src/HOL/Library/BigO.thy
author wenzelm
Wed Sep 12 13:42:28 2012 +0200 (2012-09-12)
changeset 49322 fbb320d02420
parent 47445 69e96e5500df
child 54230 b1d955791529
permissions -rw-r--r--
tuned headers;
     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    22 \item Lemmas involving @{text "sumr"} have been replaced by more general lemmas 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
    38   bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set"  ("(1O'(_'))") where
    39   "O(f::('a => 'b)) =
    40       {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
    41 
    42 lemma bigo_pos_const: "(EX (c::'a::linordered_idom). 
    43     ALL x. (abs (h x)) <= (c * (abs (f x))))
    44       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
    45   apply auto
    46   apply (case_tac "c = 0")
    47   apply simp
    48   apply (rule_tac x = "1" in exI)
    49   apply simp
    50   apply (rule_tac x = "abs c" in exI)
    51   apply auto
    52   apply (subgoal_tac "c * abs(f x) <= abs c * abs (f x)")
    53   apply (erule_tac x = x in allE)
    54   apply force
    55   apply (rule mult_right_mono)
    56   apply (rule abs_ge_self)
    57   apply (rule abs_ge_zero)
    58   done
    59 
    60 lemma bigo_alt_def: "O(f) = 
    61     {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
    62   by (auto simp add: bigo_def bigo_pos_const)
    63 
    64 lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
    65   apply (auto simp add: bigo_alt_def)
    66   apply (rule_tac x = "ca * c" in exI)
    67   apply (rule conjI)
    68   apply (rule mult_pos_pos)
    69   apply (assumption)+
    70   apply (rule allI)
    71   apply (drule_tac x = "xa" in spec)+
    72   apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))")
    73   apply (erule order_trans)
    74   apply (simp add: mult_ac)
    75   apply (rule mult_left_mono, assumption)
    76   apply (rule order_less_imp_le, assumption)
    77   done
    78 
    79 lemma bigo_refl [intro]: "f : O(f)"
    80   apply(auto simp add: bigo_def)
    81   apply(rule_tac x = 1 in exI)
    82   apply simp
    83   done
    84 
    85 lemma bigo_zero: "0 : O(g)"
    86   apply (auto simp add: bigo_def func_zero)
    87   apply (rule_tac x = 0 in exI)
    88   apply auto
    89   done
    90 
    91 lemma bigo_zero2: "O(%x.0) = {%x.0}"
    92   by (auto simp add: bigo_def) 
    93 
    94 lemma bigo_plus_self_subset [intro]: 
    95   "O(f) + O(f) <= O(f)"
    96   apply (auto simp add: bigo_alt_def set_plus_def)
    97   apply (rule_tac x = "c + ca" in exI)
    98   apply auto
    99   apply (simp add: ring_distribs func_plus)
   100   apply (rule order_trans)
   101   apply (rule abs_triangle_ineq)
   102   apply (rule add_mono)
   103   apply force
   104   apply force
   105 done
   106 
   107 lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
   108   apply (rule equalityI)
   109   apply (rule bigo_plus_self_subset)
   110   apply (rule set_zero_plus2) 
   111   apply (rule bigo_zero)
   112   done
   113 
   114 lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
   115   apply (rule subsetI)
   116   apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
   117   apply (subst bigo_pos_const [symmetric])+
   118   apply (rule_tac x = 
   119     "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
   120   apply (rule conjI)
   121   apply (rule_tac x = "c + c" in exI)
   122   apply (clarsimp)
   123   apply (auto)
   124   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
   125   apply (erule_tac x = xa in allE)
   126   apply (erule order_trans)
   127   apply (simp)
   128   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   129   apply (erule order_trans)
   130   apply (simp add: ring_distribs)
   131   apply (rule mult_left_mono)
   132   apply (simp add: abs_triangle_ineq)
   133   apply (simp add: order_less_le)
   134   apply (rule mult_nonneg_nonneg)
   135   apply auto
   136   apply (rule_tac x = "%n. if (abs (f n)) <  abs (g n) then x n else 0" 
   137      in exI)
   138   apply (rule conjI)
   139   apply (rule_tac x = "c + c" in exI)
   140   apply auto
   141   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
   142   apply (erule_tac x = xa in allE)
   143   apply (erule order_trans)
   144   apply (simp)
