src/HOL/Hyperreal/Series.thy
author nipkow
Wed Mar 02 12:06:15 2005 +0100 (2005-03-02)
changeset 15561 045a07ac35a7
parent 15546 5188ce7316b7
child 16733 236dfafbeb63
permissions -rw-r--r--
another reorganization of setsums and intervals
paulson@10751
     1
(*  Title       : Series.thy
paulson@10751
     2
    Author      : Jacques D. Fleuriot
paulson@10751
     3
    Copyright   : 1998  University of Cambridge
paulson@14416
     4
paulson@14416
     5
Converted to Isar and polished by lcp
nipkow@15539
     6
Converted to setsum and polished yet more by TNN
paulson@10751
     7
*) 
paulson@10751
     8
paulson@14416
     9
header{*Finite Summation and Infinite Series*}
paulson@10751
    10
nipkow@15131
    11
theory Series
nipkow@15140
    12
imports SEQ Lim
nipkow@15131
    13
begin
nipkow@15561
    14
nipkow@15539
    15
declare atLeastLessThan_iff[iff]
nipkow@15561
    16
declare setsum_op_ivl_Suc[simp]
paulson@10751
    17
paulson@10751
    18
constdefs
nipkow@15543
    19
   sums  :: "(nat => real) => real => bool"     (infixr "sums" 80)
nipkow@15536
    20
   "f sums s  == (%n. setsum f {0..<n}) ----> s"
paulson@10751
    21
paulson@14416
    22
   summable :: "(nat=>real) => bool"
paulson@14416
    23
   "summable f == (\<exists>s. f sums s)"
paulson@14416
    24
paulson@14416
    25
   suminf   :: "(nat=>real) => real"
nipkow@15539
    26
   "suminf f == SOME s. f sums s"
paulson@14416
    27
nipkow@15546
    28
syntax
nipkow@15546
    29
  "_suminf" :: "idt => real => real"    ("\<Sum>_. _" [0, 10] 10)
nipkow@15546
    30
translations
nipkow@15546
    31
  "\<Sum>i. b" == "suminf (%i. b)"
nipkow@15546
    32
paulson@14416
    33
nipkow@15539
    34
lemma sumr_diff_mult_const:
nipkow@15539
    35
 "setsum f {0..<n} - (real n*r) = setsum (%i. f i - r) {0..<n::nat}"
nipkow@15536
    36
by (simp add: diff_minus setsum_addf real_of_nat_def)
nipkow@15536
    37
nipkow@15542
    38
lemma real_setsum_nat_ivl_bounded:
nipkow@15542
    39
     "(!!p. p < n \<Longrightarrow> f(p) \<le> K)
nipkow@15542
    40
      \<Longrightarrow> setsum f {0..<n::nat} \<le> real n * K"
nipkow@15542
    41
using setsum_bounded[where A = "{0..<n}"]
nipkow@15542
    42
by (auto simp:real_of_nat_def)
paulson@14416
    43
nipkow@15539
    44
(* Generalize from real to some algebraic structure? *)
nipkow@15539
    45
lemma sumr_minus_one_realpow_zero [simp]:
nipkow@15543
    46
  "(\<Sum>i=0..<2*n. (-1) ^ Suc i) = (0::real)"
paulson@15251
    47
by (induct "n", auto)
paulson@14416
    48
nipkow@15539
    49
(* FIXME this is an awful lemma! *)
nipkow@15539
    50
lemma sumr_one_lb_realpow_zero [simp]:
nipkow@15539
    51
  "(\<Sum>n=Suc 0..<n. f(n) * (0::real) ^ n) = 0"
paulson@15251
    52
apply (induct "n")
paulson@14416
    53
apply (case_tac [2] "n", auto)
paulson@14416
    54
done
paulson@14416
    55
nipkow@15543
    56
lemma sumr_group:
nipkow@15539
    57
     "(\<Sum>m=0..<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {0 ..< n * k}"
nipkow@15543
    58
apply (subgoal_tac "k = 0 | 0 < k", auto)
paulson@15251
    59
apply (induct "n")
nipkow@15539
    60
apply (simp_all add: setsum_add_nat_ivl add_commute)
