src/HOL/Library/BigO.thy
author wenzelm
Wed Jul 13 14:28:15 2016 +0200 (2016-07-13)
changeset 63473 151bb79536a7
parent 63462 c1fe30f2bc32
child 63485 ea8dfb0ed10e
permissions -rw-r--r--
misc tuning and modernization;
wenzelm@16932
     1
(*  Title:      HOL/Library/BigO.thy
avigad@16908
     2
    Authors:    Jeremy Avigad and Kevin Donnelly
avigad@16908
     3
*)
avigad@16908
     4
wenzelm@60500
     5
section \<open>Big O notation\<close>
avigad@16908
     6
avigad@16908
     7
theory BigO
wenzelm@63462
     8
  imports Complex_Main Function_Algebras Set_Algebras
avigad@16908
     9
begin
avigad@16908
    10
wenzelm@60500
    11
text \<open>
wenzelm@63473
    12
  This library is designed to support asymptotic ``big O'' calculations,
wenzelm@63473
    13
  i.e.~reasoning with expressions of the form \<open>f = O(g)\<close> and \<open>f = g + O(h)\<close>.
wenzelm@63473
    14
  An earlier version of this library is described in detail in @{cite
wenzelm@63473
    15
  "Avigad-Donnelly"}.
wenzelm@63473
    16
wenzelm@63473
    17
  The main changes in this version are as follows:
wenzelm@17199
    18
wenzelm@63473
    19
    \<^item> We have eliminated the \<open>O\<close> operator on sets. (Most uses of this seem
wenzelm@63473
    20
      to be inessential.)
wenzelm@63473
    21
    \<^item> We no longer use \<open>+\<close> as output syntax for \<open>+o\<close>
wenzelm@63473
    22
    \<^item> Lemmas involving \<open>sumr\<close> have been replaced by more general lemmas
wenzelm@63473
    23
      involving `\<open>setsum\<close>.
wenzelm@63473
    24
    \<^item> The library has been expanded, with e.g.~support for expressions of
wenzelm@63473
    25
      the form \<open>f < g + O(h)\<close>.
wenzelm@17199
    26
wenzelm@63473
    27
  Note also since the Big O library includes rules that demonstrate set
wenzelm@63473
    28
  inclusion, to use the automated reasoners effectively with the library one
wenzelm@63473
    29
  should redeclare the theorem \<open>subsetI\<close> as an intro rule, rather than as an
wenzelm@63473
    30
  \<open>intro!\<close> rule, for example, using \<^theory_text>\<open>declare subsetI [del, intro]\<close>.
wenzelm@60500
    31
\<close>
avigad@16908
    32
wenzelm@63473
    33
wenzelm@60500
    34
subsection \<open>Definitions\<close>
avigad@16908
    35
wenzelm@55821
    36
definition bigo :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(1O'(_'))")
wenzelm@61945
    37
  where "O(f:: 'a \<Rightarrow> 'b) = {h. \<exists>c. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>}"
avigad@16908
    38
wenzelm@55821
    39
lemma bigo_pos_const:
wenzelm@61945
    40
  "(\<exists>c::'a::linordered_idom. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>) \<longleftrightarrow>
wenzelm@61945
    41
    (\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
avigad@16908
    42
  apply auto
avigad@16908
    43
  apply (case_tac "c = 0")
wenzelm@63473
    44
   apply simp
wenzelm@63473
    45
   apply (rule_tac x = "1" in exI)
wenzelm@63473
    46
   apply simp
wenzelm@61945
    47
  apply (rule_tac x = "\<bar>c\<bar>" in exI)
avigad@16908
    48
  apply auto
wenzelm@61945
    49
  apply (subgoal_tac "c * \<bar>f x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>")
wenzelm@63473
    50
   apply (erule_tac x = x in allE)
wenzelm@63473
    51
   apply force
avigad@16908
    52
  apply (rule mult_right_mono)
wenzelm@63473
    53
   apply (rule abs_ge_self)
avigad@16908
    54
  apply (rule abs_ge_zero)
wenzelm@22665
    55
  done
avigad@16908
    56
wenzelm@61945
    57
lemma bigo_alt_def: "O(f) = {h. \<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)}"
wenzelm@22665
    58
  by (auto simp add: bigo_def bigo_pos_const)
avigad@16908
    59
wenzelm@55821
    60
lemma bigo_elt_subset [intro]: "f \<in> O(g) \<Longrightarrow> O(f) \<le> O(g)"
avigad@16908
    61
  apply (auto simp add: bigo_alt_def)
avigad@16908
    62
  apply (rule_tac x = "ca * c" in exI)
avigad@16908
    63
  apply (rule conjI)
wenzelm@63473
    64
   apply simp
avigad@16908
    65
  apply (rule allI)
avigad@16908
    66
  apply (drule_tac x = "xa" in spec)+
wenzelm@61945
    67
  apply (subgoal_tac "ca * \<bar>f xa\<bar> \<le> ca * (c * \<bar>g xa\<bar>)")
wenzelm@63473
    68
   apply (erule order_trans)
wenzelm@63473
    69
   apply (simp add: ac_simps)
avigad@16908
    70
  apply (rule mult_left_mono, assumption)
avigad@16908
    71
  apply (rule order_less_imp_le, assumption)
wenzelm@22665
    72
  done
avigad@16908
    73
wenzelm@55821
    74
lemma bigo_refl [intro]: "f \<in> O(f)"
wenzelm@63473
    75
  apply (auto simp add: bigo_def)
wenzelm@63473
    76
  apply (rule_tac x = 1 in exI)
avigad@16908
    77
  apply simp
wenzelm@22665
    78
  done
avigad@16908
    79
wenzelm@55821
    80
lemma bigo_zero: "0 \<in> O(g)"
avigad@16908
    81
  apply (auto simp add: bigo_def func_zero)
avigad@16908
    82
  apply (rule_tac x = 0 in exI)
avigad@16908
    83
  apply auto
wenzelm@22665
    84
  done
avigad@16908
    85
wenzelm@55821
    86
lemma bigo_zero2: "O(\<lambda>x. 0) = {\<lambda>x. 0}"
wenzelm@55821
    87
  by (auto simp add: bigo_def)
avigad@16908
    88
wenzelm@55821
    89
lemma bigo_plus_self_subset [intro]: "O(f) + O(f) \<subseteq> O(f)"
berghofe@26814
    90
  apply (auto simp add: bigo_alt_def set_plus_def)
avigad@16908
    91
  apply (rule_tac x = "c + ca" in exI)
avigad@16908
    92
  apply auto
nipkow@23477
    93
  apply (simp add: ring_distribs func_plus)
avigad@16908
    94
  apply (rule order_trans)
wenzelm@63473
    95
   apply (rule abs_triangle_ineq)
avigad@16908
    96
  apply (rule add_mono)
wenzelm@63473
    97
   apply force
avigad@16908
    98
  apply force
wenzelm@55821
    99
  done
avigad@16908
   100
krauss@47445
   101
lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
avigad@16908
   102
  apply (rule equalityI)
wenzelm@63473
   103
   apply (rule bigo_plus_self_subset)
wenzelm@55821
   104
  apply (rule set_zero_plus2)
avigad@16908
   105
  apply (rule bigo_zero)
wenzelm@22665
   106
  done
avigad@16908
   107
wenzelm@55821
   108
lemma bigo_plus_subset [intro]: "O(f + g) \<subseteq> O(f) + O(g)"
