src/HOL/Library/BigO.thy
author wenzelm
Sat Mar 01 13:05:46 2014 +0100 (2014-03-01)
changeset 55821 44055f07cbd8
parent 54863 82acc20ded73
child 56536 aefb4a8da31f
permissions -rw-r--r--
more symbols, less parentheses;
     1 (*  Title:      HOL/Library/BigO.thy
     2     Authors:    Jeremy Avigad and Kevin Donnelly
     3 *)
     4 
     5 header {* Big O notation *}
     6 
     7 theory BigO
     8 imports Complex_Main Function_Algebras Set_Algebras
     9 begin
    10 
    11 text {*
    12 This library is designed to support asymptotic ``big O'' calculations,
    13 i.e.~reasoning with expressions of the form $f = O(g)$ and $f = g +
    14 O(h)$.  An earlier version of this library is described in detail in
    15 \cite{Avigad-Donnelly}.
    16 
    17 The main changes in this version are as follows:
    18 \begin{itemize}
    19 \item We have eliminated the @{text O} operator on sets. (Most uses of this seem
    20   to be inessential.)
    21 \item We no longer use @{text "+"} as output syntax for @{text "+o"}
    22 \item Lemmas involving @{text "sumr"} have been replaced by more general lemmas
    23   involving `@{text "setsum"}.
    24 \item The library has been expanded, with e.g.~support for expressions of
    25   the form @{text "f < g + O(h)"}.
    26 \end{itemize}
    27 
    28 Note also since the Big O library includes rules that demonstrate set
    29 inclusion, to use the automated reasoners effectively with the library
    30 one should redeclare the theorem @{text "subsetI"} as an intro rule,
    31 rather than as an @{text "intro!"} rule, for example, using
    32 \isa{\isakeyword{declare}}~@{text "subsetI [del, intro]"}.
    33 *}
    34 
    35 subsection {* Definitions *}
    36 
    37 definition bigo :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) set"  ("(1O'(_'))")
    38   where "O(f:: 'a \<Rightarrow> 'b) = {h. \<exists>c. \<forall>x. abs (h x) \<le> c * abs (f x)}"
    39 
    40 lemma bigo_pos_const:
    41   "(\<exists>c::'a::linordered_idom. \<forall>x. abs (h x) \<le> c * abs (f x)) \<longleftrightarrow>
    42     (\<exists>c. 0 < c \<and> (\<forall>x. abs (h x) \<le> c * abs (f x)))"
    43   apply auto
    44   apply (case_tac "c = 0")
    45   apply simp
    46   apply (rule_tac x = "1" in exI)
    47   apply simp
    48   apply (rule_tac x = "abs c" in exI)
    49   apply auto
    50   apply (subgoal_tac "c * abs (f x) \<le> abs c * abs (f x)")
    51   apply (erule_tac x = x in allE)
    52   apply force
    53   apply (rule mult_right_mono)
    54   apply (rule abs_ge_self)
    55   apply (rule abs_ge_zero)
    56   done
    57 
    58 lemma bigo_alt_def: "O(f) = {h. \<exists>c. 0 < c \<and> (\<forall>x. abs (h x) \<le> c * abs (f x))}"
    59   by (auto simp add: bigo_def bigo_pos_const)
    60 
    61 lemma bigo_elt_subset [intro]: "f \<in> O(g) \<Longrightarrow> O(f) \<le> O(g)"
    62   apply (auto simp add: bigo_alt_def)
    63   apply (rule_tac x = "ca * c" in exI)
    64   apply (rule conjI)
    65   apply (rule mult_pos_pos)
    66   apply assumption+
    67   apply (rule allI)
    68   apply (drule_tac x = "xa" in spec)+
    69   apply (subgoal_tac "ca * abs (f xa) \<le> ca * (c * abs (g xa))")
    70   apply (erule order_trans)
    71   apply (simp add: mult_ac)
    72   apply (rule mult_left_mono, assumption)
    73   apply (rule order_less_imp_le, assumption)
    74   done
    75 
    76 lemma bigo_refl [intro]: "f \<in> O(f)"
    77   apply(auto simp add: bigo_def)
    78   apply(rule_tac x = 1 in exI)
    79   apply simp
    80   done
    81 
    82 lemma bigo_zero: "0 \<in> O(g)"
    83   apply (auto simp add: bigo_def func_zero)
    84   apply (rule_tac x = 0 in exI)
    85   apply auto