paulson@14416
    61
done
nipkow@15539
    62
paulson@14416
    63
paulson@14416
    64
subsection{* Infinite Sums, by the Properties of Limits*}
paulson@14416
    65
paulson@14416
    66
(*----------------------
paulson@14416
    67
   suminf is the sum   
paulson@14416
    68
 ---------------------*)
paulson@14416
    69
lemma sums_summable: "f sums l ==> summable f"
paulson@14416
    70
by (simp add: sums_def summable_def, blast)
paulson@14416
    71
paulson@14416
    72
lemma summable_sums: "summable f ==> f sums (suminf f)"
paulson@14416
    73
apply (simp add: summable_def suminf_def)
paulson@14416
    74
apply (blast intro: someI2)
paulson@14416
    75
done
paulson@14416
    76
paulson@14416
    77
lemma summable_sumr_LIMSEQ_suminf: 
nipkow@15539
    78
     "summable f ==> (%n. setsum f {0..<n}) ----> (suminf f)"
paulson@14416
    79
apply (simp add: summable_def suminf_def sums_def)
paulson@14416
    80
apply (blast intro: someI2)
paulson@14416
    81
done
paulson@14416
    82
paulson@14416
    83
(*-------------------
paulson@14416
    84
    sum is unique                    
paulson@14416
    85
 ------------------*)
paulson@14416
    86
lemma sums_unique: "f sums s ==> (s = suminf f)"
paulson@14416
    87
apply (frule sums_summable [THEN summable_sums])
paulson@14416
    88
apply (auto intro!: LIMSEQ_unique simp add: sums_def)
paulson@14416
    89
done
paulson@14416
    90
paulson@14416
    91
lemma series_zero: 
nipkow@15539
    92
     "(\<forall>m. n \<le> m --> f(m) = 0) ==> f sums (setsum f {0..<n})"
nipkow@15537
    93
apply (simp add: sums_def LIMSEQ_def diff_minus[symmetric], safe)
paulson@14416
    94
apply (rule_tac x = n in exI)
nipkow@15542
    95
apply (clarsimp simp add:setsum_diff[symmetric] cong:setsum_ivl_cong)
paulson@14416
    96
done
paulson@14416
    97
nipkow@15539
    98
paulson@14416
    99
lemma sums_mult: "x sums x0 ==> (%n. c * x(n)) sums (c * x0)"
nipkow@15536
   100
by (auto simp add: sums_def setsum_mult [symmetric]
paulson@14416
   101
         intro!: LIMSEQ_mult intro: LIMSEQ_const)
paulson@14416
   102
paulson@14416
   103
lemma sums_divide: "x sums x' ==> (%n. x(n)/c) sums (x'/c)"
paulson@14416
   104
by (simp add: real_divide_def sums_mult mult_commute [of _ "inverse c"])
paulson@14416
   105
paulson@14416
   106
lemma sums_diff: "[| x sums x0; y sums y0 |] ==> (%n. x n - y n) sums (x0-y0)"
nipkow@15536
   107
by (auto simp add: sums_def setsum_subtractf intro: LIMSEQ_diff)
paulson@14416
   108
nipkow@15546
   109
lemma suminf_mult: "summable f ==> suminf f * c = (\<Sum>n. f n * c)"
paulson@14416
   110
by (auto intro!: sums_unique sums_mult summable_sums simp add: mult_commute)
paulson@14416
   111
nipkow@15546
   112
lemma suminf_mult2: "summable f ==> c * suminf f  = (\<Sum>n. c * f n)"
paulson@14416
   113
by (auto intro!: sums_unique sums_mult summable_sums)
paulson@14416
   114
paulson@14416
   115
lemma suminf_diff:
paulson@14416
   116
     "[| summable f; summable g |]   
nipkow@15546
   117
      ==> suminf f - suminf g  = (\<Sum>n. f n - g n)"
paulson@14416
   118
by (auto intro!: sums_diff sums_unique summable_sums)
paulson@14416
   119
paulson@14416
