src/HOL/Hyperreal/Series.thy
author nipkow
Mon Feb 21 15:04:10 2005 +0100 (2005-02-21)
changeset 15539 333a88244569
parent 15537 5538d3244b4d
child 15542 ee6cd48cf840
permissions -rw-r--r--
comprehensive cleanup, replacing sumr by setsum
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
paulson@10751
    14
nipkow@15539
    15
(* FIXME why not globally? *)
nipkow@15536
    16
declare atLeastLessThan_empty[simp];
nipkow@15539
    17
declare atLeastLessThan_iff[iff]
paulson@10751
    18
paulson@10751
    19
constdefs
paulson@14416
    20
   sums  :: "[nat=>real,real] => bool"     (infixr "sums" 80)
nipkow@15536
    21
   "f sums s  == (%n. setsum f {0..<n}) ----> s"
paulson@10751
    22
paulson@14416
    23
   summable :: "(nat=>real) => bool"
paulson@14416
    24
   "summable f == (\<exists>s. f sums s)"
paulson@14416
    25
paulson@14416
    26
   suminf   :: "(nat=>real) => real"
nipkow@15539
    27
   "suminf f == SOME s. f sums s"
paulson@14416
    28
nipkow@15539
    29
lemma setsum_Suc[simp]:
nipkow@15536
    30
  "setsum f {m..<Suc n} = (if n < m then 0 else setsum f {m..<n} + f(n))"
nipkow@15536
    31
by (simp add: atLeastLessThanSuc add_commute)
paulson@14416
    32
nipkow@15536
    33
(*
paulson@14416
    34
lemma sumr_add: "sumr m n f + sumr m n g = sumr m n (%n. f n + g n)"
paulson@15047
    35
by (simp add: setsum_addf)
paulson@14416
    36
paulson@15047
    37
lemma sumr_mult: "r * sumr m n (f::nat=>real) = sumr m n (%n. r * f n)"
paulson@15047
    38
by (simp add: setsum_mult)
paulson@14416
    39
paulson@14416
    40
lemma sumr_split_add [rule_format]:
paulson@15047
    41
     "n < p --> sumr 0 n f + sumr n p f = sumr 0 p (f::nat=>real)"
paulson@15251
    42
apply (induct "p", auto)
paulson@14416
    43
apply (rename_tac k) 
paulson@14416
    44
apply (subgoal_tac "n=k", auto) 
paulson@14416
    45
done
nipkow@15536
    46
nipkow@15536
    47
lemma sumr_split_add: "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15536
    48
  setsum f {m..<n} + setsum f {n..<p} = setsum f {m..<p::nat}"
nipkow@15536
    49
by (simp add:setsum_Un_disjoint[symmetric] ivl_disj_int ivl_disj_un)
paulson@14416
    50
paulson@15047
    51
lemma sumr_split_add_minus:
nipkow@15537
    52
fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
nipkow@15537
    53
shows "\<lbrakk> m \<le> n; n \<le> p \<rbrakk> \<Longrightarrow>
nipkow@15537
    54
  setsum f {m..<p} - setsum f {m..<n} = setsum f {n..<p}"
nipkow@15537
    55
using sumr_split_add [of m n p f,symmetric]
paulson@14416
    56
apply (simp add: add_ac)
paulson@14416
    57
done
nipkow@15539
    58
*)
paulson@14416
    59
nipkow@15539
    60
lemma sumr_diff_mult_const:
nipkow@15539
    61
 "setsum f {0..<n} - (real n*r) = setsum (%i. f i - r) {0..<n::nat}"
nipkow@15536
    62
by (simp add: diff_minus setsum_addf real_of_nat_def)
nipkow@15536
    63
nipkow@15536
    64
(*
paulson@15047
    65
lemma sumr_rabs: "abs(sumr m n  (f::nat=>real)) \<le> sumr m n (%i. abs(f i))"
paulson@15047
    66
by (simp add: setsum_abs)
paulson@14416
    67
paulson@15047
    68
lemma sumr_rabs_ge_zero [iff]: "0 \<le> sumr m n (%n. abs (f n))"
paulson@15047
    69
by (simp add: setsum_abs_ge_zero)
paulson@14416
    70
paulson@15047
    71
text{*Just a congruence rule*}
paulson@15047
    72
lemma sumr_fun_eq:
paulson@15047
    73
     "(\<forall>r. m \<le> r & r < n --> f r = g r) ==> sumr m n f = sumr m n g"
paulson@15047
    74
by (auto intro: setsum_cong) 
paulson@14416
    75
paulson@15047
    76
lemma sumr_less_bounds_zero [simp]: "n < m ==> sumr m n f = 0"
paulson@15047
    77
by (simp add: atLeastLessThan_empty)
paulson@14416
    78
paulson@14416
    79
lemma sumr_minus: "sumr m n (%i. - f i) = - sumr m n f"
paulson@15047
    80
by (simp add: Finite_Set.setsum_negf)
nipkow@15539
    81
nipkow@15539
    82
lemma sumr_shift_bounds:
nipkow@15539
    83
  "setsum f {m+k..<n+k} = setsum (%i. f(i + k)){m..<n::nat}"
nipkow@15539
    84