    86   done
    87 
    88 lemma bigo_zero2: "O(\<lambda>x. 0) = {\<lambda>x. 0}"
    89   by (auto simp add: bigo_def)
    90 
    91 lemma bigo_plus_self_subset [intro]: "O(f) + O(f) \<subseteq> O(f)"
    92   apply (auto simp add: bigo_alt_def set_plus_def)
    93   apply (rule_tac x = "c + ca" in exI)
    94   apply auto
    95   apply (simp add: ring_distribs func_plus)
    96   apply (rule order_trans)
    97   apply (rule abs_triangle_ineq)
    98   apply (rule add_mono)
    99   apply force
   100   apply force
   101   done
   102 
   103 lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
   104   apply (rule equalityI)
   105   apply (rule bigo_plus_self_subset)
   106   apply (rule set_zero_plus2)
   107   apply (rule bigo_zero)
   108   done
   109 
   110 lemma bigo_plus_subset [intro]: "O(f + g) \<subseteq> O(f) + O(g)"
   111   apply (rule subsetI)
   112   apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
   113   apply (subst bigo_pos_const [symmetric])+
   114   apply (rule_tac x = "\<lambda>n. if abs (g n) \<le> (abs (f n)) then x n else 0" in exI)
   115   apply (rule conjI)
   116   apply (rule_tac x = "c + c" in exI)
   117   apply (clarsimp)
   118   apply (auto)
   119   apply (subgoal_tac "c * abs (f xa + g xa) \<le> (c + c) * abs (f xa)")
   120   apply (erule_tac x = xa in allE)
   121   apply (erule order_trans)
   122   apply (simp)
   123   apply (subgoal_tac "c * abs (f xa + g xa) \<le> c * (abs (f xa) + abs (g xa))")
   124   apply (erule order_trans)
   125   apply (simp add: ring_distribs)
   126   apply (rule mult_left_mono)
   127   apply (simp add: abs_triangle_ineq)
   128   apply (simp add: order_less_le)
   129   apply (rule mult_nonneg_nonneg)
   130   apply auto
   131   apply (rule_tac x = "\<lambda>n. if (abs (f n)) < abs (g n) then x n else 0" in exI)
   132   apply (rule conjI)
   133   apply (rule_tac x = "c + c" in exI)
   134   apply auto
   135   apply (subgoal_tac "c * abs (f xa + g xa) \<le> (c + c) * abs (g xa)")
   136   apply (erule_tac x = xa in allE)
   137   apply (erule order_trans)
   138   apply simp
   139   apply (subgoal_tac "c * abs (f xa + g xa) \<le> c * (abs (f xa) + abs (g xa))")
   140   apply (erule order_trans)
   141   apply (simp add: ring_distribs)
   142   apply (rule mult_left_mono)
   143   apply (rule abs_triangle_ineq)
   144   apply (simp add: order_less_le)
   145   apply (rule mult_nonneg_nonneg)
   146   apply (erule order_less_imp_le)
   147   apply simp
   148   done
   149 
   150 lemma bigo_plus_subset2 [intro]: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
   151   apply (subgoal_tac "A + B \<subseteq> O(f) + O(f)")
   152   apply (erule order_trans)
   153   apply simp
   154   apply (auto del: subsetI simp del: bigo_plus_idemp)
   155   done
   156 
   157 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)"
   158   apply (rule equalityI)
   159   apply (rule bigo_plus_subset)
   160   apply (simp add: bigo_alt_def set_plus_def func_plus)
   161   apply clarify
   162   apply (rule_tac x = "max c ca" in exI)
   163   apply (rule conjI)
   164   apply (subgoal_tac "c \<le> max c ca")
   165   apply (erule order_less_le_trans)
   166   apply assumption
   167   apply (rule max.cobounded1)
   168   apply clarify
   169   apply (drule_tac x = "xa" in spec)+
   170   apply (subgoal_tac "0 \<le> f xa + g xa")
   171   apply (simp add: ring_distribs)
   172   apply (subgoal_tac "abs (a xa + b xa) \<le> abs (a xa) + abs (b xa)")
   173   apply (subgoal_tac "abs (a xa) + abs (b xa) \<le> max c ca * f xa + max c ca * g xa")
   174   apply force
   175   apply (rule add_mono)
   176   apply (subgoal_tac "c * f xa \<le> max c ca * f xa")