   120
lemma sums_minus: "x sums x0 ==> (%n. - x n) sums - x0"
nipkow@15536
   121
by (auto simp add: sums_def intro!: LIMSEQ_minus simp add: setsum_negf)
paulson@14416
   122
paulson@14416
   123
lemma sums_group:
nipkow@15539
   124
     "[|summable f; 0 < k |] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)"
paulson@14416
   125
apply (drule summable_sums)
nipkow@15543
   126
apply (auto simp add: sums_def LIMSEQ_def sumr_group)
paulson@14416
   127
apply (drule_tac x = r in spec, safe)
paulson@14416
   128
apply (rule_tac x = no in exI, safe)
paulson@14416
   129
apply (drule_tac x = "n*k" in spec)
paulson@14416
   130
apply (auto dest!: not_leE)
paulson@14416
   131
apply (drule_tac j = no in less_le_trans, auto)
paulson@14416
   132
done
paulson@14416
   133
paulson@14416
   134
lemma sumr_pos_lt_pair_lemma:
nipkow@15539
   135
  "[|\<forall>d. - f (n + (d + d)) < (f (Suc (n + (d + d))) :: real) |]
nipkow@15539
   136
   ==> setsum f {0..<n+Suc(Suc 0)} \<le> setsum f {0..<Suc(Suc 0) * Suc no + n}"
paulson@15251
   137
apply (induct "no", auto)
paulson@15251
   138
apply (drule_tac x = "Suc no" in spec)
nipkow@15539
   139
apply (simp add: add_ac)
paulson@14416
   140
done
paulson@10751
   141
paulson@10751
   142
paulson@14416
   143
lemma sumr_pos_lt_pair:
paulson@15234
   144
     "[|summable f; 
paulson@15234
   145
        \<forall>d. 0 < (f(n + (Suc(Suc 0) * d))) + f(n + ((Suc(Suc 0) * d) + 1))|]  
nipkow@15539
   146
      ==> setsum f {0..<n} < suminf f"
paulson@14416
   147
apply (drule summable_sums)
paulson@14416
   148
apply (auto simp add: sums_def LIMSEQ_def)
paulson@15234
   149
apply (drule_tac x = "f (n) + f (n + 1)" in spec)
paulson@15085
   150
apply (auto iff: real_0_less_add_iff)
paulson@15085
   151
   --{*legacy proof: not necessarily better!*}
paulson@14416
   152
apply (rule_tac [2] ccontr, drule_tac [2] linorder_not_less [THEN iffD1])
paulson@14416
   153
apply (frule_tac [2] no=no in sumr_pos_lt_pair_lemma) 
paulson@14416
   154
apply (drule_tac x = 0 in spec, simp)
paulson@14416
   155
apply (rotate_tac 1, drule_tac x = "Suc (Suc 0) * (Suc no) + n" in spec)
paulson@14416
   156
apply (safe, simp)
nipkow@15539
   157
apply (subgoal_tac "suminf f + (f (n) + f (n + 1)) \<le>
nipkow@15539
   158
 setsum f {0 ..< Suc (Suc 0) * (Suc no) + n}")
nipkow@15539
   159
apply (rule_tac [2] y = "setsum f {0..<n+ Suc (Suc 0)}" in order_trans)
paulson@14416
   160
prefer 3 apply assumption
nipkow@15539
   161
apply (rule_tac [2] y = "setsum f {0..<n} + (f (n) + f (n + 1))" in order_trans)
paulson@14416
   162
apply simp_all 
nipkow@15539
   163
apply (subgoal_tac "suminf f \<le> setsum f {0..< Suc (Suc 0) * (Suc no) + n}")
paulson@14416
   164
apply (rule_tac [2] y = "suminf f + (f (n) + f (n + 1))" in order_trans)
nipkow@15539
   165
prefer 3 apply simp
paulson@14416
   166
apply (drule_tac [2] x = 0 in spec)
paulson@14416
   167
 prefer 2 apply simp 
nipkow@15539
   168
apply (subgoal_tac "0 \<le> setsum f {0 ..< Suc (Suc 0) * Suc no + n} + - suminf f")
nipkow@15539
   169
apply (simp add: abs_if)
paulson@14416
   170
apply (auto simp add: linorder_not_less [symmetric])