by (induct "n", auto)
nipkow@15536
    85
*)
paulson@14416
    86
nipkow@15539
    87
(* Generalize from real to some algebraic structure? *)
nipkow@15539
    88
lemma sumr_minus_one_realpow_zero [simp]:
nipkow@15539
    89
  "setsum (%i. (-1) ^ Suc i) {0..<2*n} = (0::real)"
paulson@15251
    90
by (induct "n", auto)
paulson@14416
    91
nipkow@15539
    92
(*
nipkow@15539
    93
lemma sumr_interval_const2:
nipkow@15539
    94
     "[|\<forall>n\<ge>m. f n = r; m \<le> k|]
nipkow@15539
    95
      ==> sumr m k f = (real (k - m) * r)"
nipkow@15539
    96
apply (induct "k", auto)
paulson@15251
    97
apply (drule_tac x = k in spec)
paulson@14416
    98
apply (auto dest!: le_imp_less_or_eq)
paulson@15047
    99
apply (simp add: left_distrib real_of_nat_Suc split: nat_diff_split)
paulson@14416
   100
done
nipkow@15539
   101
*)
nipkow@15539
   102
(* FIXME split in tow steps
nipkow@15539
   103
lemma setsum_nat_set_real_const:
nipkow@15539
   104
  "(\<And>n. n\<in>A \<Longrightarrow> f n = r) \<Longrightarrow> setsum f A = real(card A) * r" (is "PROP ?P")
nipkow@15539
   105
proof (cases "finite A")
nipkow@15539
   106
  case True
nipkow@15539
   107
  thus "PROP ?P"
nipkow@15539
   108
  proof induct
nipkow@15539
   109
    case empty thus ?case by simp
nipkow@15539
   110
  next
nipkow@15539
   111
    case insert thus ?case by(simp add: left_distrib real_of_nat_Suc)
nipkow@15539
   112
  qed
nipkow@15539
   113
next
nipkow@15539
   114
  case False thus "PROP ?P" by (simp add: setsum_def)
nipkow@15539
   115
qed
nipkow@15539
   116
 *)
paulson@14416
   117
nipkow@15539
   118
(*
nipkow@15539
   119
lemma sumr_le:
nipkow@15539
   120
     "[|\<forall>n\<ge>m. 0 \<le> (f n::real); m < k|] ==> setsum f {0..<m} \<le> setsum f {0..<k::nat}"
nipkow@15539
   121
apply (induct "k")
nipkow@15539
   122
apply (auto simp add: less_Suc_eq_le)
nipkow@15539
   123
apply (drule_tac x = k in spec, safe)
nipkow@15539
   124
apply (drule le_imp_less_or_eq, safe)
nipkow@15539
   125
apply (arith)
nipkow@15539
   126
apply (drule_tac a = "sumr 0 m f" in order_refl [THEN add_mono], auto)
paulson@14416
   127
done
paulson@14416
   128
paulson@15251
   129
lemma sumr_le:
nipkow@15360
   130
     "[|\<forall>n\<ge>m. 0 \<le> f n; m < k|] ==> sumr 0 m f \<le> sumr 0 k f"
paulson@15251
   131
apply (induct "k")
paulson@14416
   132
apply (auto simp add: less_Suc_eq_le)
paulson@15251
   133
apply (drule_tac x = k in spec, safe)
paulson@14416
   134
apply (drule le_imp_less_or_eq, safe)
paulson@15047
   135
apply (arith) 
paulson@14416
   136
apply (drule_tac a = "sumr 0 m f" in order_refl [THEN add_mono], auto)
paulson@14416
   137
done
paulson@14416
   138
paulson@14416
   139
lemma sumr_le2 [rule_format (no_asm)]:
paulson@14416
   140
     "(\<forall>r. m \<le> r & r < n --> f r \<le> g r) --> sumr m n f \<le> sumr m n g"
paulson@15251
   141
apply (induct "n")
paulson@14416
   142
apply (auto intro: add_mono simp add: le_def)
paulson@14416
   143
done
nipkow@15539
   144
*)
paulson@14416
   145
nipkow@15539
   146
(*
nipkow@15360
   147
lemma sumr_ge_zero: "(\<forall>n\<ge>m. 0 \<le> f n) --> 0 \<le> sumr m n f"
paulson@15251
   148
apply (induct "n", auto)
paulson@14416
   149
apply (drule_tac x = n in spec, arith)
paulson@14416
   150
done
nipkow@15539
   151
*)
paulson@14416
   152
nipkow@15539
   153
(*
paulson@14416
   154
lemma rabs_sumr_rabs_cancel [simp]:
paulson@15229
   155
     "abs (sumr m n (%k. abs (f k))) = (sumr m n (%k. abs (f k)))"
paulson@15251
   156
by (induct "n", simp_all add: add_increasing)
paulson@14416
   157
paulson@14416
   158
lemma sumr_zero [rule_format]:
nipkow@15360
   159
     "\<forall>n \<ge> N. f n = 0 ==> N \<le> m --> sumr m n f = 0"
paulson@15251
   160
by (induct "n", auto)
nipkow@15539
   161
*)
paulson@14416
   162
paulson@14416
   163
lemma Suc_le_imp_diff_ge2:
nipkow@15539
   164
     "[|\<forall>n \<ge> N. f (Suc n) = 0; Suc N \<le> m|] ==> setsum f {m..<n} = 0"