avigad@16908
   109
  apply (rule subsetI)
berghofe@26814
   110
  apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
avigad@16908
   111
  apply (subst bigo_pos_const [symmetric])+
wenzelm@61945
   112
  apply (rule_tac x = "\<lambda>n. if \<bar>g n\<bar> \<le> \<bar>f n\<bar> then x n else 0" in exI)
avigad@16908
   113
  apply (rule conjI)
wenzelm@63473
   114
   apply (rule_tac x = "c + c" in exI)
wenzelm@63473
   115
   apply (clarsimp)
wenzelm@63473
   116
   apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>f xa\<bar>")
wenzelm@63473
   117
    apply (erule_tac x = xa in allE)
wenzelm@63473
   118
    apply (erule order_trans)
wenzelm@63473
   119
    apply (simp)
wenzelm@63473
   120
   apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
wenzelm@63473
   121
    apply (erule order_trans)
wenzelm@63473
   122
    apply (simp add: ring_distribs)
wenzelm@63473
   123
   apply (rule mult_left_mono)
wenzelm@63473
   124
    apply (simp add: abs_triangle_ineq)
wenzelm@63473
   125
   apply (simp add: order_less_le)
wenzelm@61945
   126
  apply (rule_tac x = "\<lambda>n. if \<bar>f n\<bar> < \<bar>g n\<bar> then x n else 0" in exI)
avigad@16908
   127
  apply (rule conjI)
wenzelm@63473
   128
   apply (rule_tac x = "c + c" in exI)
wenzelm@63473
   129
   apply auto
wenzelm@61945
   130
  apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>g xa\<bar>")
wenzelm@63473
   131
   apply (erule_tac x = xa in allE)
wenzelm@63473
   132
   apply (erule order_trans)
wenzelm@63473
   133
   apply simp
wenzelm@61945
   134
  apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
wenzelm@63473
   135
   apply (erule order_trans)
wenzelm@63473
   136
   apply (simp add: ring_distribs)
avigad@16908
   137
  apply (rule mult_left_mono)
wenzelm@63473
   138
   apply (rule abs_triangle_ineq)
avigad@16908
   139
  apply (simp add: order_less_le)
wenzelm@22665
   140
  done
avigad@16908
   141
wenzelm@55821
   142
lemma bigo_plus_subset2 [intro]: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
wenzelm@55821
   143
  apply (subgoal_tac "A + B \<subseteq> O(f) + O(f)")
wenzelm@63473
   144
   apply (erule order_trans)
wenzelm@63473
   145
   apply simp
avigad@16908
   146
  apply (auto del: subsetI simp del: bigo_plus_idemp)
wenzelm@22665
   147
  done
avigad@16908
   148
wenzelm@55821
   149
lemma bigo_plus_eq: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> O(f + g) = O(f) + O(g)"
avigad@16908
   150
  apply (rule equalityI)
wenzelm@63473
   151
   apply (rule bigo_plus_subset)
berghofe@26814
   152
  apply (simp add: bigo_alt_def set_plus_def func_plus)
avigad@16908
   153
  apply clarify
avigad@16908
   154
  apply (rule_tac x = "max c ca" in exI)
avigad@16908
   155
  apply (rule conjI)
wenzelm@63473
   156
   apply (subgoal_tac "c \<le> max c ca")
wenzelm@63473
   157
    apply (erule order_less_le_trans)
wenzelm@63473
   158
    apply assumption
wenzelm@63473
   159
   apply (rule max.cobounded1)
avigad@16908
   160
  apply clarify
avigad@16908
   161
  apply (drule_tac x = "xa" in spec)+
wenzelm@55821
   162
  apply (subgoal_tac "0 \<le> f xa + g xa")
wenzelm@63473
   163
   apply (simp add: ring_distribs)
wenzelm@63473
   164
   apply (subgoal_tac "\<bar>a xa + b xa\<bar> \<le> \<bar>a xa\<bar> + \<bar>b xa\<bar>")
wenzelm@63473
   165
    apply (subgoal_tac "\<bar>a xa\<bar> + \<bar>b xa\<bar> \<le> max c ca * f xa + max c ca * g xa")
wenzelm@63473
   166
     apply force
wenzelm@63473
   167
    apply (rule add_mono)
wenzelm@63473
   168
     apply (subgoal_tac "c * f xa \<le> max c ca * f xa")
wenzelm@63473
   169
      apply force
wenzelm@63473
   170
     apply (rule mult_right_mono)
wenzelm@63473
   171
      apply (rule max.cobounded1)
wenzelm@63473
   172
     apply assumption
wenzelm@63473
   173
    apply (subgoal_tac "ca * g xa \<le> max c ca * g xa")
wenzelm@63473
   174
     apply force
wenzelm@63473
   175
    apply (rule mult_right_mono)
wenzelm@63473
   176
     apply (rule max.cobounded2)
wenzelm@63473
   177
    apply assumption
wenzelm@63473
   178
   apply (rule abs_triangle_ineq)
avigad@16908
   179
  apply (rule add_nonneg_nonneg)
wenzelm@63473
   180
   apply assumption+
wenzelm@22665
   181
  done
avigad@16908
   182
wenzelm@55821
   183
lemma bigo_bounded_alt: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> c * g x \<Longrightarrow> f \<in> O(g)"
avigad@16908
   184
  apply (auto simp add: bigo_def)
wenzelm@61945
   185
  apply (rule_tac x = "\<bar>c\<bar>" in exI)
avigad@16908
   186
  apply auto
avigad@16908
   187
  apply (drule_tac x = x in spec)+
avigad@16908
   188
  apply (simp add: abs_mult [symmetric])
wenzelm@22665
   189
  done
avigad@16908
   190
wenzelm@55821
   191
lemma bigo_bounded: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> g x \<Longrightarrow> f \<in> O(g)"
avigad@16908
   192
  apply (erule bigo_bounded_alt [of f 1 g])
avigad@16908
   193
  apply simp
wenzelm@22665
   194
  done
avigad@16908
   195
wenzelm@55821
   196
lemma bigo_bounded2: "\<forall>x. lb x \<le> f x \<Longrightarrow> \<forall>x. f x \<le> lb x + g x \<Longrightarrow> f \<in> lb +o O(g)"
avigad@16908
   197
  apply (rule set_minus_imp_plus)
avigad@16908
   198
  apply (rule bigo_bounded)
wenzelm@63473
   199
   apply (auto simp add: fun_Compl_def func_plus)
avigad@16908
   200
  apply (drule_tac x = x in spec)+
avigad@16908
   201
  apply force
wenzelm@22665
   202
  done
avigad@16908
   203
wenzelm@61945
   204
lemma bigo_abs: "(\<lambda>x. \<bar>f x\<bar>) =o O(f)"
avigad@16908
   205
  apply (unfold bigo_def)
avigad@16908
   206
  apply auto
avigad@16908
   207
  apply (rule_tac x = 1 in exI)
avigad@16908
   208
  apply auto
wenzelm@22665
   209
  done
avigad@16908
   210
wenzelm@61945
   211
lemma bigo_abs2: "f =o O(\<lambda>x. \<bar>f x\<bar>)"
avigad@16908
   212
  apply (unfold bigo_def)
avigad@16908
   213
  apply auto
avigad@16908
   214
  apply (rule_tac x = 1 in exI)
avigad@16908
   215
  apply auto
wenzelm@22665
   216
  done
avigad@16908
   217
wenzelm@61945
   218
lemma bigo_abs3: "O(f) = O(\<lambda>x. \<bar>f x\<bar>)"
avigad@16908
   219
  apply (rule equalityI)
wenzelm@63473