paulson@14416
   171
done
paulson@14416
   172
paulson@15085
   173
text{*A summable series of positive terms has limit that is at least as
paulson@15085
   174
great as any partial sum.*}
paulson@14416
   175
paulson@14416
   176
lemma series_pos_le: 
nipkow@15539
   177
     "[| summable f; \<forall>m \<ge> n. 0 \<le> f(m) |] ==> setsum f {0..<n} \<le> suminf f"
paulson@14416
   178
apply (drule summable_sums)
paulson@14416
   179
apply (simp add: sums_def)
nipkow@15539
   180
apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const)
nipkow@15539
   181
apply (erule LIMSEQ_le, blast)
nipkow@15539
   182
apply (rule_tac x = n in exI, clarify)
nipkow@15539
   183
apply (rule setsum_mono2)
nipkow@15539
   184
apply auto
paulson@14416
   185
done
paulson@14416
   186
paulson@14416
   187
lemma series_pos_less:
nipkow@15539
   188
     "[| summable f; \<forall>m \<ge> n. 0 < f(m) |] ==> setsum f {0..<n} < suminf f"
nipkow@15539
   189
apply (rule_tac y = "setsum f {0..<Suc n}" in order_less_le_trans)
paulson@14416
   190
apply (rule_tac [2] series_pos_le, auto)
paulson@14416
   191
apply (drule_tac x = m in spec, auto)
paulson@14416
   192
done
paulson@14416
   193
paulson@15085
   194
text{*Sum of a geometric progression.*}
paulson@14416
   195
nipkow@15539
   196
lemma sumr_geometric:
nipkow@15539
   197
 "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::real)"
paulson@15251
   198
apply (induct "n", auto)
paulson@14416
   199
apply (rule_tac c1 = "x - 1" in real_mult_right_cancel [THEN iffD1])
nipkow@15539
   200
apply (auto simp add: mult_assoc left_distrib)
paulson@15234
   201
apply (simp add: right_distrib diff_minus mult_commute)
paulson@14416
   202
done
paulson@14416
   203
paulson@14416
   204
lemma geometric_sums: "abs(x) < 1 ==> (%n. x ^ n) sums (1/(1 - x))"
paulson@14416
   205
apply (case_tac "x = 1")
paulson@15234
   206
apply (auto dest!: LIMSEQ_rabs_realpow_zero2 
paulson@15234
   207
        simp add: sumr_geometric sums_def diff_minus add_divide_distrib)
paulson@14416
   208
apply (subgoal_tac "1 / (1 + -x) = 0/ (x - 1) + - 1/ (x - 1) ")
paulson@14416
   209
apply (erule ssubst)
paulson@14416
   210
apply (rule LIMSEQ_add, rule LIMSEQ_divide)
paulson@15234
   211
apply (auto intro: LIMSEQ_const simp add: diff_minus minus_divide_right LIMSEQ_rabs_realpow_zero2)
paulson@14416
   212
done
paulson@14416
   213
paulson@15085
   214
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
paulson@15085
   215
nipkow@15539
   216
lemma summable_convergent_sumr_iff:
nipkow@15539
   217
 "summable f = convergent (%n. setsum f {0..<n})"
paulson@14416
   218
by (simp add: summable_def sums_def convergent_def)
paulson@14416
   219
paulson@14416
   220
lemma summable_Cauchy:
paulson@14416
   221
     "summable f =  
nipkow@15539
   222
      (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. abs(setsum f {m..<n}) < e)"
nipkow@15537
   223
apply (auto simp add: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def diff_minus[symmetric])
nipkow@15539
   224
apply (drule_tac [!] spec, auto)
paulson@14416
   225
apply (rule_tac x = M in exI)
paulson@14416
   226
apply (rule_tac [2] x = N in exI, auto)