nipkow@15539
   165
apply (rule setsum_0')
paulson@14416
   166
apply (case_tac "n", auto)
nipkow@15539
   167
apply(erule_tac x = "a - 1" in allE)
nipkow@15539
   168
apply (simp split:nat_diff_split)
paulson@14416
   169
done
paulson@14416
   170
nipkow@15539
   171
(* FIXME this is an awful lemma! *)
nipkow@15539
   172
lemma sumr_one_lb_realpow_zero [simp]:
nipkow@15539
   173
  "(\<Sum>n=Suc 0..<n. f(n) * (0::real) ^ n) = 0"
paulson@15251
   174
apply (induct "n")
paulson@14416
   175
apply (case_tac [2] "n", auto)
paulson@14416
   176
done
nipkow@15536
   177
(*
paulson@14416
   178
lemma sumr_diff: "sumr m n f - sumr m n g = sumr m n (%n. f n - g n)"
nipkow@15536
   179
by (simp add: diff_minus setsum_addf setsum_negf)
nipkow@15536
   180
*)
nipkow@15539
   181
(*
paulson@14416
   182
lemma sumr_subst [rule_format (no_asm)]:
paulson@14416
   183
     "(\<forall>p. m \<le> p & p < m+n --> (f p = g p)) --> sumr m n f = sumr m n g"
paulson@15251
   184
by (induct "n", auto)
nipkow@15539
   185
*)
paulson@14416
   186
nipkow@15539
   187
lemma setsum_bounded:
nipkow@15539
   188
  assumes le: "\<And>i. i\<in>A \<Longrightarrow> f i \<le> (K::'a::{comm_semiring_1_cancel, pordered_ab_semigroup_add})"
nipkow@15539
   189
  shows "setsum f A \<le> of_nat(card A) * K"
nipkow@15539
   190
proof (cases "finite A")
nipkow@15539
   191
  case True
nipkow@15539
   192
  thus ?thesis using le setsum_mono[where K=A and g = "%x. K"] by simp
nipkow@15539
   193
next
nipkow@15539
   194
  case False thus ?thesis by (simp add: setsum_def)
nipkow@15539
   195
qed
nipkow@15539
   196
(*
paulson@14416
   197
lemma sumr_bound [rule_format (no_asm)]:
paulson@14416
   198
     "(\<forall>p. m \<le> p & p < m + n --> (f(p) \<le> K))  
nipkow@15539
   199
      --> setsum f {m..<m+n::nat} \<le> (real n * K)"
paulson@15251
   200
apply (induct "n")
paulson@14416
   201
apply (auto intro: add_mono simp add: left_distrib real_of_nat_Suc)
paulson@14416
   202
done
nipkow@15539
   203
*)
nipkow@15539
   204
(* FIXME should be generalized
paulson@14416
   205
lemma sumr_bound2 [rule_format (no_asm)]:
paulson@14416
   206
     "(\<forall>p. 0 \<le> p & p < n --> (f(p) \<le> K))  
nipkow@15539
   207
      --> setsum f {0..<n::nat} \<le> (real n * K)"
paulson@15251
   208
apply (induct "n")
paulson@15047
   209
apply (auto intro: add_mono simp add: left_distrib real_of_nat_Suc add_commute)
paulson@14416
   210
done
nipkow@15539
   211
 *)
nipkow@15539
   212
(* FIXME a bit specialized for [simp]! *)
paulson@14416
   213
lemma sumr_group [simp]:
nipkow@15539
   214
     "(\<Sum>m=0..<n::nat. setsum f {m * k ..< m*k + k}) = setsum f {0 ..< n * k}"
nipkow@15539
   215
apply (subgoal_tac "k = 0 | 0 < k", auto simp:setsum_0')
paulson@15251
   216
apply (induct "n")
nipkow@15539
   217
apply (simp_all add: setsum_add_nat_ivl add_commute)
paulson@14416
   218
done
nipkow@15539
   219
(* FIXME setsum_0[simp] *)
nipkow@15539
   220
paulson@14416
   221
paulson@14416
   222
subsection{* Infinite Sums, by the Properties of Limits*}
paulson@14416
   223
paulson@14416
   224
(*----------------------
paulson@14416
   225
   suminf is the sum   
paulson@14416
   226
 ---------------------*)
paulson@14416
   227
lemma sums_summable: "f sums l ==> summable f"
paulson@14416
   228
by (simp add: sums_def summable_def, blast)
paulson@14416
   229
paulson@14416
   230
lemma summable_sums: "summable f ==> f sums (suminf f)"
paulson@14416
   231
apply (simp add: summable_def suminf_def)
paulson@14416
   232
apply (blast intro: someI2)
paulson@14416
   233
done
paulson@14416
   234
paulson@14416
   235
lemma summable_sumr_LIMSEQ_suminf: 
nipkow@15539
   236
     "summable f ==> (%n. setsum f {0..<n}) ----> (suminf f)"
paulson@14416
   237
apply (simp add: summable_def suminf_def sums_def)
paulson@14416
   238
apply (blast intro: someI2)
paulson@14416
   239
done
paulson@14416
   240
paulson@14416
   241
