imp_less_or_eq)

apply (simp add: left_distrib real_of_nat_Suc split: nat_diff_split)

done

*)

(* FIXME split in tow steps

lemma setsum_nat_set_real_const:

  "(\<And>n. n\<in>A \<Longrightarrow> f n = r) \<Longrightarrow> setsum f A = real(card A) * r" (is "PROP ?P")

proof (cases "finite A")

  case True

  thus "PROP ?P"

  proof induct

    case empty thus ?case by simp

  next

    case insert thus ?case by(simp add: left_distrib real_of_nat_Suc)

  qed

next

  case False thus "PROP ?P" by (simp add: setsum_def)

qed

 *)

(*

lemma sumr_le:

     "[|\<forall>n\<ge>m. 0 \<le> (f n::real); m < k|] ==> setsum f {0..<m} \<le> setsum f {0..<k::nat}"

apply (induct "k")

apply (auto simp add: less_Suc_eq_le)

apply (drule_tac x = k in spec, safe)

apply (drule le_imp_less_or_eq, safe)

apply (arith)

apply (drule_tac a = "sumr 0 m f" in order_refl [THEN add_mono], auto)

done

lemma sumr_le:

     "[|\<forall>n\<ge>m. 0 \<le> f n; m < k|] ==> sumr 0 m f \<le> sumr 0 k f"

apply (induct "k")

apply (auto simp add: less_Suc_eq_le)

apply (drule_tac x = k in spec, safe)

apply (drule le_imp_less_or_eq, safe)

apply (arith) 

apply (drule_tac a = "sumr 0 m f" in order_refl [THEN add_mono], auto)

done

lemma sumr_le2 [rule_format (no_asm)]:

     "(\<forall>r. m \<le> r & r < n --> f r \<le> g r) --> sumr m n f \<le> sumr m n g"

apply (induct "n")

apply (auto intro: add_mono simp add: le_def)

done

*)

(*

lemma sumr_ge_zero: "(\<forall>n\<ge>m. 0 \<le> f n) --> 0 \<le> sumr m n f"

apply (induct "n", auto)

apply (drule_tac x = n in spec, arith)

done

*)

(*

lemma rabs_sumr_rabs_cancel [simp]:

     "abs (sumr m n (%k. abs (f k))) = (sumr m n (%k. abs (f k)))"

by (induct "n", simp_all add: add_increasing)

lemma sumr_zero [rule_format]:

     "\<forall>n \<ge> N. f n = 0 ==> N \<le> m --> sumr m n f = 0"

by (induct "n", auto)

*)

lemma Suc_le_imp_diff_ge2:

     "[|\<forall>n \<ge> N. f (Suc n) = 0; Suc N \<le> m|] ==> setsum f {m..<n} = 0"

apply (rule setsum_0')

apply (case_tac "n", auto)

apply(erule_tac x = "a - 1" in allE)

apply (simp split:nat_diff_split)

done

(* FIXME this is an awful lemma! *)

lemma sumr_one_lb_realpow_zero [simp]:

  "(\<Sum>n=Suc 0..<n. f(n) * (0::real) ^ n) = 0"

apply (induct "n")

apply (case_tac [2] "n", auto)

done

(*

lemma sumr_diff: "sumr m n f - sumr m n g = sumr m n (%n. f n - g n)"

by (simp add: diff_minus setsum_addf setsum_negf)

*)

(*

lemma sumr_subst [rule_format (no_asm)]:

     "(\<forall>p. m \<le> p & p < m+n --> (f p = g p)) --> sumr m n f = sumr m n g"

by (induct "n", auto)

*)

lemma setsum_bounded:

  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{comm_semiring_1_cancel, pordered_ab_semigroup_add})"

  shows "setsum f A \<le> of_nat(card A) * K"

proof (cases "finite A")

  case True

  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp

next

  case False thus ?thesis by (simp add: setsum_def)

qed

(*

lemma sumr_bound [rule_format (no_asm)]:

     "(\<forall>p. m \<le> p & p < m + n --> (f(p) \<le> K))  

      --> setsum f {m..<m+n::nat} \<le> (real n * K)"

apply (induct "n")

apply (auto intro: add_mono simp add: left_distrib real_of_nat_Suc)

done

*)