paulson@14416
   227
apply (cut_tac [!] m = m and n = n in less_linear, auto)
paulson@14416
   228
apply (frule le_less_trans [THEN less_imp_le], assumption)
nipkow@15360
   229
apply (drule_tac x = n in spec, simp)
paulson@14416
   230
apply (drule_tac x = m in spec)
nipkow@15539
   231
apply(simp add: setsum_diff[symmetric])
nipkow@15537
   232
apply(subst abs_minus_commute)
nipkow@15539
   233
apply(simp add: setsum_diff[symmetric])
nipkow@15539
   234
apply(simp add: setsum_diff[symmetric])
paulson@14416
   235
done
paulson@14416
   236
paulson@15085
   237
text{*Comparison test*}
paulson@15085
   238
paulson@14416
   239
lemma summable_comparison_test:
nipkow@15360
   240
     "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] ==> summable f"
paulson@14416
   241
apply (auto simp add: summable_Cauchy)
paulson@14416
   242
apply (drule spec, auto)
paulson@14416
   243
apply (rule_tac x = "N + Na" in exI, auto)
paulson@14416
   244
apply (rotate_tac 2)
paulson@14416
   245
apply (drule_tac x = m in spec)
paulson@14416
   246
apply (auto, rotate_tac 2, drule_tac x = n in spec)
nipkow@15539
   247
apply (rule_tac y = "\<Sum>k=m..<n. abs(f k)" in order_le_less_trans)
nipkow@15536
   248
apply (rule setsum_abs)
nipkow@15539
   249
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
nipkow@15539
   250
apply (auto intro: setsum_mono simp add: abs_interval_iff)
paulson@14416
   251
done
paulson@14416
   252
paulson@14416
   253
lemma summable_rabs_comparison_test:
nipkow@15360
   254
     "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] 
paulson@14416
   255
      ==> summable (%k. abs (f k))"
paulson@14416
   256
apply (rule summable_comparison_test)
nipkow@15543
   257
apply (auto)
paulson@14416
   258
done
paulson@14416
   259
paulson@15085
   260
text{*Limit comparison property for series (c.f. jrh)*}
paulson@15085
   261
paulson@14416
   262
lemma summable_le:
paulson@14416
   263
     "[|\<forall>n. f n \<le> g n; summable f; summable g |] ==> suminf f \<le> suminf g"
paulson@14416
   264
apply (drule summable_sums)+
paulson@14416
   265
apply (auto intro!: LIMSEQ_le simp add: sums_def)
paulson@14416
   266
apply (rule exI)
nipkow@15539
   267
apply (auto intro!: setsum_mono)
paulson@14416
   268
done
paulson@14416
   269
paulson@14416
   270
lemma summable_le2:
paulson@14416
   271
     "[|\<forall>n. abs(f n) \<le> g n; summable g |]  
paulson@14416
   272
      ==> summable f & suminf f \<le> suminf g"
paulson@14416
   273
apply (auto intro: summable_comparison_test intro!: summable_le)
paulson@14416
   274
apply (simp add: abs_le_interval_iff)
paulson@14416
   275
done
paulson@14416
   276
paulson@15085
   277
text{*Absolute convergence imples normal convergence*}
paulson@14416
   278
lemma summable_rabs_cancel: "summable (%n. abs (f n)) ==> summable f"
nipkow@15536
   279
apply (auto simp add: summable_Cauchy)
paulson@14416
   280
apply (drule spec, auto)
paulson@14416
   281
apply (rule_tac x = N in exI, auto)
paulson@14416
   282
apply (drule spec, auto)
nipkow@15539
   283
apply (rule_tac y = "\<Sum>n=m..<n. abs(f n)" in order_le_less_trans)
nipkow@15536
   284