   145   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   146   apply (erule order_trans)
   147   apply (simp add: ring_distribs)
   148   apply (rule mult_left_mono)
   149   apply (rule abs_triangle_ineq)
   150   apply (simp add: order_less_le)
   151   apply (rule mult_nonneg_nonneg)
   152   apply (erule order_less_imp_le)
   153   apply simp
   154   done
   155 
   156 lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A + B <= O(f)"
   157   apply (subgoal_tac "A + B <= O(f) + O(f)")
   158   apply (erule order_trans)
   159   apply simp
   160   apply (auto del: subsetI simp del: bigo_plus_idemp)
   161   done
   162 
   163 lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==> 
   164     O(f + g) = O(f) + O(g)"
   165   apply (rule equalityI)
   166   apply (rule bigo_plus_subset)
   167   apply (simp add: bigo_alt_def set_plus_def func_plus)
   168   apply clarify
   169   apply (rule_tac x = "max c ca" in exI)
   170   apply (rule conjI)
   171   apply (subgoal_tac "c <= max c ca")
   172   apply (erule order_less_le_trans)
   173   apply assumption
   174   apply (rule le_maxI1)
   175   apply clarify
   176   apply (drule_tac x = "xa" in spec)+
   177   apply (subgoal_tac "0 <= f xa + g xa")
   178   apply (simp add: ring_distribs)
   179   apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
   180   apply (subgoal_tac "abs(a xa) + abs(b xa) <= 
   181       max c ca * f xa + max c ca * g xa")
   182   apply (force)
   183   apply (rule add_mono)
   184   apply (subgoal_tac "c * f xa <= max c ca * f xa")
   185   apply (force)
   186   apply (rule mult_right_mono)
   187   apply (rule le_maxI1)
   188   apply assumption
   189   apply (subgoal_tac "ca * g xa <= max c ca * g xa")
   190   apply (force)
   191   apply (rule mult_right_mono)
   192   apply (rule le_maxI2)
   193   apply assumption
   194   apply (rule abs_triangle_ineq)
   195   apply (rule add_nonneg_nonneg)
   196   apply assumption+
   197   done
   198 
   199 lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
   200     f : O(g)" 
   201   apply (auto simp add: bigo_def)
   202   apply (rule_tac x = "abs c" in exI)
   203   apply auto
   204   apply (drule_tac x = x in spec)+
   205   apply (simp add: abs_mult [symmetric])
   206   done
   207 
   208 lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> 
   209     f : O(g)" 
   210   apply (erule bigo_bounded_alt [of f 1 g])
   211   apply simp
   212   done
   213 
   214 lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
   215     f : lb +o O(g)"
   216   apply (rule set_minus_imp_plus)
   217   apply (rule bigo_bounded)
   218   apply (auto simp add: diff_minus fun_Compl_def func_plus)
   219   apply (drule_tac x = x in spec)+
   220   apply force
   221   apply (drule_tac x = x in spec)+
   222   apply force
   223   done
   224 
   225 lemma bigo_abs: "(%x. abs(f x)) =o O(f)" 
   226   apply (unfold bigo_def)
   227   apply auto
   228   apply (rule_tac x = 1 in exI)
   229   apply auto
   230   done
   231 
   232 lemma bigo_abs2: "f =o O(%x. abs(f x))"
   233   apply (unfold bigo_def)
   234   apply auto
   235   apply (rule_tac x = 1 in exI)
   236   apply auto
   237   done
   238 
   239 lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
   240   apply (rule equalityI)
   241   apply (rule bigo_elt_subset)
   242   apply (rule bigo_abs2)
   243   apply (rule bigo_elt_subset)
   244   apply (rule bigo_abs)
   245   done
   246 
   247 lemma bigo_abs4: "f =o g +o O(h) ==> 
   248     (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
   249   apply (drule set_plus_imp_minus)
   250   apply (rule set_minus_imp_plus)
   251   apply (subst fun_diff_def)
   252 proof -
   253   assume a: "f - g : O(h)"
   254   have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
   255     by (rule bigo_abs2)
   256   also have "... <= O(%x. abs (f x - g x))"
   257     apply (rule bigo_elt_subset)
   258     apply (rule bigo_bounded)
   259     apply force
   260     apply (rule allI)
   261     apply (rule abs_triangle_ineq3)
   262     done
   263   also have "... <= O(f - g)"
   264     apply (rule bigo_elt_subset)
   265     apply (subst fun_diff_def)
   266     apply (rule bigo_abs)
   267     done
   268   also from a have "... <= O(h)"
   269     by (rule bigo_elt_subset)
   270   finally show "(%x. abs (f x) - abs (g x)) : O(h)".