(*-------------------
paulson@14416
   242
    sum is unique                    
paulson@14416
   243
 ------------------*)
paulson@14416
   244
lemma sums_unique: "f sums s ==> (s = suminf f)"
paulson@14416
   245
apply (frule sums_summable [THEN summable_sums])
paulson@14416
   246
apply (auto intro!: LIMSEQ_unique simp add: sums_def)
paulson@14416
   247
done
paulson@14416
   248
paulson@14416
   249
(*
paulson@14416
   250
Goalw [sums_def,LIMSEQ_def] 
paulson@14416
   251
     "(\<forall>m. n \<le> Suc m --> f(m) = 0) ==> f sums (sumr 0 n f)"
paulson@14416
   252
by safe
paulson@14416
   253
by (res_inst_tac [("x","n")] exI 1);
paulson@14416
   254
by (safe THEN ftac le_imp_less_or_eq 1)
paulson@14416
   255
by safe
paulson@14416
   256
by (dres_inst_tac [("f","f")] sumr_split_add_minus 1);
paulson@14416
   257
by (ALLGOALS (Asm_simp_tac));
paulson@14416
   258
by (dtac (conjI RS sumr_interval_const) 1);
paulson@14416
   259
by Auto_tac
paulson@14416
   260
qed "series_zero";
paulson@14416
   261
next one was called series_zero2
paulson@14416
   262
**********************)
paulson@14416
   263
nipkow@15539
   264
lemma ivl_subset[simp]:
nipkow@15539
   265
 "({i..<j} \<subseteq> {m..<n}) = (j \<le> i | m \<le> i & j \<le> (n::'a::linorder))"
nipkow@15539
   266
apply(auto simp:linorder_not_le)
nipkow@15539
   267
apply(rule ccontr)
nipkow@15539
   268
apply(frule subsetCE[where c = n])
nipkow@15539
   269
apply(auto simp:linorder_not_le)
nipkow@15539
   270
done
nipkow@15539
   271
nipkow@15539
   272
lemma ivl_diff[simp]:
nipkow@15539
   273
 "i \<le> n \<Longrightarrow> {i..<m} - {i..<n} = {n..<(m::'a::linorder)}"
nipkow@15539
   274
by(auto)
nipkow@15539
   275
nipkow@15539
   276
nipkow@15539
   277
(* FIXME the last step should work w/o Ball_def, ideally just with
nipkow@15539
   278
   setsum_0 and setsum_cong. Currently the simplifier does not simplify
nipkow@15539
   279
   the premise x:{i..<j} that ball_cong (or a modified version of setsum_0')
nipkow@15539
   280
   generates. FIX simplifier??? *)
paulson@14416
   281
lemma series_zero: 
nipkow@15539
   282
     "(\<forall>m. n \<le> m --> f(m) = 0) ==> f sums (setsum f {0..<n})"
nipkow@15537
   283
apply (simp add: sums_def LIMSEQ_def diff_minus[symmetric], safe)
paulson@14416
   284
apply (rule_tac x = n in exI)
nipkow@15539
   285
apply (clarsimp simp add:setsum_diff[symmetric] setsum_0' Ball_def)
paulson@14416
   286
done
paulson@14416
   287
nipkow@15539
   288
paulson@14416
   289
lemma sums_mult: "x sums x0 ==> (%n. c * x(n)) sums (c * x0)"
nipkow@15536
   290
by (auto simp add: sums_def setsum_mult [symmetric]
paulson@14416
   291
         intro!: LIMSEQ_mult intro: LIMSEQ_const)
paulson@14416
   292
paulson@14416
   293
lemma sums_divide: "x sums x' ==> (%n. x(n)/c) sums (x'/c)"
paulson@14416
   294
by (simp add: real_divide_def sums_mult mult_commute [of _ "inverse c"])
paulson@14416
   295
paulson@14416
   296
lemma sums_diff: "[| x sums x0; y sums y0 |] ==> (%n. x n - y n) sums (x0-y0)"
nipkow@15536
   297
by (auto simp add: sums_def setsum_subtractf intro: LIMSEQ_diff)
paulson@14416
   298
paulson@14416
   299
lemma suminf_mult: "summable f ==> suminf f * c = suminf(%n. f n * c)"
paulson@14416
   300
by (auto intro!: sums_unique sums_mult summable_sums simp add: mult_commute)
paulson@14416
   301
paulson@14416
   302
lemma suminf_mult2: "summable f ==> c * suminf f  = suminf(%n. c * f n)"
paulson@14416
   303
by (auto intro!: sums_unique sums_mult summable_sums)
paulson@14416
   304
paulson@14416
   305
lemma suminf_diff:
paulson@14416
   306
     "[| summable f; summable g |]   
paulson@14416
   307
      ==> suminf f - suminf g  = suminf(%n. f n - g n)"
paulson@14416
   308
by (auto intro!: sums_diff sums_unique summable_sums)
paulson@14416
   309
paulson@14416
   310
lemma sums_minus: "x sums x0 ==> (%n. - x n) sums - x0"
nipkow@15536
   311
by (auto simp add: sums_def intro!: LIMSEQ_minus simp add: setsum_negf)