   220
   apply (rule bigo_elt_subset)
wenzelm@63473
   221
   apply (rule bigo_abs2)
avigad@16908
   222
  apply (rule bigo_elt_subset)
avigad@16908
   223
  apply (rule bigo_abs)
wenzelm@22665
   224
  done
avigad@16908
   225
wenzelm@61945
   226
lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o (\<lambda>x. \<bar>g x\<bar>) +o O(h)"
avigad@16908
   227
  apply (drule set_plus_imp_minus)
avigad@16908
   228
  apply (rule set_minus_imp_plus)
berghofe@26814
   229
  apply (subst fun_diff_def)
avigad@16908
   230
proof -
wenzelm@63473
   231
  assume *: "f - g \<in> O(h)"
wenzelm@61945
   232
  have "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) =o O(\<lambda>x. \<bar>\<bar>f x\<bar> - \<bar>g x\<bar>\<bar>)"
avigad@16908
   233
    by (rule bigo_abs2)
wenzelm@61945
   234
  also have "\<dots> \<subseteq> O(\<lambda>x. \<bar>f x - g x\<bar>)"
avigad@16908
   235
    apply (rule bigo_elt_subset)
avigad@16908
   236
    apply (rule bigo_bounded)
wenzelm@63473
   237
     apply force
avigad@16908
   238
    apply (rule allI)
avigad@16908
   239
    apply (rule abs_triangle_ineq3)
avigad@16908
   240
    done
wenzelm@55821
   241
  also have "\<dots> \<subseteq> O(f - g)"
avigad@16908
   242
    apply (rule bigo_elt_subset)
berghofe@26814
   243
    apply (subst fun_diff_def)
avigad@16908
   244
    apply (rule bigo_abs)
avigad@16908
   245
    done
wenzelm@63473
   246
  also from * have "\<dots> \<subseteq> O(h)"
avigad@16908
   247
    by (rule bigo_elt_subset)
wenzelm@61945
   248
  finally show "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) \<in> O(h)".
avigad@16908
   249
qed
avigad@16908
   250
wenzelm@61945
   251
lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o O(g)"
wenzelm@63473
   252
  by (auto simp: bigo_def)
avigad@16908
   253
wenzelm@63473
   254
lemma bigo_elt_subset2 [intro]:
wenzelm@63473
   255
  assumes *: "f \<in> g +o O(h)"
wenzelm@63473
   256
  shows "O(f) \<subseteq> O(g) + O(h)"
avigad@16908
   257
proof -
wenzelm@63473
   258
  note *
wenzelm@63473
   259
  also have "g +o O(h) \<subseteq> O(g) + O(h)"
avigad@16908
   260
    by (auto del: subsetI)
wenzelm@61945
   261
  also have "\<dots> = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)"
wenzelm@63473
   262
    by (subst bigo_abs3 [symmetric])+ (rule refl)
wenzelm@61945
   263
  also have "\<dots> = O((\<lambda>x. \<bar>g x\<bar>) + (\<lambda>x. \<bar>h x\<bar>))"
wenzelm@55821
   264
    by (rule bigo_plus_eq [symmetric]) auto
wenzelm@55821
   265
  finally have "f \<in> \<dots>" .
wenzelm@55821
   266
  then have "O(f) \<subseteq> \<dots>"
avigad@16908
   267
    by (elim bigo_elt_subset)
wenzelm@61945
   268
  also have "\<dots> = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)"
avigad@16908
   269
    by (rule bigo_plus_eq, auto)
avigad@16908
   270
  finally show ?thesis
avigad@16908
   271
    by (simp add: bigo_abs3 [symmetric])
avigad@16908
   272
qed
avigad@16908
   273
wenzelm@55821
   274
lemma bigo_mult [intro]: "O(f)*O(g) \<subseteq> O(f * g)"
avigad@16908
   275
  apply (rule subsetI)
avigad@16908
   276
  apply (subst bigo_def)
berghofe@26814
   277
  apply (auto simp add: bigo_alt_def set_times_def func_times)
avigad@16908
   278
  apply (rule_tac x = "c * ca" in exI)
wenzelm@55821
   279
  apply (rule allI)
wenzelm@55821
   280
  apply (erule_tac x = x in allE)+
wenzelm@61945
   281
  apply (subgoal_tac "c * ca * \<bar>f x * g x\<bar> = (c * \<bar>f x\<bar>) * (ca * \<bar>g x\<bar>)")
wenzelm@63473
   282
   apply (erule ssubst)
wenzelm@63473
   283
   apply (subst abs_mult)
wenzelm@63473
   284
   apply (rule mult_mono)
wenzelm@63473
   285
      apply assumption+
wenzelm@63473
   286
    apply auto
haftmann@57514
   287
  apply (simp add: ac_simps abs_mult)
wenzelm@22665
   288
  done
avigad@16908
   289
wenzelm@55821
   290
lemma bigo_mult2 [intro]: "f *o O(g) \<subseteq> O(f * g)"
avigad@16908
   291
  apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
avigad@16908
   292
  apply (rule_tac x = c in exI)
avigad@16908
   293
  apply auto
avigad@16908
   294
  apply (drule_tac x = x in spec)
wenzelm@61945
   295
  apply (subgoal_tac "\<bar>f x\<bar> * \<bar>b x\<bar> \<le> \<bar>f x\<bar> * (c * \<bar>g x\<bar>)")
wenzelm@63473
   296
   apply (force simp add: ac_simps)
avigad@16908
   297
  apply (rule mult_left_mono, assumption)
avigad@16908
   298
  apply (rule abs_ge_zero)
wenzelm@22665
   299
  done
avigad@16908
   300
wenzelm@55821
   301
lemma bigo_mult3: "f \<in> O(h) \<Longrightarrow> g \<in> O(j) \<Longrightarrow> f * g \<in> O(h * j)"
avigad@16908
   302
  apply (rule subsetD)
wenzelm@63473
   303
   apply (rule bigo_mult)
avigad@16908
   304
  apply (erule set_times_intro, assumption)
wenzelm@22665
   305
  done
avigad@16908
   306
wenzelm@55821
   307
lemma bigo_mult4 [intro]: "f \<in> k +o O(h) \<Longrightarrow> g * f \<in> (g * k) +o O(g * h)"
avigad@16908
   308
  apply (drule set_plus_imp_minus)
avigad@16908
   309
  apply (rule set_minus_imp_plus)
avigad@16908
   310
  apply (drule bigo_mult3 [where g = g and j = g])
wenzelm@63473
   311
   apply (auto simp add: algebra_simps)
wenzelm@22665
   312
  done
avigad@16908
   313
wenzelm@41528
   314
lemma bigo_mult5:
wenzelm@55821
   315
  fixes f :: "'a \<Rightarrow> 'b::linordered_field"
wenzelm@55821
   316
  assumes "\<forall>x. f x \<noteq> 0"
wenzelm@55821
   317
  shows "O(f * g) \<subseteq> f *o O(g)"
wenzelm@41528
   318
proof
wenzelm@41528
   319
  fix h
wenzelm@55821
   320
  assume "h \<in> O(f * g)"
wenzelm@55821
   321
  then have "(\<lambda>x. 1 / (f x)) * h \<in> (\<lambda>x. 1 / f x) *o O(f * g)"
wenzelm@41528
   322
    by auto
wenzelm@55821
   323
  also have "\<dots> \<subseteq> O((\<lambda>x. 1 / f x) * (f * g))"
wenzelm@41528
   324
    by (rule bigo_mult2)
wenzelm@55821
   325
  also have "(\<lambda>x. 1 / f x) * (f * g) = g"
wenzelm@55821
   326
    apply (simp add: func_times)
wenzelm@41528
   327
    apply (rule ext)
haftmann@57514
   328
    apply (simp add: assms nonzero_divide_eq_eq ac_simps)
wenzelm@41528
   329
    done
wenzelm@55821
   330
  finally have "(\<lambda>x. (1::'b) / f x) * h \<in> O(g)" .