(* FIXME should be generalized

lemma sumr_bound2 [rule_format (no_asm)]:

     "(\<forall>p. 0 \<le> p & p < n --> (f(p) \<le> K))  

      --> setsum f {0..<n::nat} \<le> (real n * K)"

apply (induct "n")

apply (auto intro: add_mono simp add: left_distrib real_of_nat_Suc add_commute)

done

 *)

(* FIXME a bit specialized for [simp]! *)

lemma sumr_group [simp]:

     "(\<Sum>m=0..<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {0 ..< n * k}"

apply (subgoal_tac "k = 0 | 0 < k", auto simp:setsum_0')

apply (induct "n")

apply (simp_all add: setsum_add_nat_ivl add_commute)

done

(* FIXME setsum_0[simp] *)

subsection{* Infinite Sums, by the Properties of Limits*}

(*----------------------

   suminf is the sum   

 ---------------------*)

lemma sums_summable: "f sums l ==> summable f"

by (simp add: sums_def summable_def, blast)

lemma summable_sums: "summable f ==> f sums (suminf f)"

apply (simp add: summable_def suminf_def)

apply (blast intro: someI2)

done

lemma summable_sumr_LIMSEQ_suminf: 

     "summable f ==> (%n. setsum f {0..<n}) ----> (suminf f)"

apply (simp add: summable_def suminf_def sums_def)

apply (blast intro: someI2)

done

(*-------------------

    sum is unique                    

 ------------------*)

lemma sums_unique: "f sums s ==> (s = suminf f)"

apply (frule sums_summable [THEN summable_sums])

apply (auto intro!: LIMSEQ_unique simp add: sums_def)

done

(*

Goalw [sums_def,LIMSEQ_def] 

     "(\<forall>m. n \<le> Suc m --> f(m) = 0) ==> f sums (sumr 0 n f)"

by safe

by (res_inst_tac [("x","n")] exI 1);

by (safe THEN ftac le_imp_less_or_eq 1)

by safe

by (dres_inst_tac [("f","f")] sumr_split_add_minus 1);

by (ALLGOALS (Asm_simp_tac));

by (dtac (conjI RS sumr_interval_const) 1);

by Auto_tac

qed "series_zero";

next one was called series_zero2

**********************)

lemma ivl_subset[simp]:

 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"

apply(auto simp:linorder_not_le)

apply(rule ccontr)

apply(frule subsetCE[where c = n])

apply(auto simp:linorder_not_le)

done

lemma ivl_diff[simp]:

 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"

by(auto)

(* FIXME the last step should work w/o Ball_def, ideally just with

   setsum_0 and setsum_cong. Currently the simplifier does not simplify

   the premise x:{i..<j} that ball_cong (or a modified version of setsum_0')

   generates. FIX simplifier??? *)

lemma series_zero: 

     "(\<forall>m. n \<le> m --> f(m) = 0) ==> f sums (setsum f {0..<n})"

apply (simp add: sums_def LIMSEQ_def diff_minus[symmetric], safe)

apply (rule_tac x = n in exI)

apply (clarsimp simp add:setsum_diff[symmetric] setsum_0' Ball_def)

done

lemma sums_mult: "x sums x0 ==> (%n. c * x(n)) sums (c * x0)"

by (auto simp add: sums_def setsum_mult [symmetric]

         intro!: LIMSEQ_mult intro: LIMSEQ_const)

lemma sums_divide: "x sums x' ==> (%n. x(n)/c) sums (x'/c)"

by (simp add: real_divide_def sums_mult mult_commute [of _ "inverse c"])

lemma sums_diff: "[| x sums x0; y sums y0 |] ==> (%n. x n - y n) sums (x0-y0)"

by (auto simp add: sums_def setsum_subtractf intro: LIMSEQ_diff)

lemma suminf_mult: "summable f ==> suminf f * c = suminf(%n. f n * c)"

by (auto intro!: sums_unique sums_mult summable_sums simp add: mult_commute)

lemma suminf_mult2: "summable f ==> c * suminf f  = suminf(%n. c * f n)"

by (auto intro!: sums_unique sums_mult summable_sums)

lemma suminf_diff:

     "[| summable f; summable g |]   

      ==> suminf f - suminf g  = suminf(%n. f n - g n)"

by (auto intro!: sums_diff sums_unique summable_sums)

lemma sums_minus: "x sums x0 ==> (%n. - x n) sums - x0"

by (auto simp add: sums_def intro!: LIMSEQ_minus simp add: setsum_negf)

lemma sums_group:

     "[|summable f; 0 < k |] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)"

apply (drule summable_sums)

apply (auto simp add: sums_def LIMSEQ_def)

apply (drule_tac x = r in spec, safe)

apply (rule_tac x = no in exI, safe)

apply (drule_tac x = "n*k" in spec)

apply (auto dest!: not_leE)

apply (drule_tac j = no in less_le_trans, auto)

done

lemma sumr_pos_lt_pair_lemma:

  "[|\<forall>d. - f (n + (d + d)) < (f (Suc (n + (d + d))) :: real) |]

   ==> setsum f {0..<n+Suc(Suc 0)} \<le> setsum f {0..<Suc(Suc 0) * Suc no + n}"

apply (induct "no", auto)

apply (drule_tac x = "Suc no" in spec)

apply (simp add: add_ac)

done

lemma sumr_pos_lt_pair:

     "[|summable f; 

        \<forall>d. 0 < (f(n + (Suc(Suc 0) * d))) + f(n + ((Suc(Suc 0) * d) + 1))|]  

      ==> setsum f {0..<n} < suminf f"

apply (drule summable_sums)

apply (auto simp add: sums_def LIMSEQ_def)

apply (drule_tac x = "f (n) + f (n + 1)" in spec)

apply (auto iff: real_0_less_add_iff)

   --{*legacy proof: not necessarily better!*}

apply (rule_tac [2] ccontr, drule_tac [2] linorder_not_less [THEN iffD1])

apply (frule_tac [2] no=no in sumr_pos_lt_pair_lemma) 

apply (drule_tac x = 0 in spec, simp)

apply (rotate_tac 1, drule_tac x = "Suc (Suc 0) * (Suc no) + n" in spec)

apply (safe, simp)

apply (subgoal_tac "suminf f + (f (n) + f (n + 1)) \<le>

 setsum f {0 ..< Suc (Suc 0) * (Suc no) + n}")

apply (rule_tac [2] y = "setsum f {0..<n+ Suc (Suc 0)}" in order_trans)

prefer 3 apply assumption

apply (rule_tac [2] y = "setsum f {0..<n} + (f (n) + f (n + 1))" in order_trans)

apply simp_all 

apply (subgoal_tac "suminf f \<le> setsum f {0..< Suc (Suc 0) * (Suc no) + n}")

apply (rule_tac [2] y = "suminf f + (f (n) + f (n + 1))" in order_trans)

prefer 3 apply simp

apply (drule_tac [2] x = 0 in spec)

 prefer 2 apply simp 

apply (subgoal_tac "0 \<le> setsum f {0 ..< Suc (Suc 0) * Suc no + n} + - suminf f")

apply (simp add: abs_if)

apply (auto simp add: linorder_not_less [symmetric])

done

text{*A summable series of positive terms has limit that is at least as

great as any partial sum.*}

lemma series_pos_le: 

     "[| summable f; \<forall>m \<ge> n. 0 \<le> f(m) |] ==> setsum f {0..<n} \<le> suminf f"

apply (drule summable_sums)

apply (simp add: sums_def)

apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const)

apply (erule LIMSEQ_le, blast)

apply (rule_tac x = n in exI, clarify)

apply (rule setsum_mono2)

apply auto

done

lemma series_pos_less:

     "[| summable f; \<forall>m \<ge> n. 0 < f(m) |] ==> setsum f {0..<n} < suminf f"

apply (rule_tac y = "setsum f {0..<Suc n}" in order_less_le_trans)

apply (rule_tac [2] series_pos_le, auto)

apply (drule_tac x = m in spec, auto)

done

text{*Sum of a geometric progression.*}

lemma sumr_geometric:

 "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::real)"

apply (induct "n", auto)

apply (rule_tac c1 = "x - 1" in real_mult_right_cancel [THEN iffD1])

apply (auto simp add: mult_assoc left_distrib)

apply (simp add: right_distrib diff_minus mult_commute)

done

lemma geometric_sums: "abs(x) < 1 ==> (%n. x ^ n) sums (1/(1 - x))"

apply (case_tac "x = 1")

apply (auto dest!: LIMSEQ_rabs_realpow_zero2 

        simp add: sumr_geometric sums_def diff_minus add_divide_distrib)

apply (subgoal_tac "1 / (1 + -x) = 0/ (x - 1) + - 1/ (x - 1) ")

apply (erule ssubst)

apply (rule LIMSEQ_add, rule LIMSEQ_divide)

apply (auto intro: LIMSEQ_const simp add: diff_minus minus_divide_right LIMSEQ_rabs_realpow_zero2)

done

text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}

lemma summable_convergent_sumr_iff:

 "summable f = convergent (%n. setsum f {0..<n})"

by (simp add: summable_def sums_def convergent_def)

lemma summable_Cauchy:

     "summable f =  

      (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. abs(setsum f {m..<n}) < e)"

apply (auto simp add: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def diff_minus[symmetric])

apply (drule_tac [!] spec, auto)

apply (rule_tac x = M in exI)

apply (rule_tac [2] x = N in exI, auto)

apply (cut_tac [!] m = m and n = n in less_linear, auto)

apply (frule le_less_trans [THEN less_imp_le], assumption)

apply (drule_tac x = n in spec, simp)

apply (drule_tac x = m in spec)

apply(simp add: setsum_diff[symmetric])

apply(subst abs_minus_commute)

apply(simp add: setsum_diff[symmetric])

apply(simp add: setsum_diff[symmetric])

done

(* FIXME move ivl_ lemmas out of this theory *)

text{*Comparison test*}

lemma summable_comparison_test:

     "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] ==> summable f"

apply (auto simp add: summable_Cauchy)

apply (drule spec, auto)

apply (rule_tac x = "N + Na" in exI, auto)

apply (rotate_tac 2)

apply (drule_tac x = m in spec)

apply (auto, rotate_tac 2, drule_tac x = n in spec)

apply (rule_tac y = "\<Sum>k=m..<n. abs(f k)" in order_le_less_trans)

apply (rule setsum_abs)

apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)

apply (auto intro: setsum_mono simp add: abs_interval_iff)

done

lemma summable_rabs_comparison_test:

     "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] 

      ==> summable (%k. abs (f k))"

apply (rule summable_comparison_test)

apply (auto simp add: abs_idempotent)

done

text{*Limit comparison property for series (c.f. jrh)*}

lemma summable_le:

     "[|\<forall>n. f n \<le> g n; summable f; summable g |] ==> suminf f \<le> suminf g"

apply (drule summable_sums)+

apply (auto intro!: LIMSEQ_le simp add: sums_def)

apply (rule exI)

apply (auto intro!: setsum_mono)

done

lemma summable_le2:

     "[|\<forall>n. abs(f n) \<le> g n; summable g |]  

      ==> summable f & suminf f \<le> suminf g"

apply (auto intro: summable_comparison_test intro!: summable_le)

apply (simp add: abs_le_interval_iff)

done

text{*Absolute convergence imples normal convergence*}

lemma summable_rabs_cancel: "summable (%n. abs (f n)) ==> summable f"

apply (auto simp add: summable_Cauchy)

apply (drule spec, auto)

apply (rule_tac x = N in exI, auto)

apply (drule spec, auto)

apply (rule_tac y = "\<Sum>n=m..<n. abs(f n)" in order_le_less_trans)

apply (auto)

done

text{*Absolute convergence of series*}

lemma summable_rabs:

     "summable (%n. abs (f n)) ==> abs(suminf f) \<le> suminf (%n. abs(f n))"

by (auto intro: LIMSEQ_le LIMSEQ_imp_rabs summable_rabs_cancel summable_sumr_LIMSEQ_suminf)

subsection{* The Ratio Test*}

lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"

apply (drule order_le_imp_less_or_eq, auto)

apply (subgoal_tac "0 \<le> c * abs y")

apply (simp add: zero_le_mult_iff, arith)

done

lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"

apply (drule le_imp_less_or_eq)

apply (auto dest: less_imp_Suc_add)

done

lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"

by (auto simp add: le_Suc_ex)