paulson@14416
   312
paulson@14416
   313
lemma sums_group:
nipkow@15539
   314
     "[|summable f; 0 < k |] ==> (%n. setsum f {n*k..<n*k+k}) sums (suminf f)"
paulson@14416
   315
apply (drule summable_sums)
paulson@14416
   316
apply (auto simp add: sums_def LIMSEQ_def)
paulson@14416
   317
apply (drule_tac x = r in spec, safe)
paulson@14416
   318
apply (rule_tac x = no in exI, safe)
paulson@14416
   319
apply (drule_tac x = "n*k" in spec)
paulson@14416
   320
apply (auto dest!: not_leE)
paulson@14416
   321
apply (drule_tac j = no in less_le_trans, auto)
paulson@14416
   322
done
paulson@14416
   323
paulson@14416
   324
lemma sumr_pos_lt_pair_lemma:
nipkow@15539
   325
  "[|\<forall>d. - f (n + (d + d)) < (f (Suc (n + (d + d))) :: real) |]
nipkow@15539
   326
   ==> setsum f {0..<n+Suc(Suc 0)} \<le> setsum f {0..<Suc(Suc 0) * Suc no + n}"
paulson@15251
   327
apply (induct "no", auto)
paulson@15251
   328
apply (drule_tac x = "Suc no" in spec)
nipkow@15539
   329
apply (simp add: add_ac)
paulson@14416
   330
done
paulson@10751
   331
paulson@10751
   332
paulson@14416
   333
lemma sumr_pos_lt_pair:
paulson@15234
   334
     "[|summable f; 
paulson@15234
   335
        \<forall>d. 0 < (f(n + (Suc(Suc 0) * d))) + f(n + ((Suc(Suc 0) * d) + 1))|]  
nipkow@15539
   336
      ==> setsum f {0..<n} < suminf f"
paulson@14416
   337
apply (drule summable_sums)
paulson@14416
   338
apply (auto simp add: sums_def LIMSEQ_def)
paulson@15234
   339
apply (drule_tac x = "f (n) + f (n + 1)" in spec)
paulson@15085
   340
apply (auto iff: real_0_less_add_iff)
paulson@15085
   341
   --{*legacy proof: not necessarily better!*}
paulson@14416
   342
apply (rule_tac [2] ccontr, drule_tac [2] linorder_not_less [THEN iffD1])
paulson@14416
   343
apply (frule_tac [2] no=no in sumr_pos_lt_pair_lemma) 
paulson@14416
   344
apply (drule_tac x = 0 in spec, simp)
paulson@14416
   345
apply (rotate_tac 1, drule_tac x = "Suc (Suc 0) * (Suc no) + n" in spec)
paulson@14416
   346
apply (safe, simp)
nipkow@15539
   347
apply (subgoal_tac "suminf f + (f (n) + f (n + 1)) \<le>
nipkow@15539
   348
 setsum f {0 ..< Suc (Suc 0) * (Suc no) + n}")
nipkow@15539
   349
apply (rule_tac [2] y = "setsum f {0..<n+ Suc (Suc 0)}" in order_trans)
paulson@14416
   350
prefer 3 apply assumption
nipkow@15539
   351
apply (rule_tac [2] y = "setsum f {0..<n} + (f (n) + f (n + 1))" in order_trans)
paulson@14416
   352
apply simp_all 
nipkow@15539
   353
apply (subgoal_tac "suminf f \<le> setsum f {0..< Suc (Suc 0) * (Suc no) + n}")
paulson@14416
   354
apply (rule_tac [2] y = "suminf f + (f (n) + f (n + 1))" in order_trans)
nipkow@15539
   355
prefer 3 apply simp
paulson@14416
   356
apply (drule_tac [2] x = 0 in spec)
paulson@14416
   357
 prefer 2 apply simp 
nipkow@15539
   358
apply (subgoal_tac "0 \<le> setsum f {0 ..< Suc (Suc 0) * Suc no + n} + - suminf f")
nipkow@15539
   359
apply (simp add: abs_if)
paulson@14416
   360
apply (auto simp add: linorder_not_less [symmetric])
paulson@14416
   361
done
paulson@14416
   362
paulson@15085
   363
text{*A summable series of positive terms has limit that is at least as
paulson@15085
   364
great as any partial sum.*}
paulson@14416
   365
paulson@14416
   366
lemma series_pos_le: 
nipkow@15539
   367
     "[| summable f; \<forall>m \<ge> n. 0 \<le> f(m) |] ==> setsum f {0..<n} \<le> suminf f"
paulson@14416
   368
apply (drule summable_sums)
paulson@14416
   369
apply (simp add: sums_def)
nipkow@15539
   370
apply (cut_tac k = "setsum f {0..<n}" in LIMSEQ_const)
nipkow@15539
   371
apply (erule LIMSEQ_le, blast)
nipkow@15539
   372
apply (rule_tac x = n in exI, clarify)
nipkow@15539
   373
apply (rule setsum_mono2)
nipkow@15539
   374
apply auto
paulson@14416
   375
done
paulson@14416
   376
paulson@14416
   377
lemma series_pos_less:
nipkow@15539
   378
     "[| summable f; \<forall>m \<ge> n. 0 < f(m) |] ==> setsum f {0..<n} < suminf f"
nipkow@15539