wenzelm@55821
   331
  then have "f * ((\<lambda>x. (1::'b) / f x) * h) \<in> f *o O(g)"
wenzelm@41528
   332
    by auto
wenzelm@55821
   333
  also have "f * ((\<lambda>x. (1::'b) / f x) * h) = h"
wenzelm@55821
   334
    apply (simp add: func_times)
wenzelm@41528
   335
    apply (rule ext)
haftmann@57514
   336
    apply (simp add: assms nonzero_divide_eq_eq ac_simps)
wenzelm@41528
   337
    done
wenzelm@55821
   338
  finally show "h \<in> f *o O(g)" .
avigad@16908
   339
qed
avigad@16908
   340
wenzelm@63473
   341
lemma bigo_mult6: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = f *o O(g)"
wenzelm@63473
   342
  for f :: "'a \<Rightarrow> 'b::linordered_field"
avigad@16908
   343
  apply (rule equalityI)
wenzelm@63473
   344
   apply (erule bigo_mult5)
avigad@16908
   345
  apply (rule bigo_mult2)
wenzelm@22665
   346
  done
avigad@16908
   347
wenzelm@63473
   348
lemma bigo_mult7: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<subseteq> O(f) * O(g)"
wenzelm@63473
   349
  for f :: "'a \<Rightarrow> 'b::linordered_field"
avigad@16908
   350
  apply (subst bigo_mult6)
wenzelm@63473
   351
   apply assumption
avigad@16908
   352
  apply (rule set_times_mono3)
avigad@16908
   353
  apply (rule bigo_refl)
wenzelm@22665
   354
  done
avigad@16908
   355
wenzelm@63473
   356
lemma bigo_mult8: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f) * O(g)"
wenzelm@63473
   357
  for f :: "'a \<Rightarrow> 'b::linordered_field"
avigad@16908
   358
  apply (rule equalityI)
wenzelm@63473
   359
   apply (erule bigo_mult7)
avigad@16908
   360
  apply (rule bigo_mult)
wenzelm@22665
   361
  done
avigad@16908
   362
wenzelm@55821
   363
lemma bigo_minus [intro]: "f \<in> O(g) \<Longrightarrow> - f \<in> O(g)"
berghofe@26814
   364
  by (auto simp add: bigo_def fun_Compl_def)
avigad@16908
   365
wenzelm@55821
   366
lemma bigo_minus2: "f \<in> g +o O(h) \<Longrightarrow> - f \<in> -g +o O(h)"
avigad@16908
   367
  apply (rule set_minus_imp_plus)
avigad@16908
   368
  apply (drule set_plus_imp_minus)
avigad@16908
   369
  apply (drule bigo_minus)
haftmann@54230
   370
  apply simp
wenzelm@22665
   371
  done
avigad@16908
   372
wenzelm@55821
   373
lemma bigo_minus3: "O(- f) = O(f)"
wenzelm@41528
   374
  by (auto simp add: bigo_def fun_Compl_def)
avigad@16908
   375
wenzelm@63473
   376
lemma bigo_plus_absorb_lemma1:
wenzelm@63473
   377
  assumes *: "f \<in> O(g)"
wenzelm@63473
   378
  shows "f +o O(g) \<subseteq> O(g)"
avigad@16908
   379
proof -
wenzelm@63473
   380
  have "f \<in> O(f)" by auto
wenzelm@63473
   381
  then have "f +o O(g) \<subseteq> O(f) + O(g)"
wenzelm@63473
   382
    by (auto del: subsetI)
wenzelm@63473
   383
  also have "\<dots> \<subseteq> O(g) + O(g)"
avigad@16908
   384
  proof -
wenzelm@63473
   385
    from * have "O(f) \<subseteq> O(g)"
avigad@16908
   386
      by (auto del: subsetI)
wenzelm@63473
   387
    then show ?thesis
wenzelm@63473
   388
      by (auto del: subsetI)
avigad@16908
   389
  qed
wenzelm@63473
   390
  also have "\<dots> \<subseteq> O(g)" by simp
wenzelm@63473
   391
  finally show ?thesis .
avigad@16908
   392
qed
avigad@16908
   393
wenzelm@63473
   394
lemma bigo_plus_absorb_lemma2:
wenzelm@63473
   395
  assumes *: "f \<in> O(g)"
wenzelm@63473
   396
  shows "O(g) \<subseteq> f +o O(g)"
avigad@16908
   397
proof -
wenzelm@63473
   398
  from * have "- f \<in> O(g)"
wenzelm@63473
   399
    by auto
wenzelm@63473
   400
  then have "- f +o O(g) \<subseteq> O(g)"
wenzelm@63473
   401
    by (elim bigo_plus_absorb_lemma1)
wenzelm@63473
   402
  then have "f +o (- f +o O(g)) \<subseteq> f +o O(g)"
wenzelm@63473
   403
    by auto
wenzelm@63473
   404
  also have "f +o (- f +o O(g)) = O(g)"
wenzelm@63473
   405
    by (simp add: set_plus_rearranges)
wenzelm@63473
   406
  finally show ?thesis .