apply (auto)
paulson@14416
   285
done
paulson@14416
   286
paulson@15085
   287
text{*Absolute convergence of series*}
paulson@14416
   288
lemma summable_rabs:
nipkow@15546
   289
     "summable (%n. abs (f n)) ==> abs(suminf f) \<le> (\<Sum>n. abs(f n))"
nipkow@15536
   290
by (auto intro: LIMSEQ_le LIMSEQ_imp_rabs summable_rabs_cancel summable_sumr_LIMSEQ_suminf)
paulson@14416
   291
paulson@14416
   292
paulson@14416
   293
subsection{* The Ratio Test*}
paulson@14416
   294
paulson@14416
   295
lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
paulson@14416
   296
apply (drule order_le_imp_less_or_eq, auto)
paulson@14416
   297
apply (subgoal_tac "0 \<le> c * abs y")
paulson@14416
   298
apply (simp add: zero_le_mult_iff, arith)
paulson@14416
   299
done
paulson@14416
   300
paulson@14416
   301
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"
paulson@14416
   302
apply (drule le_imp_less_or_eq)
paulson@14416
   303
apply (auto dest: less_imp_Suc_add)
paulson@14416
   304
done
paulson@14416
   305
paulson@14416
   306
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"
paulson@14416
   307
by (auto simp add: le_Suc_ex)
paulson@14416
   308
paulson@14416
   309
(*All this trouble just to get 0<c *)
paulson@14416
   310
lemma ratio_test_lemma2:
nipkow@15360
   311
     "[| \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  
paulson@14416
   312
      ==> 0 < c | summable f"
paulson@14416
   313
apply (simp (no_asm) add: linorder_not_le [symmetric])
paulson@14416
   314
apply (simp add: summable_Cauchy)
nipkow@15543
   315
apply (safe, subgoal_tac "\<forall>n. N < n --> f (n) = 0")
nipkow@15543
   316
 prefer 2
nipkow@15543
   317
 apply clarify
nipkow@15543
   318
 apply(erule_tac x = "n - 1" in allE)
nipkow@15543
   319
 apply (simp add:diff_Suc split:nat.splits)
nipkow@15543
   320
 apply (blast intro: rabs_ratiotest_lemma)
paulson@14416
   321
apply (rule_tac x = "Suc N" in exI, clarify)
nipkow@15543
   322
apply(simp cong:setsum_ivl_cong)
paulson@14416
   323
done
paulson@14416
   324
paulson@14416
   325
lemma ratio_test:
nipkow@15360
   326
     "[| c < 1; \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  
paulson@14416
   327
      ==> summable f"
paulson@14416
   328
apply (frule ratio_test_lemma2, auto)
paulson@15234
   329
apply (rule_tac g = "%n. (abs (f N) / (c ^ N))*c ^ n" 
paulson@15234
   330
       in summable_comparison_test)
paulson@14416
   331
apply (rule_tac x = N in exI, safe)
paulson@14416
   332
apply (drule le_Suc_ex_iff [THEN iffD1])
paulson@14416
   333
apply (auto simp add: power_add realpow_not_zero)
nipkow@15539
   334
apply (induct_tac "na", auto)
paulson@14416
   335
apply (rule_tac y = "c*abs (f (N + n))" in order_trans)
paulson@14416
   336
apply (auto intro: mult_right_mono simp add: summable_def)
paulson@14416
   337
apply (simp add: mult_ac)
paulson@15234
   338
apply (rule_tac x = "abs (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
paulson@15234
   339
apply (rule sums_divide) 
paulson@15234
   340
apply (rule sums_mult) 
paulson@15234
   341
apply (auto intro!: geometric_sums)
paulson@14416
   342
done
paulson@14416
   343
paulson@14416