   379
apply (rule_tac y = "setsum f {0..<Suc n}" in order_less_le_trans)
paulson@14416
   380
apply (rule_tac [2] series_pos_le, auto)
paulson@14416
   381
apply (drule_tac x = m in spec, auto)
paulson@14416
   382
done
paulson@14416
   383
paulson@15085
   384
text{*Sum of a geometric progression.*}
paulson@14416
   385
nipkow@15539
   386
lemma sumr_geometric:
nipkow@15539
   387
 "x ~= 1 ==> (\<Sum>i=0..<n. x ^ i) = (x ^ n - 1) / (x - 1::real)"
paulson@15251
   388
apply (induct "n", auto)
paulson@14416
   389
apply (rule_tac c1 = "x - 1" in real_mult_right_cancel [THEN iffD1])
nipkow@15539
   390
apply (auto simp add: mult_assoc left_distrib)
paulson@15234
   391
apply (simp add: right_distrib diff_minus mult_commute)
paulson@14416
   392
done
paulson@14416
   393
paulson@14416
   394
lemma geometric_sums: "abs(x) < 1 ==> (%n. x ^ n) sums (1/(1 - x))"
paulson@14416
   395
apply (case_tac "x = 1")
paulson@15234
   396
apply (auto dest!: LIMSEQ_rabs_realpow_zero2 
paulson@15234
   397
        simp add: sumr_geometric sums_def diff_minus add_divide_distrib)
paulson@14416
   398
apply (subgoal_tac "1 / (1 + -x) = 0/ (x - 1) + - 1/ (x - 1) ")
paulson@14416
   399
apply (erule ssubst)
paulson@14416
   400
apply (rule LIMSEQ_add, rule LIMSEQ_divide)
paulson@15234
   401
apply (auto intro: LIMSEQ_const simp add: diff_minus minus_divide_right LIMSEQ_rabs_realpow_zero2)
paulson@14416
   402
done
paulson@14416
   403
paulson@15085
   404
text{*Cauchy-type criterion for convergence of series (c.f. Harrison)*}
paulson@15085
   405
nipkow@15539
   406
lemma summable_convergent_sumr_iff:
nipkow@15539
   407
 "summable f = convergent (%n. setsum f {0..<n})"
paulson@14416
   408
by (simp add: summable_def sums_def convergent_def)
paulson@14416
   409
paulson@14416
   410
lemma summable_Cauchy:
paulson@14416
   411
     "summable f =  
nipkow@15539
   412
      (\<forall>e > 0. \<exists>N. \<forall>m \<ge> N. \<forall>n. abs(setsum f {m..<n}) < e)"
nipkow@15537
   413
apply (auto simp add: summable_convergent_sumr_iff Cauchy_convergent_iff [symmetric] Cauchy_def diff_minus[symmetric])
nipkow@15539
   414
apply (drule_tac [!] spec, auto)
paulson@14416
   415
apply (rule_tac x = M in exI)
paulson@14416
   416
apply (rule_tac [2] x = N in exI, auto)
paulson@14416
   417
apply (cut_tac [!] m = m and n = n in less_linear, auto)
paulson@14416
   418
apply (frule le_less_trans [THEN less_imp_le], assumption)
nipkow@15360
   419
apply (drule_tac x = n in spec, simp)
paulson@14416
   420
apply (drule_tac x = m in spec)
nipkow@15539
   421
apply(simp add: setsum_diff[symmetric])
nipkow@15537
   422
apply(subst abs_minus_commute)
nipkow@15539
   423
apply(simp add: setsum_diff[symmetric])
nipkow@15539
   424
apply(simp add: setsum_diff[symmetric])
paulson@14416
   425
done
nipkow@15539
   426
(* FIXME move ivl_ lemmas out of this theory *)
paulson@14416
   427
paulson@15085
   428
text{*Comparison test*}
paulson@15085
   429
paulson@14416
   430
lemma summable_comparison_test:
nipkow@15360
   431
     "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] ==> summable f"
paulson@14416
   432
apply (auto simp add: summable_Cauchy)
paulson@14416
   433
apply (drule spec, auto)
paulson@14416
   434
apply (rule_tac x = "N + Na" in exI, auto)
paulson@14416
   435
apply (rotate_tac 2)
paulson@14416
   436
apply (drule_tac x = m in spec)
paulson@14416
   437
apply (auto, rotate_tac 2, drule_tac x = n in spec)
nipkow@15539
   438
apply (rule_tac y = "\<Sum>k=m..<n. abs(f k)" in order_le_less_trans)
nipkow@15536
   439
apply (rule setsum_abs)
nipkow@15539
   440
apply (rule_tac y = "setsum g {m..<n}" in order_le_less_trans)
nipkow@15539
   441
apply (auto intro: setsum_mono simp add: abs_interval_iff)
paulson@14416
   442
done
paulson@14416
   443
paulson@14416
   444
lemma summable_rabs_comparison_test:
nipkow@15360
   445
     "[| \<exists>N. \<forall>n \<ge> N. abs(f n) \<le> g n; summable g |] 