   177   apply force
   178   apply (rule mult_right_mono)
   179   apply (rule max.cobounded1)
   180   apply assumption
   181   apply (subgoal_tac "ca * g xa \<le> max c ca * g xa")
   182   apply force
   183   apply (rule mult_right_mono)
   184   apply (rule max.cobounded2)
   185   apply assumption
   186   apply (rule abs_triangle_ineq)
   187   apply (rule add_nonneg_nonneg)
   188   apply assumption+
   189   done
   190 
   191 lemma bigo_bounded_alt: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> c * g x \<Longrightarrow> f \<in> O(g)"
   192   apply (auto simp add: bigo_def)
   193   apply (rule_tac x = "abs c" in exI)
   194   apply auto
   195   apply (drule_tac x = x in spec)+
   196   apply (simp add: abs_mult [symmetric])
   197   done
   198 
   199 lemma bigo_bounded: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> g x \<Longrightarrow> f \<in> O(g)"
   200   apply (erule bigo_bounded_alt [of f 1 g])
   201   apply simp
   202   done
   203 
   204 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)"
   205   apply (rule set_minus_imp_plus)
   206   apply (rule bigo_bounded)
   207   apply (auto simp add: fun_Compl_def func_plus)
   208   apply (drule_tac x = x in spec)+
   209   apply force
   210   apply (drule_tac x = x in spec)+
   211   apply force
   212   done
   213 
   214 lemma bigo_abs: "(\<lambda>x. abs (f x)) =o O(f)"
   215   apply (unfold bigo_def)
   216   apply auto
   217   apply (rule_tac x = 1 in exI)
   218   apply auto
   219   done
   220 
   221 lemma bigo_abs2: "f =o O(\<lambda>x. abs (f x))"
   222   apply (unfold bigo_def)
   223   apply auto
   224   apply (rule_tac x = 1 in exI)
   225   apply auto
   226   done
   227 
   228 lemma bigo_abs3: "O(f) = O(\<lambda>x. abs (f x))"
   229   apply (rule equalityI)
   230   apply (rule bigo_elt_subset)
   231   apply (rule bigo_abs2)
   232   apply (rule bigo_elt_subset)
   233   apply (rule bigo_abs)
   234   done
   235 
   236 lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. abs (f x)) =o (\<lambda>x. abs (g x)) +o O(h)"
   237   apply (drule set_plus_imp_minus)
   238   apply (rule set_minus_imp_plus)
   239   apply (subst fun_diff_def)
   240 proof -
   241   assume a: "f - g \<in> O(h)"
   242   have "(\<lambda>x. abs (f x) - abs (g x)) =o O(\<lambda>x. abs (abs (f x) - abs (g x)))"
   243     by (rule bigo_abs2)
   244   also have "\<dots> \<subseteq> O(\<lambda>x. abs (f x - g x))"
   245     apply (rule bigo_elt_subset)
   246     apply (rule bigo_bounded)
   247     apply force
   248     apply (rule allI)
   249     apply (rule abs_triangle_ineq3)
   250     done
   251   also have "\<dots> \<subseteq> O(f - g)"
   252     apply (rule bigo_elt_subset)
   253     apply (subst fun_diff_def)
   254     apply (rule bigo_abs)
   255     done
   256   also from a have "\<dots> \<subseteq> O(h)"
   257     by (rule bigo_elt_subset)
   258   finally show "(\<lambda>x. abs (f x) - abs (g x)) \<in> O(h)".
   259 qed
   260 
   261 lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. abs (f x)) =o O(g)"
   262   by (unfold bigo_def, auto)
   263 
   264 lemma bigo_elt_subset2 [intro]: "f \<in> g +o O(h) \<Longrightarrow> O(f) \<subseteq> O(g) + O(h)"
   265 proof -
   266   assume "f \<in> g +o O(h)"
   267   also have "\<dots> \<subseteq> O(g) + O(h)"
   268     by (auto del: subsetI)
   269   also have "\<dots> = O(\<lambda>x. abs (g x)) + O(\<lambda>x. abs (h x))"
   270     apply (subst bigo_abs3 [symmetric])+
   271     apply (rule refl)
   272     done
   273   also have "\<dots> = O((\<lambda>x. abs (g x)) + (\<lambda>x. abs (h x)))"
   274     by (rule bigo_plus_eq [symmetric]) auto
   275   finally have "f \<in> \<dots>" .