   271 qed
   272 
   273 lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" 
   274   by (unfold bigo_def, auto)
   275 
   276 lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) + O(h)"
   277 proof -
   278   assume "f : g +o O(h)"
   279   also have "... <= O(g) + O(h)"
   280     by (auto del: subsetI)
   281   also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
   282     apply (subst bigo_abs3 [symmetric])+
   283     apply (rule refl)
   284     done
   285   also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
   286     by (rule bigo_plus_eq [symmetric], auto)
   287   finally have "f : ...".
   288   then have "O(f) <= ..."
   289     by (elim bigo_elt_subset)
   290   also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
   291     by (rule bigo_plus_eq, auto)
   292   finally show ?thesis
   293     by (simp add: bigo_abs3 [symmetric])
   294 qed
   295 
   296 lemma bigo_mult [intro]: "O(f)*O(g) <= O(f * g)"
   297   apply (rule subsetI)
   298   apply (subst bigo_def)
   299   apply (auto simp add: bigo_alt_def set_times_def func_times)
   300   apply (rule_tac x = "c * ca" in exI)
   301   apply(rule allI)
   302   apply(erule_tac x = x in allE)+
   303   apply(subgoal_tac "c * ca * abs(f x * g x) = 
   304       (c * abs(f x)) * (ca * abs(g x))")
   305   apply(erule ssubst)
   306   apply (subst abs_mult)
   307   apply (rule mult_mono)
   308   apply assumption+
   309   apply (rule mult_nonneg_nonneg)
   310   apply auto
   311   apply (simp add: mult_ac abs_mult)
   312   done
   313 
   314 lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
   315   apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
   316   apply (rule_tac x = c in exI)
   317   apply auto
   318   apply (drule_tac x = x in spec)
   319   apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
   320   apply (force simp add: mult_ac)
   321   apply (rule mult_left_mono, assumption)
   322   apply (rule abs_ge_zero)
   323   done
   324 
   325 lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
   326   apply (rule subsetD)
   327   apply (rule bigo_mult)
   328   apply (erule set_times_intro, assumption)
   329   done
   330 
   331 lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
   332   apply (drule set_plus_imp_minus)
   333   apply (rule set_minus_imp_plus)
   334   apply (drule bigo_mult3 [where g = g and j = g])
   335   apply (auto simp add: algebra_simps)
   336   done
   337 
   338 lemma bigo_mult5:
   339   assumes "ALL x. f x ~= 0"
   340   shows "O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
   341 proof
   342   fix h
   343   assume "h : O(f * g)"
   344   then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
   345     by auto
   346   also have "... <= O((%x. 1 / f x) * (f * g))"
   347     by (rule bigo_mult2)
   348   also have "(%x. 1 / f x) * (f * g) = g"
   349     apply (simp add: func_times) 
   350     apply (rule ext)
   351     apply (simp add: assms nonzero_divide_eq_eq mult_ac)
   352     done
   353   finally have "(%x. (1::'b) / f x) * h : O(g)" .
   354   then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
   355     by auto
   356   also have "f * ((%x. (1::'b) / f x) * h) = h"
   357     apply (simp add: func_times) 
   358     apply (rule ext)
   359     apply (simp add: assms nonzero_divide_eq_eq mult_ac)
   360     done
   361   finally show "h : f *o O(g)" .