paulson@14416
   446
      ==> summable (%k. abs (f k))"
paulson@14416
   447
apply (rule summable_comparison_test)
paulson@14416
   448
apply (auto simp add: abs_idempotent)
paulson@14416
   449
done
paulson@14416
   450
paulson@15085
   451
text{*Limit comparison property for series (c.f. jrh)*}
paulson@15085
   452
paulson@14416
   453
lemma summable_le:
paulson@14416
   454
     "[|\<forall>n. f n \<le> g n; summable f; summable g |] ==> suminf f \<le> suminf g"
paulson@14416
   455
apply (drule summable_sums)+
paulson@14416
   456
apply (auto intro!: LIMSEQ_le simp add: sums_def)
paulson@14416
   457
apply (rule exI)
nipkow@15539
   458
apply (auto intro!: setsum_mono)
paulson@14416
   459
done
paulson@14416
   460
paulson@14416
   461
lemma summable_le2:
paulson@14416
   462
     "[|\<forall>n. abs(f n) \<le> g n; summable g |]  
paulson@14416
   463
      ==> summable f & suminf f \<le> suminf g"
paulson@14416
   464
apply (auto intro: summable_comparison_test intro!: summable_le)
paulson@14416
   465
apply (simp add: abs_le_interval_iff)
paulson@14416
   466
done
paulson@14416
   467
paulson@15085
   468
text{*Absolute convergence imples normal convergence*}
paulson@14416
   469
lemma summable_rabs_cancel: "summable (%n. abs (f n)) ==> summable f"
nipkow@15536
   470
apply (auto simp add: summable_Cauchy)
paulson@14416
   471
apply (drule spec, auto)
paulson@14416
   472
apply (rule_tac x = N in exI, auto)
paulson@14416
   473
apply (drule spec, auto)
nipkow@15539
   474
apply (rule_tac y = "\<Sum>n=m..<n. abs(f n)" in order_le_less_trans)
nipkow@15536
   475
apply (auto)
paulson@14416
   476
done
paulson@14416
   477
paulson@15085
   478
text{*Absolute convergence of series*}
paulson@14416
   479
lemma summable_rabs:
paulson@14416
   480
     "summable (%n. abs (f n)) ==> abs(suminf f) \<le> suminf (%n. abs(f n))"
nipkow@15536
   481
by (auto intro: LIMSEQ_le LIMSEQ_imp_rabs summable_rabs_cancel summable_sumr_LIMSEQ_suminf)
paulson@14416
   482
paulson@14416
   483
paulson@14416
   484
subsection{* The Ratio Test*}
paulson@14416
   485
paulson@14416
   486
lemma rabs_ratiotest_lemma: "[| c \<le> 0; abs x \<le> c * abs y |] ==> x = (0::real)"
paulson@14416
   487
apply (drule order_le_imp_less_or_eq, auto)
paulson@14416
   488
apply (subgoal_tac "0 \<le> c * abs y")
paulson@14416
   489
apply (simp add: zero_le_mult_iff, arith)
paulson@14416
   490
done
paulson@14416
   491
paulson@14416
   492
lemma le_Suc_ex: "(k::nat) \<le> l ==> (\<exists>n. l = k + n)"
paulson@14416
   493
apply (drule le_imp_less_or_eq)
paulson@14416
   494
apply (auto dest: less_imp_Suc_add)
paulson@14416
   495
done
paulson@14416
   496
paulson@14416
   497
lemma le_Suc_ex_iff: "((k::nat) \<le> l) = (\<exists>n. l = k + n)"
paulson@14416
   498
by (auto simp add: le_Suc_ex)
paulson@14416
   499
paulson@14416
   500
(*All this trouble just to get 0<c *)
paulson@14416
   501
lemma ratio_test_lemma2:
nipkow@15360
   502
     "[| \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  
paulson@14416
   503
      ==> 0 < c | summable f"
paulson@14416
   504
apply (simp (no_asm) add: linorder_not_le [symmetric])
paulson@14416
   505
apply (simp add: summable_Cauchy)
paulson@14416
   506
apply (safe, subgoal_tac "\<forall>n. N \<le> n --> f (Suc n) = 0")
paulson@14416
   507
prefer 2 apply (blast intro: rabs_ratiotest_lemma)
paulson@14416
   508
apply (rule_tac x = "Suc N" in exI, clarify)
paulson@14416
   509
apply (drule_tac n=n in Suc_le_imp_diff_ge2, auto) 
paulson@14416
   510
done
paulson@14416
   511
paulson@14416
   512
lemma ratio_test:
nipkow@15360
   513
     "[| c < 1; \<forall>n \<ge> N. abs(f(Suc n)) \<le> c*abs(f n) |]  
paulson@14416
   514
      ==> summable f"
paulson@14416
   515
apply (frule ratio_test_lemma2, auto)
paulson@15234
   516
apply (rule_tac g = "%n. (abs (f N) / (c ^ N))*c ^ n" 
paulson@15234
   517
       in summable_comparison_test)
paulson@14416
   518