   276   then have "O(f) \<subseteq> \<dots>"
   277     by (elim bigo_elt_subset)
   278   also have "\<dots> = O(\<lambda>x. abs (g x)) + O(\<lambda>x. abs (h x))"
   279     by (rule bigo_plus_eq, auto)
   280   finally show ?thesis
   281     by (simp add: bigo_abs3 [symmetric])
   282 qed
   283 
   284 lemma bigo_mult [intro]: "O(f)*O(g) \<subseteq> O(f * g)"
   285   apply (rule subsetI)
   286   apply (subst bigo_def)
   287   apply (auto simp add: bigo_alt_def set_times_def func_times)
   288   apply (rule_tac x = "c * ca" in exI)
   289   apply (rule allI)
   290   apply (erule_tac x = x in allE)+
   291   apply (subgoal_tac "c * ca * abs (f x * g x) = (c * abs (f x)) * (ca * abs (g x))")
   292   apply (erule ssubst)
   293   apply (subst abs_mult)
   294   apply (rule mult_mono)
   295   apply assumption+
   296   apply (rule mult_nonneg_nonneg)
   297   apply auto
   298   apply (simp add: mult_ac abs_mult)
   299   done
   300 
   301 lemma bigo_mult2 [intro]: "f *o O(g) \<subseteq> O(f * g)"
   302   apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
   303   apply (rule_tac x = c in exI)
   304   apply auto
   305   apply (drule_tac x = x in spec)
   306   apply (subgoal_tac "abs (f x) * abs (b x) \<le> abs (f x) * (c * abs (g x))")
   307   apply (force simp add: mult_ac)
   308   apply (rule mult_left_mono, assumption)
   309   apply (rule abs_ge_zero)
   310   done
   311 
   312 lemma bigo_mult3: "f \<in> O(h) \<Longrightarrow> g \<in> O(j) \<Longrightarrow> f * g \<in> O(h * j)"
   313   apply (rule subsetD)
   314   apply (rule bigo_mult)
   315   apply (erule set_times_intro, assumption)
   316   done
   317 
   318 lemma bigo_mult4 [intro]: "f \<in> k +o O(h) \<Longrightarrow> g * f \<in> (g * k) +o O(g * h)"
   319   apply (drule set_plus_imp_minus)
   320   apply (rule set_minus_imp_plus)
   321   apply (drule bigo_mult3 [where g = g and j = g])
   322   apply (auto simp add: algebra_simps)
   323   done
   324 
   325 lemma bigo_mult5:
   326   fixes f :: "'a \<Rightarrow> 'b::linordered_field"
   327   assumes "\<forall>x. f x \<noteq> 0"
   328   shows "O(f * g) \<subseteq> f *o O(g)"
   329 proof
   330   fix h
   331   assume "h \<in> O(f * g)"
   332   then have "(\<lambda>x. 1 / (f x)) * h \<in> (\<lambda>x. 1 / f x) *o O(f * g)"
   333     by auto
   334   also have "\<dots> \<subseteq> O((\<lambda>x. 1 / f x) * (f * g))"
   335     by (rule bigo_mult2)
   336   also have "(\<lambda>x. 1 / f x) * (f * g) = g"
   337     apply (simp add: func_times)
   338     apply (rule ext)
   339     apply (simp add: assms nonzero_divide_eq_eq mult_ac)
   340     done
   341   finally have "(\<lambda>x. (1::'b) / f x) * h \<in> O(g)" .
   342   then have "f * ((\<lambda>x. (1::'b) / f x) * h) \<in> f *o O(g)"
   343     by auto
   344   also have "f * ((\<lambda>x. (1::'b) / f x) * h) = h"
   345     apply (simp add: func_times)
   346     apply (rule ext)
   347     apply (simp add: assms nonzero_divide_eq_eq mult_ac)
   348     done
   349   finally show "h \<in> f *o O(g)" .
   350 qed
   351 
   352 lemma bigo_mult6:
   353   fixes f :: "'a \<Rightarrow> 'b::linordered_field"
   354   shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = f *o O(g)"
   355   apply (rule equalityI)
   356   apply (erule bigo_mult5)
   357   apply (rule bigo_mult2)
   358   done
   359 
   360 lemma bigo_mult7:
   361   fixes f :: "'a \<Rightarrow> 'b::linordered_field"
   362   shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<subseteq> O(f) * O(g)"
   363   apply (subst bigo_mult6)
   364   apply assumption
   365   apply (rule set_times_mono3)
   366   apply (rule bigo_refl)
   367   done
   368 
   369 lemma bigo_mult8:
   370   fixes f :: "'a \<Rightarrow> 'b::linordered_field"
   371   shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f) * O(g)"
   372   apply (rule equalityI)
   373   apply (erule bigo_mult7)
   374   apply (rule bigo_mult)
   375   done
   376 
   377 lemma bigo_minus [intro]: "f \<in> O(g) \<Longrightarrow> - f \<in> O(g)"
   378   by (auto simp add: bigo_def fun_Compl_def)
   379 
   380 lemma bigo_minus2: "f \<in> g +o O(h) \<Longrightarrow> - f \<in> -g +o O(h)"
   381   apply (rule set_minus_imp_plus)
   382   apply (drule set_plus_imp_minus)
   383   apply (drule bigo_minus)
   384   apply simp
   385   done
   386 
   387 lemma bigo_minus3: "O(- f) = O(f)"
   388   by (auto simp add: bigo_def fun_Compl_def)
   389 
   390 lemma bigo_plus_absorb_lemma1: "f \<in> O(g) \<Longrightarrow> f +o O(g) \<subseteq> O(g)"
   391 proof -
   392   assume a: "f \<in> O(g)"
   393   show "f +o O(g) \<subseteq> O(g)"
   394   proof -
   395     have "f \<in> O(f)" by auto
   396     then have "f +o O(g) \<subseteq> O(f) + O(g)"
   397       by (auto del: subsetI)
   398     also have "\<dots> \<subseteq> O(g) + O(g)"
   399     proof -
   400       from a have "O(f) \<subseteq> O(g)" by (auto del: subsetI)
   401       thus ?thesis by (auto del: subsetI)
   402     qed
   403     also have "\<dots> \<subseteq> O(g)" by simp
   404     finally show ?thesis .