   362 qed
   363 
   364 lemma bigo_mult6: "ALL x. f x ~= 0 ==>
   365     O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)"
   366   apply (rule equalityI)
   367   apply (erule bigo_mult5)
   368   apply (rule bigo_mult2)
   369   done
   370 
   371 lemma bigo_mult7: "ALL x. f x ~= 0 ==>
   372     O(f * g) <= O(f::'a => ('b::linordered_field)) * O(g)"
   373   apply (subst bigo_mult6)
   374   apply assumption
   375   apply (rule set_times_mono3)
   376   apply (rule bigo_refl)
   377   done
   378 
   379 lemma bigo_mult8: "ALL x. f x ~= 0 ==>
   380     O(f * g) = O(f::'a => ('b::linordered_field)) * O(g)"
   381   apply (rule equalityI)
   382   apply (erule bigo_mult7)
   383   apply (rule bigo_mult)
   384   done
   385 
   386 lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
   387   by (auto simp add: bigo_def fun_Compl_def)
   388 
   389 lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
   390   apply (rule set_minus_imp_plus)
   391   apply (drule set_plus_imp_minus)
   392   apply (drule bigo_minus)
   393   apply (simp add: diff_minus)
   394   done
   395 
   396 lemma bigo_minus3: "O(-f) = O(f)"
   397   by (auto simp add: bigo_def fun_Compl_def)
   398 
   399 lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
   400 proof -
   401   assume a: "f : O(g)"
   402   show "f +o O(g) <= O(g)"
   403   proof -
   404     have "f : O(f)" by auto
   405     then have "f +o O(g) <= O(f) + O(g)"
   406       by (auto del: subsetI)
   407     also have "... <= O(g) + O(g)"
   408     proof -
   409       from a have "O(f) <= O(g)" by (auto del: subsetI)
   410       thus ?thesis by (auto del: subsetI)
   411     qed
   412     also have "... <= O(g)" by simp
   413     finally show ?thesis .
   414   qed
   415 qed
   416 
   417 lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
   418 proof -
   419   assume a: "f : O(g)"
   420   show "O(g) <= f +o O(g)"
   421   proof -
   422     from a have "-f : O(g)" by auto
   423     then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1)
   424     then have "f +o (-f +o O(g)) <= f +o O(g)" by auto
   425     also have "f +o (-f +o O(g)) = O(g)"
   426       by (simp add: set_plus_rearranges)
   427     finally show ?thesis .
   428   qed
   429 qed
   430 
   431 lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
   432   apply (rule equalityI)
   433   apply (erule bigo_plus_absorb_lemma1)
   434   apply (erule bigo_plus_absorb_lemma2)
   435   done
   436 
   437 lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
   438   apply (subgoal_tac "f +o A <= f +o O(g)")
   439   apply force+
   440   done
   441 
   442 lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
   443   apply (subst set_minus_plus [symmetric])
   444   apply (subgoal_tac "g - f = - (f - g)")
   445   apply (erule ssubst)
   446   apply (rule bigo_minus)
   447   apply (subst set_minus_plus)
   448   apply assumption
   449   apply  (simp add: diff_minus add_ac)
   450   done
   451 
   452 lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
   453   apply (rule iffI)
   454   apply (erule bigo_add_commute_imp)+
   455   done
   456 
   457 lemma bigo_const1: "(%x. c) : O(%x. 1)"
   458   by (auto simp add: bigo_def mult_ac)
   459 
   460 lemma bigo_const2 [intro]: "O(%x. c) <= O(%x. 1)"
   461   apply (rule bigo_elt_subset)
   462   apply (rule bigo_const1)
   463   done
   464 
   465 lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
   466   apply (simp add: bigo_def)
   467   apply (rule_tac x = "abs(inverse c)" in exI)
   468   apply (simp add: abs_mult [symmetric])
   469   done
   470 
   471 lemma bigo_const4: "(c::'a::linordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
   472   by (rule bigo_elt_subset, rule bigo_const3, assumption)
   473 
   474 lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ==> 
   475     O(%x. c) = O(%x. 1)"
   476   by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
   477 
   478 lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
   479   apply (simp add: bigo_def)
   480   apply (rule_tac x = "abs(c)" in exI)
   481   apply (auto simp add: abs_mult [symmetric])
   482   done
   483 