apply (rule_tac x = N in exI, safe)
paulson@14416
   519
apply (drule le_Suc_ex_iff [THEN iffD1])
paulson@14416
   520
apply (auto simp add: power_add realpow_not_zero)
nipkow@15539
   521
apply (induct_tac "na", auto)
paulson@14416
   522
apply (rule_tac y = "c*abs (f (N + n))" in order_trans)
paulson@14416
   523
apply (auto intro: mult_right_mono simp add: summable_def)
paulson@14416
   524
apply (simp add: mult_ac)
paulson@15234
   525
apply (rule_tac x = "abs (f N) * (1/ (1 - c)) / (c ^ N)" in exI)
paulson@15234
   526
apply (rule sums_divide) 
paulson@15234
   527
apply (rule sums_mult) 
paulson@15234
   528
apply (auto intro!: geometric_sums)
paulson@14416
   529
done
paulson@14416
   530
paulson@14416
   531
paulson@15085
   532
text{*Differentiation of finite sum*}
paulson@14416
   533
paulson@14416
   534
lemma DERIV_sumr [rule_format (no_asm)]:
paulson@14416
   535
     "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))  
nipkow@15539
   536
      --> DERIV (%x. \<Sum>n=m..<n::nat. f n x) x :> (\<Sum>r=m..<n. f' r x)"
paulson@15251
   537
apply (induct "n")
paulson@14416
   538
apply (auto intro: DERIV_add)
paulson@14416
   539
done
paulson@14416
   540
paulson@14416
   541
ML
paulson@14416
   542
{*
paulson@14416
   543
val sums_def = thm"sums_def";
paulson@14416
   544
val summable_def = thm"summable_def";
paulson@14416
   545
val suminf_def = thm"suminf_def";
paulson@14416
   546
paulson@14416
   547
val sumr_minus_one_realpow_zero = thm "sumr_minus_one_realpow_zero";
paulson@14416
   548
val Suc_le_imp_diff_ge2 = thm "Suc_le_imp_diff_ge2";
paulson@14416
   549
val sumr_one_lb_realpow_zero = thm "sumr_one_lb_realpow_zero";
paulson@14416
   550
val sumr_group = thm "sumr_group";
paulson@14416
   551
val sums_summable = thm "sums_summable";
paulson@14416
   552
val summable_sums = thm "summable_sums";
paulson@14416
   553
val summable_sumr_LIMSEQ_suminf = thm "summable_sumr_LIMSEQ_suminf";
paulson@14416
   554
val sums_unique = thm "sums_unique";
paulson@14416
   555
val series_zero = thm "series_zero";
paulson@14416
   556
val sums_mult = thm "sums_mult";
paulson@14416
   557
val sums_divide = thm "sums_divide";
paulson@14416
   558
val sums_diff = thm "sums_diff";
paulson@14416
   559
val suminf_mult = thm "suminf_mult";
paulson@14416
   560
val suminf_mult2 = thm "suminf_mult2";
paulson@14416
   561
val suminf_diff = thm "suminf_diff";
paulson@14416
   562
val sums_minus = thm "sums_minus";
paulson@14416
   563
val sums_group = thm "sums_group";
paulson@14416
   564
val sumr_pos_lt_pair_lemma = thm "sumr_pos_lt_pair_lemma";
paulson@14416
   565
val sumr_pos_lt_pair = thm "sumr_pos_lt_pair";
paulson@14416
   566
val series_pos_le = thm "series_pos_le";
paulson@14416
   567
val series_pos_less = thm "series_pos_less";
paulson@14416
   568
val sumr_geometric = thm "sumr_geometric";
paulson@14416
   569
val geometric_sums = thm "geometric_sums";
paulson@14416
   570
val summable_convergent_sumr_iff = thm "summable_convergent_sumr_iff";
paulson@14416
   571
val summable_Cauchy = thm "summable_Cauchy";
paulson@14416
   572
val summable_comparison_test = thm "summable_comparison_test";
paulson@14416
   573
val summable_rabs_comparison_test = thm "summable_rabs_comparison_test";
paulson@14416
   574
val summable_le = thm "summable_le";
paulson@14416
   575
val summable_le2 = thm "summable_le2";
paulson@14416
   576
val summable_rabs_cancel = thm "summable_rabs_cancel";
paulson@14416
   577
val summable_rabs = thm "summable_rabs";
paulson@14416
   578
val rabs_ratiotest_lemma = thm "rabs_ratiotest_lemma";
paulson@14416
   579
val le_Suc_ex = thm "le_Suc_ex";
paulson@14416
   580
val le_Suc_ex_iff = thm "le_Suc_ex_iff";
paulson@14416
   581
val ratio_test_lemma2 = thm "ratio_test_lemma2";
paulson@14416
   582
val ratio_test = thm "ratio_test";
paulson@14416
   583
val DERIV_sumr = thm "DERIV_sumr";
paulson@14416
   584
*}
paulson@14416
   585
paulson@14416
   586
end