   405   qed
   406 qed
   407 
   408 lemma bigo_plus_absorb_lemma2: "f \<in> O(g) \<Longrightarrow> O(g) \<subseteq> f +o O(g)"
   409 proof -
   410   assume a: "f \<in> O(g)"
   411   show "O(g) \<subseteq> f +o O(g)"
   412   proof -
   413     from a have "- f \<in> O(g)"
   414       by auto
   415     then have "- f +o O(g) \<subseteq> O(g)"
   416       by (elim bigo_plus_absorb_lemma1)
   417     then have "f +o (- f +o O(g)) \<subseteq> f +o O(g)"
   418       by auto
   419     also have "f +o (- f +o O(g)) = O(g)"
   420       by (simp add: set_plus_rearranges)
   421     finally show ?thesis .
   422   qed
   423 qed
   424 
   425 lemma bigo_plus_absorb [simp]: "f \<in> O(g) \<Longrightarrow> f +o O(g) = O(g)"
   426   apply (rule equalityI)
   427   apply (erule bigo_plus_absorb_lemma1)
   428   apply (erule bigo_plus_absorb_lemma2)
   429   done
   430 
   431 lemma bigo_plus_absorb2 [intro]: "f \<in> O(g) \<Longrightarrow> A \<subseteq> O(g) \<Longrightarrow> f +o A \<subseteq> O(g)"
   432   apply (subgoal_tac "f +o A \<subseteq> f +o O(g)")
   433   apply force+
   434   done
   435 
   436 lemma bigo_add_commute_imp: "f \<in> g +o O(h) \<Longrightarrow> g \<in> f +o O(h)"
   437   apply (subst set_minus_plus [symmetric])
   438   apply (subgoal_tac "g - f = - (f - g)")
   439   apply (erule ssubst)
   440   apply (rule bigo_minus)
   441   apply (subst set_minus_plus)
   442   apply assumption
   443   apply (simp add: add_ac)
   444   done
   445 
   446 lemma bigo_add_commute: "f \<in> g +o O(h) \<longleftrightarrow> g \<in> f +o O(h)"
   447   apply (rule iffI)
   448   apply (erule bigo_add_commute_imp)+
   449   done
   450 
   451 lemma bigo_const1: "(\<lambda>x. c) \<in> O(\<lambda>x. 1)"
   452   by (auto simp add: bigo_def mult_ac)
   453 
   454 lemma bigo_const2 [intro]: "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)"
   455   apply (rule bigo_elt_subset)
   456   apply (rule bigo_const1)
   457   done
   458 
   459 lemma bigo_const3:
   460   fixes c :: "'a::linordered_field"
   461   shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)"
   462   apply (simp add: bigo_def)
   463   apply (rule_tac x = "abs (inverse c)" in exI)
   464   apply (simp add: abs_mult [symmetric])
   465   done
   466 
   467 lemma bigo_const4:
   468   fixes c :: "'a::linordered_field"
   469   shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)"
   470   apply (rule bigo_elt_subset)
   471   apply (rule bigo_const3)
   472   apply assumption
   473   done
   474 
   475 lemma bigo_const [simp]:
   476   fixes c :: "'a::linordered_field"
   477   shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)"
   478   apply (rule equalityI)
   479   apply (rule bigo_const2)
   480   apply (rule bigo_const4)
   481   apply assumption
   482   done
   483 
   484 lemma bigo_const_mult1: "(\<lambda>x. c * f x) \<in> O(f)"
   485   apply (simp add: bigo_def)
   486   apply (rule_tac x = "abs c" in exI)
   487   apply (auto simp add: abs_mult [symmetric])
   488   done
   489 
   490 lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<subseteq> O(f)"
   491   apply (rule bigo_elt_subset)
   492   apply (rule bigo_const_mult1)
   493   done
   494 
   495 lemma bigo_const_mult3:
   496   fixes c :: "'a::linordered_field"
   497   shows "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
   498   apply (simp add: bigo_def)
   499   apply (rule_tac x = "abs (inverse c)" in exI)
   500   apply (simp add: abs_mult [symmetric] mult_assoc [symmetric])
   501   done
   502 
   503 lemma bigo_const_mult4:
   504   fixes c :: "'a::linordered_field"
   505   shows "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)"
   506   apply (rule bigo_elt_subset)
   507   apply (rule bigo_const_mult3)
   508   apply assumption
   509   done
   510 
   511 lemma bigo_const_mult [simp]:
   512   fixes c :: "'a::linordered_field"
   513   shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)"
   514   apply (rule equalityI)
   515   apply (rule bigo_const_mult2)
   516   apply (erule bigo_const_mult4)
   517   done
   518 
   519 lemma bigo_const_mult5 [simp]:
   520   fixes c :: "'a::linordered_field"
   521   shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
   522   apply (auto del: subsetI)
   523   apply (rule order_trans)
   524   apply (rule bigo_mult2)
   525   apply (simp add: func_times)
   526   apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
   527   apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
   528   apply (simp add: mult_assoc [symmetric] abs_mult)
   529   apply (rule_tac x = "abs (inverse c) * ca" in exI)
   530   apply (rule allI)
   531   apply (subst mult_assoc)
   532   apply (rule mult_left_mono)
   533   apply (erule spec)
   534   apply force
   535   done
   536 
   537 lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) \<subseteq> O(f)"
   538   apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
   539   apply (rule_tac x = "ca * abs c" in exI)
   540   apply (rule allI)
   541   apply (subgoal_tac "ca * abs c * abs (f x) = abs c * (ca * abs (f x))")
   542   apply (erule ssubst)
   543   apply (subst abs_mult)
   544   apply (rule mult_left_mono)
   545   apply (erule spec)
   546   apply simp
   547   apply(simp add: mult_ac)
   548   done
   549 
   550 lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
   551 proof -
   552   assume "f =o O(g)"
   553   then have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
   554     by auto
   555   also have "(\<lambda>x. c) * f = (\<lambda>x. c * f x)"
   556     by (simp add: func_times)
   557   also have "(\<lambda>x. c) *o O(g) \<subseteq> O(g)"
   558     by (auto del: subsetI)
   559   finally show ?thesis .