avigad@16908
   407
qed
avigad@16908
   408
wenzelm@55821
   409
lemma bigo_plus_absorb [simp]: "f \<in> O(g) \<Longrightarrow> f +o O(g) = O(g)"
avigad@16908
   410
  apply (rule equalityI)
wenzelm@63473
   411
   apply (erule bigo_plus_absorb_lemma1)
avigad@16908
   412
  apply (erule bigo_plus_absorb_lemma2)
wenzelm@22665
   413
  done
avigad@16908
   414
wenzelm@55821
   415
lemma bigo_plus_absorb2 [intro]: "f \<in> O(g) \<Longrightarrow> A \<subseteq> O(g) \<Longrightarrow> f +o A \<subseteq> O(g)"
wenzelm@55821
   416
  apply (subgoal_tac "f +o A \<subseteq> f +o O(g)")
wenzelm@63473
   417
   apply force+
wenzelm@22665
   418
  done
avigad@16908
   419
wenzelm@55821
   420
lemma bigo_add_commute_imp: "f \<in> g +o O(h) \<Longrightarrow> g \<in> f +o O(h)"
avigad@16908
   421
  apply (subst set_minus_plus [symmetric])
avigad@16908
   422
  apply (subgoal_tac "g - f = - (f - g)")
wenzelm@63473
   423
   apply (erule ssubst)
wenzelm@63473
   424
   apply (rule bigo_minus)
wenzelm@63473
   425
   apply (subst set_minus_plus)
wenzelm@63473
   426
   apply assumption
haftmann@57514
   427
  apply (simp add: ac_simps)
wenzelm@22665
   428
  done
avigad@16908
   429
wenzelm@55821
   430
lemma bigo_add_commute: "f \<in> g +o O(h) \<longleftrightarrow> g \<in> f +o O(h)"
avigad@16908
   431
  apply (rule iffI)
wenzelm@63473
   432
   apply (erule bigo_add_commute_imp)+
wenzelm@22665
   433
  done
avigad@16908
   434
wenzelm@55821
   435
lemma bigo_const1: "(\<lambda>x. c) \<in> O(\<lambda>x. 1)"
haftmann@57514
   436
  by (auto simp add: bigo_def ac_simps)
avigad@16908
   437
wenzelm@55821
   438
lemma bigo_const2 [intro]: "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)"
avigad@16908
   439
  apply (rule bigo_elt_subset)
avigad@16908
   440
  apply (rule bigo_const1)
wenzelm@22665
   441
  done
avigad@16908
   442
wenzelm@63473
   443
lemma bigo_const3: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)"
wenzelm@63473
   444
  for c :: "'a::linordered_field"
avigad@16908
   445
  apply (simp add: bigo_def)
wenzelm@61945
   446
  apply (rule_tac x = "\<bar>inverse c\<bar>" in exI)
avigad@16908
   447
  apply (simp add: abs_mult [symmetric])
wenzelm@22665
   448
  done
avigad@16908
   449
wenzelm@63473
   450
lemma bigo_const4: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)"
wenzelm@63473
   451
  for c :: "'a::linordered_field"
wenzelm@55821
   452
  apply (rule bigo_elt_subset)
wenzelm@55821
   453
  apply (rule bigo_const3)
wenzelm@55821
   454
  apply assumption
wenzelm@55821
   455
  done
avigad@16908
   456
wenzelm@63473
   457
lemma bigo_const [simp]: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)"
wenzelm@63473
   458
  for c :: "'a::linordered_field"
wenzelm@55821
   459
  apply (rule equalityI)
wenzelm@63473
   460
   apply (rule bigo_const2)
wenzelm@55821
   461
  apply (rule bigo_const4)
wenzelm@55821
   462
  apply assumption
wenzelm@55821
   463
  done
avigad@16908
   464
wenzelm@55821
   465
lemma bigo_const_mult1: "(\<lambda>x. c * f x) \<in> O(f)"
avigad@16908
   466
  apply (simp add: bigo_def)
wenzelm@61945
   467
  apply (rule_tac x = "\<bar>c\<bar>" in exI)
avigad@16908
   468
  apply (auto simp add: abs_mult [symmetric])
wenzelm@22665
   469
  done
avigad@16908
   470
wenzelm@55821
   471
lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<subseteq> O(f)"
wenzelm@55821
   472
  apply (rule bigo_elt_subset)
wenzelm@55821
   473
  apply (rule bigo_const_mult1)
wenzelm@55821
   474
  done
avigad@16908
   475
wenzelm@63473
   476
lemma bigo_const_mult3: "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
wenzelm@63473
   477
  for c :: "'a::linordered_field"
avigad@16908
   478
  apply (simp add: bigo_def)
wenzelm@61945
   479
  apply (rule_tac x = "\<bar>inverse c\<bar>" in exI)
haftmann@59867
   480
  apply (simp add: abs_mult mult.assoc [symmetric])
wenzelm@22665
   481
  done
avigad@16908
   482
wenzelm@63473
   483
lemma bigo_const_mult4: "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)"
wenzelm@63473
   484
  for c :: "'a::linordered_field"
wenzelm@55821
   485
  apply (rule bigo_elt_subset)
wenzelm@55821
   486
  apply (rule bigo_const_mult3)
wenzelm@55821
   487
  apply assumption
wenzelm@55821
   488
  done
avigad@16908
   489
wenzelm@63473
   490
lemma bigo_const_mult [simp]: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)"
wenzelm@63473
   491
  for c :: "'a::linordered_field"
wenzelm@55821
   492
  apply (rule equalityI)
wenzelm@63473
   493
   apply (rule bigo_const_mult2)
wenzelm@55821
   494
  apply (erule bigo_const_mult4)
wenzelm@55821
   495
  done
avigad@16908
   496
wenzelm@63473
   497
lemma bigo_const_mult5 [simp]: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
wenzelm@63473
   498
  for c :: "'a::linordered_field"
avigad@16908
   499
  apply (auto del: subsetI)
wenzelm@63473
   500
   apply (rule order_trans)
wenzelm@63473
   501
    apply (rule bigo_mult2)
wenzelm@63473
   502
   apply (simp add: func_times)
wenzelm@41528
   503
  apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
wenzelm@55821
   504
  apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
haftmann@57512
   505
  apply (simp add: mult.assoc [symmetric] abs_mult)
wenzelm@61945
   506
  apply (rule_tac x = "\<bar>inverse c\<bar> * ca" in exI)
haftmann@59867
   507
  apply auto
wenzelm@22665
   508
  done
avigad@16908
   509
wenzelm@55821
   510
lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) \<subseteq> O(f)"
wenzelm@41528
   511
  apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
wenzelm@61945
   512
  apply (rule_tac x = "ca * \<bar>c\<bar>" in exI)
avigad@16908
   513
  apply (rule allI)
wenzelm@61945
   514
  apply (subgoal_tac "ca * \<bar>c\<bar> * \<bar>f x\<bar> = \<bar>c\<bar> * (ca * \<bar>f x\<bar>)")
wenzelm@63473
   515
   apply (erule ssubst)
wenzelm@63473
   516
   apply (subst abs_mult)
wenzelm@63473
   517
   apply (rule mult_left_mono)
wenzelm@63473
   518
    apply (erule spec)
wenzelm@63473
   519
   apply simp
wenzelm@63473
   520
  apply (simp add: ac_simps)
wenzelm@22665
   521
  done
avigad@16908
   522
wenzelm@63473
   523
lemma bigo_const_mult7 [intro]:
wenzelm@63473
   524
  assumes *: "f =o O(g)"
wenzelm@63473
   525
  shows "(\<lambda>x. c * f x) =o O(g)"
avigad@16908
   526
proof -
wenzelm@63473
   527
  from * have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
avigad@16908
   528
    by auto
wenzelm@55821
   529
  also have "(\<lambda>x. c) * f = (\<lambda>x. c * f x)"
avigad@16908
   530
    by (simp add: func_times)
wenzelm@55821
   531
  also have "(\<lambda>x. c) *o O(g) \<subseteq> O(g)"
avigad@16908
   532
    by (auto del: subsetI)
avigad@16908
   533
  finally show ?thesis .