(*All this trouble just to get 0<c *)

lemma ratio_test_lemma2:

     "[| \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  

      ==> 0 < c | summable f"

apply (simp (no_asm) add: linorder_not_le [symmetric])

apply (simp add: summable_Cauchy)

apply (safe, subgoal_tac "\<forall>n. N \<le> n --> f (Suc n) = 0")

prefer 2 apply (blast intro: rabs_ratiotest_lemma)

apply (rule_tac x = "Suc N" in exI, clarify)

apply (drule_tac n=n in Suc_le_imp_diff_ge2, auto) 

done

lemma ratio_test:

     "[| c < 1; \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  

      ==> summable f"

apply (frule ratio_test_lemma2, auto)

apply (rule_tac g = "%n. (abs (f N) / (c ^ N))*c ^ n" 

       in summable_comparison_test)

apply (rule_tac x = N in exI, safe)

apply (drule le_Suc_ex_iff [THEN iffD1])

apply (auto simp add: power_add realpow_not_zero)

apply (induct_tac "na", auto)

apply (rule_tac y = "c*abs (f (N + n))" in order_trans)

apply (auto intro: mult_right_mono simp add: summable_def)

apply (simp add: mult_ac)

apply (rule_tac x = "abs (f N) * (1/ (1 - c)) / (c ^ N)" in exI)

apply (rule sums_divide) 

apply (rule sums_mult) 

apply (auto intro!: geometric_sums)

done

text{*Differentiation of finite sum*}

lemma DERIV_sumr [rule_format (no_asm)]:

     "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))  

      --> DERIV (%x. \<Sum>n=m..<n::nat. f n x) x :> (\<Sum>r=m..<n. f' r x)"

apply (induct "n")

apply (auto intro: DERIV_add)

done

ML

{*

val sums_def = thm"sums_def";

val summable_def = thm"summable_def";

val suminf_def = thm"suminf_def";

val sumr_minus_one_realpow_zero = thm "sumr_minus_one_realpow_zero";

val Suc_le_imp_diff_ge2 = thm "Suc_le_imp_diff_ge2";

val sumr_one_lb_realpow_zero = thm "sumr_one_lb_realpow_zero";

val sumr_group = thm "sumr_group";

val sums_summable = thm "sums_summable";

val summable_sums = thm "summable_sums";

val summable_sumr_LIMSEQ_suminf = thm "summable_sumr_LIMSEQ_suminf";

val sums_unique = thm "sums_unique";

val series_zero = thm "series_zero";

val sums_mult = thm "sums_mult";

val sums_divide = thm "sums_divide";

val sums_diff = thm "sums_diff";

val suminf_mult = thm "suminf_mult";

val suminf_mult2 = thm "suminf_mult2";

val suminf_diff = thm "suminf_diff";

val sums_minus = thm "sums_minus";

val sums_group = thm "sums_group";

val sumr_pos_lt_pair_lemma = thm "sumr_pos_lt_pair_lemma";

val sumr_pos_lt_pair = thm "sumr_pos_lt_pair";

val series_pos_le = thm "series_pos_le";

val series_pos_less = thm "series_pos_less";

val sumr_geometric = thm "sumr_geometric";

val geometric_sums = thm "geometric_sums";

val summable_convergent_sumr_iff = thm "summable_convergent_sumr_iff";

val summable_Cauchy = thm "summable_Cauchy";

val summable_comparison_test = thm "summable_comparison_test";

val summable_rabs_comparison_test = thm "summable_rabs_comparison_test";

val summable_le = thm "summable_le";

val summable_le2 = thm "summable_le2";

val summable_rabs_cancel = thm "summable_rabs_cancel";

val summable_rabs = thm "summable_rabs";

val rabs_ratiotest_lemma = thm "rabs_ratiotest_lemma";

val le_Suc_ex = thm "le_Suc_ex";

val le_Suc_ex_iff = thm "le_Suc_ex_iff";

val ratio_test_lemma2 = thm "ratio_test_lemma2";

val ratio_test = thm "ratio_test";

val DERIV_sumr = thm "DERIV_sumr";

*}

end