   560 qed
   561 
   562 lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f (k x)) =o O(\<lambda>x. g (k x))"
   563   unfolding bigo_def by auto
   564 
   565 lemma bigo_compose2: "f =o g +o O(h) \<Longrightarrow>
   566     (\<lambda>x. f (k x)) =o (\<lambda>x. g (k x)) +o O(\<lambda>x. h(k x))"
   567   apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus)
   568   apply (drule bigo_compose1)
   569   apply (simp add: fun_diff_def)
   570   done
   571 
   572 
   573 subsection {* Setsum *}
   574 
   575 lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 \<le> h x y \<Longrightarrow>
   576     \<exists>c. \<forall>x. \<forall>y \<in> A x. abs (f x y) \<le> c * (h x y) \<Longrightarrow>
   577       (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
   578   apply (auto simp add: bigo_def)
   579   apply (rule_tac x = "abs c" in exI)
   580   apply (subst abs_of_nonneg) back back
   581   apply (rule setsum_nonneg)
   582   apply force
   583   apply (subst setsum_right_distrib)
   584   apply (rule allI)
   585   apply (rule order_trans)
   586   apply (rule setsum_abs)
   587   apply (rule setsum_mono)
   588   apply (rule order_trans)
   589   apply (drule spec)+
   590   apply (drule bspec)+
   591   apply assumption+
   592   apply (drule bspec)
   593   apply assumption+
   594   apply (rule mult_right_mono)
   595   apply (rule abs_ge_self)
   596   apply force
   597   done
   598 
   599 lemma bigo_setsum1: "\<forall>x y. 0 \<le> h x y \<Longrightarrow>
   600     \<exists>c. \<forall>x y. abs (f x y) \<le> c * h x y \<Longrightarrow>
   601       (\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
   602   apply (rule bigo_setsum_main)
   603   apply force
   604   apply clarsimp
   605   apply (rule_tac x = c in exI)
   606   apply force
   607   done
   608 
   609 lemma bigo_setsum2: "\<forall>y. 0 \<le> h y \<Longrightarrow>
   610     \<exists>c. \<forall>y. abs (f y) \<le> c * (h y) \<Longrightarrow>
   611       (\<lambda>x. \<Sum>y \<in> A x. f y) =o O(\<lambda>x. \<Sum>y \<in> A x. h y)"
   612   by (rule bigo_setsum1) auto
   613 
   614 lemma bigo_setsum3: "f =o O(h) \<Longrightarrow>
   615     (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o O(\<lambda>x. \<Sum>y \<in> A x. abs (l x y * h (k x y)))"
   616   apply (rule bigo_setsum1)
   617   apply (rule allI)+
   618   apply (rule abs_ge_zero)
   619   apply (unfold bigo_def)
   620   apply auto
   621   apply (rule_tac x = c in exI)
   622   apply (rule allI)+
   623   apply (subst abs_mult)+
   624   apply (subst mult_left_commute)
   625   apply (rule mult_left_mono)
   626   apply (erule spec)
   627   apply (rule abs_ge_zero)
   628   done
   629 
   630 lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow>
   631     (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
   632       (\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o
   633         O(\<lambda>x. \<Sum>y \<in> A x. abs (l x y * h (k x y)))"
   634   apply (rule set_minus_imp_plus)
   635   apply (subst fun_diff_def)
   636   apply (subst setsum_subtractf [symmetric])
   637   apply (subst right_diff_distrib [symmetric])
   638   apply (rule bigo_setsum3)
   639   apply (subst fun_diff_def [symmetric])
   640   apply (erule set_plus_imp_minus)
   641   done
   642 
   643 lemma bigo_setsum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
   644     \<forall>x. 0 \<le> h x \<Longrightarrow>
   645       (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
   646         O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
   647   apply (subgoal_tac "(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y)) =
   648       (\<lambda>x. \<Sum>y \<in> A x. abs (l x y * h (k x y)))")
   649   apply (erule ssubst)
   650   apply (erule bigo_setsum3)
   651   apply (rule ext)
   652   apply (rule setsum_cong2)
   653   apply (subst abs_of_nonneg)
   654   apply (rule mult_nonneg_nonneg)
   655   apply auto
   656   done
   657 
   658 lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
   659     \<forall>x. 0 \<le> h x \<Longrightarrow>
   660       (\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o
   661         (\<lambda>x. \<Sum>y \<in> A x. l x y * g (k x y)) +o