avigad@16908
   534
qed
avigad@16908
   535
wenzelm@55821
   536
lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f (k x)) =o O(\<lambda>x. g (k x))"
wenzelm@63473
   537
  by (auto simp: bigo_def)
avigad@16908
   538
wenzelm@63473
   539
lemma bigo_compose2: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. f (k x)) =o (\<lambda>x. g (k x)) +o O(\<lambda>x. h(k x))"
haftmann@54230
   540
  apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus)
wenzelm@55821
   541
  apply (drule bigo_compose1)
wenzelm@55821
   542
  apply (simp add: fun_diff_def)
haftmann@54230
   543
  done
avigad@16908
   544
wenzelm@22665
   545
wenzelm@60500
   546
subsection \<open>Setsum\<close>
avigad@16908
   547
wenzelm@55821
   548
lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 \<le> h x y \<Longrightarrow>
wenzelm@61945
   549
    \<exists>c. \<forall>x. \<forall>y \<in> A x. \<bar>f x y\<bar> \<le> c * h x y \<Longrightarrow>
wenzelm@55821
   550
      (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
avigad@16908
   551
  apply (auto simp add: bigo_def)
wenzelm@61945
   552
  apply (rule_tac x = "\<bar>c\<bar>" in exI)
wenzelm@17199
   553
  apply (subst abs_of_nonneg) back back
wenzelm@63473
   554
   apply (rule setsum_nonneg)
wenzelm@63473
   555
   apply force
ballarin@19279
   556
  apply (subst setsum_right_distrib)
avigad@16908
   557
  apply (rule allI)
avigad@16908
   558
  apply (rule order_trans)
wenzelm@63473
   559
   apply (rule setsum_abs)
avigad@16908
   560
  apply (rule setsum_mono)
avigad@16908
   561
  apply (rule order_trans)
wenzelm@63473
   562
   apply (drule spec)+
wenzelm@63473
   563
   apply (drule bspec)+
wenzelm@63473
   564
     apply assumption+
wenzelm@63473
   565
   apply (drule bspec)
wenzelm@63473
   566
    apply assumption+
wenzelm@55821
   567
  apply (rule mult_right_mono)
wenzelm@63473
   568
   apply (rule abs_ge_self)
avigad@16908
   569
  apply force
wenzelm@22665
   570
  done
avigad@16908
   571
wenzelm@55821
   572
lemma bigo_setsum1: "\<forall>x y. 0 \<le> h x y \<Longrightarrow>
wenzelm@61945
   573
    \<exists>c. \<forall>x y. \<bar>f x y\<bar> \<le> c * h x y \<Longrightarrow>
wenzelm@55821
   574
      (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
avigad@16908
   575
  apply (rule bigo_setsum_main)
wenzelm@63473
   576
   apply force
avigad@16908
   577
  apply clarsimp
avigad@16908
   578
  apply (rule_tac x = c in exI)
avigad@16908
   579
  apply force
wenzelm@22665
   580
  done
avigad@16908
   581
wenzelm@55821
   582
lemma bigo_setsum2: "\<forall>y. 0 \<le> h y \<Longrightarrow>
wenzelm@61945
   583
    \<exists>c. \<forall>y. \<bar>f y\<bar> \<le> c * (h y) \<Longrightarrow>
wenzelm@55821
   584
      (\<lambda>x. \<Sum>y \<in> A x. f y) =o O(\<lambda>x. \<Sum>y \<in> A x. h y)"
wenzelm@55821
   585
  by (rule bigo_setsum1) auto
avigad@16908
   586
wenzelm@55821
   587
lemma bigo_setsum3: "f =o O(h) \<Longrightarrow>
wenzelm@61945
   588
    (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)"
avigad@16908
   589
  apply (rule bigo_setsum1)
wenzelm@63473
   590
   apply (rule allI)+
wenzelm@63473
   591
   apply (rule abs_ge_zero)
avigad@16908
   592
  apply (unfold bigo_def)
avigad@16908
   593
  apply auto
avigad@16908
   594
  apply (rule_tac x = c in exI)
avigad@16908
   595
  apply (rule allI)+
avigad@16908
   596
  apply (subst abs_mult)+
haftmann@57512
   597
  apply (subst mult.left_commute)
avigad@16908
   598
  apply (rule mult_left_mono)
wenzelm@63473
   599
   apply (erule spec)
avigad@16908
   600
  apply (rule abs_ge_zero)
wenzelm@22665
   601
  done
avigad@16908
   602
wenzelm@55821
   603
lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow>
wenzelm@55821
   604
    (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
wenzelm@55821
   605
      (\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o
wenzelm@61945
   606
        O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)"
avigad@16908
   607
  apply (rule set_minus_imp_plus)
berghofe@26814
   608
  apply (subst fun_diff_def)
avigad@16908
   609
  apply (subst setsum_subtractf [symmetric])
avigad@16908
   610
  apply (subst right_diff_distrib [symmetric])
avigad@16908
   611
  apply (rule bigo_setsum3)
berghofe@26814
   612
  apply (subst fun_diff_def [symmetric])
avigad@16908
   613
  apply (erule set_plus_imp_minus)
wenzelm@22665
   614
  done
avigad@16908
   615
wenzelm@55821
   616
lemma bigo_setsum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
wenzelm@55821
   617
    \<forall>x. 0 \<le> h x \<Longrightarrow>
wenzelm@55821
   618
      (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
wenzelm@55821
   619
        O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
wenzelm@55821
   620
  apply (subgoal_tac "(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y)) =
wenzelm@61945
   621
      (\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)")
wenzelm@63473
   622
   apply (erule ssubst)
wenzelm@63473
   623
   apply (erule bigo_setsum3)
avigad@16908
   624
  apply (rule ext)
haftmann@57418
   625
  apply (rule setsum.cong)
wenzelm@63473
   626
   apply (rule refl)
avigad@16908
   627
  apply (subst abs_of_nonneg)
wenzelm@63473
   628
   apply auto
wenzelm@22665
   629
  done
avigad@16908
   630
wenzelm@55821
   631
lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
wenzelm@55821
   632
    \<forall>x. 0 \<le> h x \<Longrightarrow>
wenzelm@55821
   633
      (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
wenzelm@55821
   634
        (\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o
wenzelm@55821
   635
          O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
avigad@16908
   636
  apply (rule set_minus_imp_plus)
berghofe@26814
   637
  apply (subst fun_diff_def)