   344
paulson@15085
   345
text{*Differentiation of finite sum*}
paulson@14416
   346
paulson@14416
   347
lemma DERIV_sumr [rule_format (no_asm)]:
paulson@14416
   348
     "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))  
nipkow@15539
   349
      --> DERIV (%x. \<Sum>n=m..<n::nat. f n x) x :> (\<Sum>r=m..<n. f' r x)"
paulson@15251
   350
apply (induct "n")
paulson@14416
   351
apply (auto intro: DERIV_add)
paulson@14416
   352
done
paulson@14416
   353
paulson@14416
   354
ML
paulson@14416
   355
{*
paulson@14416
   356
val sums_def = thm"sums_def";
paulson@14416
   357
val summable_def = thm"summable_def";
paulson@14416
   358
val suminf_def = thm"suminf_def";
paulson@14416
   359
paulson@14416
   360
val sumr_minus_one_realpow_zero = thm "sumr_minus_one_realpow_zero";
paulson@14416
   361
val sumr_one_lb_realpow_zero = thm "sumr_one_lb_realpow_zero";
paulson@14416
   362
val sumr_group = thm "sumr_group";
paulson@14416
   363
val sums_summable = thm "sums_summable";
paulson@14416
   364
val summable_sums = thm "summable_sums";
paulson@14416
   365
val summable_sumr_LIMSEQ_suminf = thm "summable_sumr_LIMSEQ_suminf";
paulson@14416
   366
val sums_unique = thm "sums_unique";
paulson@14416
   367
val series_zero = thm "series_zero";
paulson@14416
   368
val sums_mult = thm "sums_mult";
paulson@14416
   369
val sums_divide = thm "sums_divide";
paulson@14416
   370
val sums_diff = thm "sums_diff";
paulson@14416
   371
val suminf_mult = thm "suminf_mult";
paulson@14416
   372
val suminf_mult2 = thm "suminf_mult2";
paulson@14416
   373
val suminf_diff = thm "suminf_diff";
paulson@14416
   374
val sums_minus = thm "sums_minus";
paulson@14416
   375
val sums_group = thm "sums_group";
paulson@14416
   376
val sumr_pos_lt_pair_lemma = thm "sumr_pos_lt_pair_lemma";
paulson@14416
   377
val sumr_pos_lt_pair = thm "sumr_pos_lt_pair";
paulson@14416
   378
val series_pos_le = thm "series_pos_le";
paulson@14416
   379
val series_pos_less = thm "series_pos_less";
paulson@14416
   380
val sumr_geometric = thm "sumr_geometric";
paulson@14416
   381
val geometric_sums = thm "geometric_sums";
paulson@14416
   382
val summable_convergent_sumr_iff = thm "summable_convergent_sumr_iff";
paulson@14416
   383
val summable_Cauchy = thm "summable_Cauchy";
paulson@14416
   384
val summable_comparison_test = thm "summable_comparison_test";
paulson@14416
   385
val summable_rabs_comparison_test = thm "summable_rabs_comparison_test";
paulson@14416
   386
val summable_le = thm "summable_le";
paulson@14416
   387
val summable_le2 = thm "summable_le2";
paulson@14416
   388
val summable_rabs_cancel = thm "summable_rabs_cancel";
paulson@14416
   389
val summable_rabs = thm "summable_rabs";
paulson@14416
   390
val rabs_ratiotest_lemma = thm "rabs_ratiotest_lemma";
paulson@14416
   391
val le_Suc_ex = thm "le_Suc_ex";
paulson@14416
   392
val le_Suc_ex_iff = thm "le_Suc_ex_iff";
paulson@14416
   393
val ratio_test_lemma2 = thm "ratio_test_lemma2";
paulson@14416
   394
val ratio_test = thm "ratio_test";
paulson@14416
   395
val DERIV_sumr = thm "DERIV_sumr";
paulson@14416
   396
*}
paulson@14416
   397
paulson@14416
   398
end