   662           O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
   663   apply (rule set_minus_imp_plus)
   664   apply (subst fun_diff_def)
   665   apply (subst setsum_subtractf [symmetric])
   666   apply (subst right_diff_distrib [symmetric])
   667   apply (rule bigo_setsum5)
   668   apply (subst fun_diff_def [symmetric])
   669   apply (drule set_plus_imp_minus)
   670   apply auto
   671   done
   672 
   673 
   674 subsection {* Misc useful stuff *}
   675 
   676 lemma bigo_useful_intro: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
   677   apply (subst bigo_plus_idemp [symmetric])
   678   apply (rule set_plus_mono2)
   679   apply assumption+
   680   done
   681 
   682 lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)"
   683   apply (subst bigo_plus_idemp [symmetric])
   684   apply (rule set_plus_intro)
   685   apply assumption+
   686   done
   687 
   688 lemma bigo_useful_const_mult:
   689   fixes c :: "'a::linordered_field"
   690   shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
   691   apply (rule subsetD)
   692   apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) \<subseteq> O(h)")
   693   apply assumption
   694   apply (rule bigo_const_mult6)
   695   apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
   696   apply (erule ssubst)
   697   apply (erule set_times_intro2)
   698   apply (simp add: func_times)
   699   done
   700 
   701 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)"
   702   apply (simp add: bigo_alt_def)
   703   apply auto
   704   apply (rule_tac x = c in exI)
   705   apply auto
   706   apply (case_tac "x = 0")
   707   apply simp
   708   apply (rule mult_nonneg_nonneg)
   709   apply force
   710   apply force
   711   apply (subgoal_tac "x = Suc (x - 1)")
   712   apply (erule ssubst) back
   713   apply (erule spec)
   714   apply simp
   715   done
   716 
   717 lemma bigo_fix2:
   718     "(\<lambda>x. f ((x::nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
   719        f 0 = g 0 \<Longrightarrow> f =o g +o O(h)"
   720   apply (rule set_minus_imp_plus)
   721   apply (rule bigo_fix)
   722   apply (subst fun_diff_def)
   723   apply (subst fun_diff_def [symmetric])
   724   apply (rule set_plus_imp_minus)
   725   apply simp
   726   apply (simp add: fun_diff_def)
   727   done
   728 
   729 
   730 subsection {* Less than or equal to *}
   731 
   732 definition lesso :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"  (infixl "<o" 70)
   733   where "f <o g = (\<lambda>x. max (f x - g x) 0)"
   734 
   735 lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) \<le> abs (f x) \<Longrightarrow> g =o O(h)"
   736   apply (unfold bigo_def)
   737   apply clarsimp
   738   apply (rule_tac x = c in exI)
   739   apply (rule allI)
   740   apply (rule order_trans)
   741   apply (erule spec)+
   742   done
   743 
   744 lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) \<le> f x \<Longrightarrow> g =o O(h)"
   745   apply (erule bigo_lesseq1)
   746   apply (rule allI)
   747   apply (drule_tac x = x in spec)
   748   apply (rule order_trans)
   749   apply assumption
   750   apply (rule abs_ge_self)
   751   done
   752 
   753 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)"
   754   apply (erule bigo_lesseq2)
   755   apply (rule allI)
   756   apply (subst abs_of_nonneg)
   757   apply (erule spec)+
   758   done
   759 
   760 lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
   761     \<forall>x. 0 \<le> g x \<Longrightarrow> \<forall>x. g x \<le> abs (f x) \<Longrightarrow> g =o O(h)"
   762   apply (erule bigo_lesseq1)
   763   apply (rule allI)
   764   apply (subst abs_of_nonneg)
   765   apply (erule spec)+
   766   done
   767 
   768 lemma bigo_lesso1: "\<forall>x. f x \<le> g x \<Longrightarrow> f <o g =o O(h)"
   769   apply (unfold lesso_def)
   770   apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
   771   apply (erule ssubst)
   772   apply (rule bigo_zero)
   773   apply (unfold func_zero)
   774   apply (rule ext)
   775   apply (simp split: split_max)
   776   done
   777 
   778 lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow>
   779     \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. k x \<le> f x \<Longrightarrow> k <o g =o O(h)"