avigad@16908
   638
  apply (subst setsum_subtractf [symmetric])
avigad@16908
   639
  apply (subst right_diff_distrib [symmetric])
avigad@16908
   640
  apply (rule bigo_setsum5)
wenzelm@63473
   641
    apply (subst fun_diff_def [symmetric])
wenzelm@63473
   642
    apply (drule set_plus_imp_minus)
wenzelm@63473
   643
    apply auto
wenzelm@22665
   644
  done
wenzelm@22665
   645
avigad@16908
   646
wenzelm@60500
   647
subsection \<open>Misc useful stuff\<close>
avigad@16908
   648
wenzelm@55821
   649
lemma bigo_useful_intro: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
avigad@16908
   650
  apply (subst bigo_plus_idemp [symmetric])
avigad@16908
   651
  apply (rule set_plus_mono2)
wenzelm@63473
   652
   apply assumption+
wenzelm@22665
   653
  done
avigad@16908
   654
wenzelm@55821
   655
lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)"
avigad@16908
   656
  apply (subst bigo_plus_idemp [symmetric])
avigad@16908
   657
  apply (rule set_plus_intro)
wenzelm@63473
   658
   apply assumption+
wenzelm@22665
   659
  done
wenzelm@55821
   660
wenzelm@63473
   661
lemma bigo_useful_const_mult: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
wenzelm@63473
   662
  for c :: "'a::linordered_field"
avigad@16908
   663
  apply (rule subsetD)
wenzelm@63473
   664
   apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) \<subseteq> O(h)")
wenzelm@63473
   665
    apply assumption
wenzelm@63473
   666
   apply (rule bigo_const_mult6)
wenzelm@55821
   667
  apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
wenzelm@63473
   668
   apply (erule ssubst)
wenzelm@63473
   669
   apply (erule set_times_intro2)
nipkow@23413
   670
  apply (simp add: func_times)
wenzelm@22665
   671
  done
avigad@16908
   672
wenzelm@55821
   673
lemma bigo_fix: "(\<lambda>x::nat. f (x + 1)) =o O(\<lambda>x. h (x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow> f =o O(h)"
avigad@16908
   674
  apply (simp add: bigo_alt_def)
avigad@16908
   675
  apply auto
avigad@16908
   676
  apply (rule_tac x = c in exI)
avigad@16908
   677
  apply auto
avigad@16908
   678
  apply (case_tac "x = 0")
wenzelm@63473
   679
   apply simp
avigad@16908
   680
  apply (subgoal_tac "x = Suc (x - 1)")
wenzelm@63473
   681
   apply (erule ssubst) back
wenzelm@63473
   682
   apply (erule spec)
avigad@16908
   683
  apply simp
wenzelm@22665
   684
  done
avigad@16908
   685
wenzelm@55821
   686
lemma bigo_fix2:
wenzelm@55821
   687
    "(\<lambda>x. f ((x::nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
wenzelm@55821
   688
       f 0 = g 0 \<Longrightarrow> f =o g +o O(h)"
avigad@16908
   689
  apply (rule set_minus_imp_plus)
avigad@16908
   690
  apply (rule bigo_fix)
wenzelm@63473
   691
   apply (subst fun_diff_def)
wenzelm@63473
   692
   apply (subst fun_diff_def [symmetric])
wenzelm@63473
   693
   apply (rule set_plus_imp_minus)
wenzelm@63473
   694
   apply simp
berghofe@26814
   695
  apply (simp add: fun_diff_def)
wenzelm@22665
   696
  done
wenzelm@22665
   697
avigad@16908
   698
wenzelm@60500
   699
subsection \<open>Less than or equal to\<close>
avigad@16908
   700
wenzelm@55821
   701
definition lesso :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"  (infixl "<o" 70)
wenzelm@55821
   702
  where "f <o g = (\<lambda>x. max (f x - g x) 0)"
avigad@16908
   703
wenzelm@61945
   704
lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> \<le> \<bar>f x\<bar> \<Longrightarrow> g =o O(h)"
avigad@16908
   705
  apply (unfold bigo_def)
avigad@16908
   706
  apply clarsimp
avigad@16908
   707
  apply (rule_tac x = c in exI)
avigad@16908
   708
  apply (rule allI)
avigad@16908
   709
  apply (rule order_trans)
wenzelm@63473
   710
   apply (erule spec)+
wenzelm@22665
   711
  done
avigad@16908
   712
wenzelm@61945
   713
lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> \<le> f x \<Longrightarrow> g =o O(h)"
avigad@16908
   714
  apply (erule bigo_lesseq1)
avigad@16908
   715
  apply (rule allI)
avigad@16908
   716
  apply (drule_tac x = x in spec)
avigad@16908
   717
  apply (rule order_trans)
wenzelm@63473
   718
   apply assumption
avigad@16908
   719
  apply (rule abs_ge_self)
wenzelm@22665
   720
  done
avigad@16908
   721
wenzelm@55821
   722
lemma bigo_lesseq3: "f =o O(h) \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> f x \<Longrightarrow> g =o O(h)"
avigad@16908
   723
  apply (erule bigo_lesseq2)
avigad@16908
   724
  apply (rule allI)
avigad@16908
   725
  apply (subst abs_of_nonneg)
wenzelm@63473
   726
   apply (erule spec)+
wenzelm@22665
   727
  done
avigad@16908
   728
wenzelm@55821
   729
lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
wenzelm@61945
   730
    \<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> \<bar>f x\<bar> \<Longrightarrow> g =o O(h)"
avigad@16908
   731
  apply (erule bigo_lesseq1)
avigad@16908
   732
  apply (rule allI)
avigad@16908
   733
  apply (subst abs_of_nonneg)
wenzelm@63473
   734
   apply (erule spec)+
wenzelm@22665
   735
  done
avigad@16908
   736
wenzelm@55821
   737
lemma bigo_lesso1: "\<forall>x. f x \<le> g x \<Longrightarrow> f <o g =o O(h)"
avigad@16908
   738
  apply (unfold lesso_def)
wenzelm@55821
   739
  apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
wenzelm@63473
   740
   apply (erule ssubst)
wenzelm@63473
   741
   apply (rule bigo_zero)
avigad@16908
   742
  apply (unfold func_zero)
avigad@16908
   743
  apply (rule ext)
avigad@16908
   744
  apply (simp split: split_max)
wenzelm@22665
   745
  done
avigad@16908
   746
wenzelm@63473
   747
lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow> \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. k x \<le> f x \<Longrightarrow> k <o g =o O(h)"
avigad@16908
   748
  apply (unfold lesso_def)
avigad@16908
   749
  apply (rule bigo_lesseq4)
wenzelm@63473
   750
    apply (erule set_plus_imp_minus)
wenzelm@63473
   751
   apply (rule allI)
wenzelm@63473
   752
   apply (rule max.cobounded2)
avigad@16908
   753
  apply (rule allI)
berghofe@26814
   754
  apply (subst fun_diff_def)
wenzelm@55821
   755
  apply (case_tac "0 \<le> k x - g x")
wenzelm@63473
   756
   apply simp
wenzelm@63473
   757
   apply (subst abs_of_nonneg)
wenzelm@63473
   758
    apply (drule_tac x = x in spec) back
wenzelm@63473
   759
    apply (simp add: algebra_simps)
wenzelm@63473
   760
   apply (subst diff_conv_add_uminus)+
wenzelm@63473
   761
   apply (rule add_right_mono)
wenzelm@63473
   762
   apply (erule spec)
wenzelm@55821
   763
  apply (rule order_trans)
wenzelm@63473
   764
   prefer 2
wenzelm@63473
   765
   apply (rule abs_ge_zero)
nipkow@29667
   766
  apply (simp add: algebra_simps)
wenzelm@22665
   767
  done
avigad@16908
   768
wenzelm@63473
   769
lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow> \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. g x \<le> k x \<Longrightarrow> f <o k =o O(h)"
avigad@16908
   770
  apply (unfold lesso_def)
avigad@16908
   771
  apply (rule bigo_lesseq4)
wenzelm@63473
   772
    apply (erule set_plus_imp_minus)
wenzelm@63473
   773
   apply (rule allI)
wenzelm@63473
   774
   apply (rule max.cobounded2)
avigad@16908
   775
  apply (rule allI)
berghofe@26814
   776
  apply (subst fun_diff_def)
wenzelm@55821
   777
  apply (case_tac "0 \<le> f x - k x")
wenzelm@63473
   778
   apply simp
wenzelm@63473
   779
   apply (subst abs_of_nonneg)
wenzelm@63473
   780
    apply (drule_tac x = x in spec) back
wenzelm@63473
   781
    apply (simp add: algebra_simps)
wenzelm@63473
   782
   apply (subst diff_conv_add_uminus)+
wenzelm@63473
   783
   apply (rule add_left_mono)
wenzelm@63473
   784
   apply (rule le_imp_neg_le)
wenzelm@63473
   785
   apply (erule spec)
wenzelm@55821
   786
  apply (rule order_trans)
wenzelm@63473
   787
   prefer 2
wenzelm@63473
   788
   apply (rule abs_ge_zero)
nipkow@29667
   789
  apply (simp add: algebra_simps)
wenzelm@22665
   790
  done
avigad@16908
   791
wenzelm@63473
   792
lemma bigo_lesso4: "f <o g =o O(k) \<Longrightarrow> g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
wenzelm@63473
   793
  for k :: "'a \<Rightarrow> 'b::linordered_field"
avigad@16908
   794
  apply (unfold lesso_def)
avigad@16908
   795
  apply (drule set_plus_imp_minus)
wenzelm@17199
   796
  apply (drule bigo_abs5) back
berghofe@26814
   797
  apply (simp add: fun_diff_def)
avigad@16908
   798
  apply (drule bigo_useful_add)
wenzelm@63473
   799
   apply assumption
wenzelm@17199
   800
  apply (erule bigo_lesseq2) back
avigad@16908
   801
  apply (rule allI)
wenzelm@55821
   802
  apply (auto simp add: func_plus fun_diff_def algebra_simps split: split_max abs_split)
wenzelm@22665
   803
  done
avigad@16908
   804
wenzelm@61945
   805
lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x \<le> g x + C * \<bar>h x\<bar>"
avigad@16908
   806
  apply (simp only: lesso_def bigo_alt_def)
avigad@16908
   807
  apply clarsimp
avigad@16908
   808
  apply (rule_tac x = c in exI)
avigad@16908
   809
  apply (rule allI)
avigad@16908
   810
  apply (drule_tac x = x in spec)
wenzelm@61945
   811
  apply (subgoal_tac "\<bar>max (f x - g x) 0\<bar> = max (f x - g x) 0")
wenzelm@63473
   812
   apply (clarsimp simp add: algebra_simps)
avigad@16908
   813
  apply (rule abs_of_nonneg)
haftmann@54863
   814
  apply (rule max.cobounded2)
wenzelm@22665
   815
  done
avigad@16908
   816
wenzelm@55821
   817
lemma lesso_add: "f <o g =o O(h) \<Longrightarrow> k <o l =o O(h) \<Longrightarrow> (f + k) <o (g + l) =o O(h)"
avigad@16908
   818
  apply (unfold lesso_def)
avigad@16908
   819
  apply (rule bigo_lesseq3)
wenzelm@63473
   820
    apply (erule bigo_useful_add)
wenzelm@63473
   821
    apply assumption
wenzelm@63473
   822
   apply (force split: split_max)
avigad@16908
   823
  apply (auto split: split_max simp add: func_plus)
wenzelm@22665
   824
  done
avigad@16908
   825
wenzelm@63473
   826
lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> 0"
wenzelm@63473
   827
  for f g :: "nat \<Rightarrow> real"
huffman@31337
   828
  apply (simp add: LIMSEQ_iff bigo_alt_def)
haftmann@29786
   829
  apply clarify
haftmann@29786
   830
  apply (drule_tac x = "r / c" in spec)
haftmann@29786
   831
  apply (drule mp)
wenzelm@63473
   832
   apply simp
haftmann@29786
   833
  apply clarify
haftmann@29786
   834
  apply (rule_tac x = no in exI)
haftmann@29786
   835
  apply (rule allI)
haftmann@29786
   836
  apply (drule_tac x = n in spec)+
haftmann@29786
   837
  apply (rule impI)
haftmann@29786
   838
  apply (drule mp)
wenzelm@63473
   839
   apply assumption
haftmann@29786
   840
  apply (rule order_le_less_trans)
wenzelm@63473
   841
   apply assumption
haftmann@29786
   842
  apply (rule order_less_le_trans)
wenzelm@63473
   843
   apply (subgoal_tac "c * \<bar>g n\<bar> < c * (r / c)")
wenzelm@63473
   844
    apply assumption
wenzelm@63473
   845
   apply (erule mult_strict_left_mono)
wenzelm@63473
   846
   apply assumption
haftmann@29786
   847
  apply simp
wenzelm@55821
   848
  done
haftmann@29786
   849
wenzelm@63473
   850
lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> g \<longlonglongrightarrow> a"
wenzelm@63473
   851
  for f g h :: "nat \<Rightarrow> real"
haftmann@29786
   852
  apply (drule set_plus_imp_minus)
haftmann@29786
   853
  apply (drule bigo_LIMSEQ1)
wenzelm@63473
   854
   apply assumption
haftmann@29786
   855
  apply (simp only: fun_diff_def)
lp15@60142
   856
  apply (erule Lim_transform2)
haftmann@29786
   857
  apply assumption
wenzelm@55821
   858
  done
haftmann@29786
   859
avigad@16908
   860
end