   780   apply (unfold lesso_def)
   781   apply (rule bigo_lesseq4)
   782   apply (erule set_plus_imp_minus)
   783   apply (rule allI)
   784   apply (rule max.cobounded2)
   785   apply (rule allI)
   786   apply (subst fun_diff_def)
   787   apply (case_tac "0 \<le> k x - g x")
   788   apply simp
   789   apply (subst abs_of_nonneg)
   790   apply (drule_tac x = x in spec) back
   791   apply (simp add: algebra_simps)
   792   apply (subst diff_conv_add_uminus)+
   793   apply (rule add_right_mono)
   794   apply (erule spec)
   795   apply (rule order_trans)
   796   prefer 2
   797   apply (rule abs_ge_zero)
   798   apply (simp add: algebra_simps)
   799   done
   800 
   801 lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow>
   802     \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. g x \<le> k x \<Longrightarrow> f <o k =o O(h)"
   803   apply (unfold lesso_def)
   804   apply (rule bigo_lesseq4)
   805   apply (erule set_plus_imp_minus)
   806   apply (rule allI)
   807   apply (rule max.cobounded2)
   808   apply (rule allI)
   809   apply (subst fun_diff_def)
   810   apply (case_tac "0 \<le> f x - k x")
   811   apply simp
   812   apply (subst abs_of_nonneg)
   813   apply (drule_tac x = x in spec) back
   814   apply (simp add: algebra_simps)
   815   apply (subst diff_conv_add_uminus)+
   816   apply (rule add_left_mono)
   817   apply (rule le_imp_neg_le)
   818   apply (erule spec)
   819   apply (rule order_trans)
   820   prefer 2
   821   apply (rule abs_ge_zero)
   822   apply (simp add: algebra_simps)
   823   done
   824 
   825 lemma bigo_lesso4:
   826   fixes k :: "'a \<Rightarrow> 'b::linordered_field"
   827   shows "f <o g =o O(k) \<Longrightarrow> g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
   828   apply (unfold lesso_def)
   829   apply (drule set_plus_imp_minus)
   830   apply (drule bigo_abs5) back
   831   apply (simp add: fun_diff_def)
   832   apply (drule bigo_useful_add)
   833   apply assumption
   834   apply (erule bigo_lesseq2) back
   835   apply (rule allI)
   836   apply (auto simp add: func_plus fun_diff_def algebra_simps split: split_max abs_split)
   837   done
   838 
   839 lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x \<le> g x + C * abs (h x)"
   840   apply (simp only: lesso_def bigo_alt_def)
   841   apply clarsimp
   842   apply (rule_tac x = c in exI)
   843   apply (rule allI)
   844   apply (drule_tac x = x in spec)
   845   apply (subgoal_tac "abs (max (f x - g x) 0) = max (f x - g x) 0")
   846   apply (clarsimp simp add: algebra_simps)
   847   apply (rule abs_of_nonneg)
   848   apply (rule max.cobounded2)
   849   done
   850 
   851 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)"
   852   apply (unfold lesso_def)
   853   apply (rule bigo_lesseq3)
   854   apply (erule bigo_useful_add)
   855   apply assumption
   856   apply (force split: split_max)
   857   apply (auto split: split_max simp add: func_plus)
   858   done
   859 
   860 lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g ----> 0 \<Longrightarrow> f ----> (0::real)"
   861   apply (simp add: LIMSEQ_iff bigo_alt_def)
   862   apply clarify
   863   apply (drule_tac x = "r / c" in spec)
   864   apply (drule mp)
   865   apply (erule divide_pos_pos)
   866   apply assumption
   867   apply clarify
   868   apply (rule_tac x = no in exI)
   869   apply (rule allI)
   870   apply (drule_tac x = n in spec)+
   871   apply (rule impI)
   872   apply (drule mp)
   873   apply assumption
   874   apply (rule order_le_less_trans)
   875   apply assumption
   876   apply (rule order_less_le_trans)
   877   apply (subgoal_tac "c * abs (g n) < c * (r / c)")
   878   apply assumption
   879   apply (erule mult_strict_left_mono)
   880   apply assumption
   881   apply simp
   882   done
   883 
   884 lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h ----> 0 \<Longrightarrow> f ----> a \<Longrightarrow> g ----> (a::real)"
   885   apply (drule set_plus_imp_minus)
   886   apply (drule bigo_LIMSEQ1)
   887   apply assumption
   888   apply (simp only: fun_diff_def)
   889   apply (erule LIMSEQ_diff_approach_zero2)
   890   apply assumption
   891   done
   892 
   893 end