   484 lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
   485   by (rule bigo_elt_subset, rule bigo_const_mult1)
   486 
   487 lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%x. c * f x)"
   488   apply (simp add: bigo_def)
   489   apply (rule_tac x = "abs(inverse c)" in exI)
   490   apply (simp add: abs_mult [symmetric] mult_assoc [symmetric])
   491   done
   492 
   493 lemma bigo_const_mult4: "(c::'a::linordered_field) ~= 0 ==> 
   494     O(f) <= O(%x. c * f x)"
   495   by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
   496 
   497 lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ==> 
   498     O(%x. c * f x) = O(f)"
   499   by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
   500 
   501 lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==> 
   502     (%x. c) *o O(f) = O(f)"
   503   apply (auto del: subsetI)
   504   apply (rule order_trans)
   505   apply (rule bigo_mult2)
   506   apply (simp add: func_times)
   507   apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
   508   apply (rule_tac x = "%y. inverse c * x y" in exI)
   509   apply (simp add: mult_assoc [symmetric] abs_mult)
   510   apply (rule_tac x = "abs (inverse c) * ca" in exI)
   511   apply (rule allI)
   512   apply (subst mult_assoc)
   513   apply (rule mult_left_mono)
   514   apply (erule spec)
   515   apply force
   516   done
   517 
   518 lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
   519   apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
   520   apply (rule_tac x = "ca * (abs c)" in exI)
   521   apply (rule allI)
   522   apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
   523   apply (erule ssubst)
   524   apply (subst abs_mult)
   525   apply (rule mult_left_mono)
   526   apply (erule spec)
   527   apply simp
   528   apply(simp add: mult_ac)
   529   done
   530 
   531 lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
   532 proof -
   533   assume "f =o O(g)"
   534   then have "(%x. c) * f =o (%x. c) *o O(g)"
   535     by auto
   536   also have "(%x. c) * f = (%x. c * f x)"
   537     by (simp add: func_times)
   538   also have "(%x. c) *o O(g) <= O(g)"
   539     by (auto del: subsetI)
   540   finally show ?thesis .
   541 qed
   542 
   543 lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
   544 by (unfold bigo_def, auto)
   545 
   546 lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o 
   547     O(%x. h(k x))"
   548   apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def
   549       func_plus)
   550   apply (erule bigo_compose1)
   551 done
   552 
   553 
   554 subsection {* Setsum *}
   555 
   556 lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==> 
   557     EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
   558       (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"  
   559   apply (auto simp add: bigo_def)
   560   apply (rule_tac x = "abs c" in exI)
   561   apply (subst abs_of_nonneg) back back
   562   apply (rule setsum_nonneg)
   563   apply force
   564   apply (subst setsum_right_distrib)
   565   apply (rule allI)
   566   apply (rule order_trans)
   567   apply (rule setsum_abs)
   568   apply (rule setsum_mono)
   569   apply (rule order_trans)
   570   apply (drule spec)+
   571   apply (drule bspec)+
   572   apply assumption+
   573   apply (drule bspec)
   574   apply assumption+
   575   apply (rule mult_right_mono) 
   576   apply (rule abs_ge_self)
   577   apply force
   578   done
   579 
   580 lemma bigo_setsum1: "ALL x y. 0 <= h x y ==> 
   581     EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
   582       (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
   583   apply (rule bigo_setsum_main)
   584   apply force
   585   apply clarsimp
   586   apply (rule_tac x = c in exI)
   587   apply force
   588   done
   589 
   590 lemma bigo_setsum2: "ALL y. 0 <= h y ==> 
   591     EX c. ALL y. abs(f y) <= c * (h y) ==>
   592       (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
   593   by (rule bigo_setsum1, auto)  
   594 
   595 lemma bigo_setsum3: "f =o O(h) ==>
   596     (%x. SUM y : A x. (l x y) * f(k x y)) =o
   597       O(%x. SUM y : A x. abs(l x y * h(k x y)))"
   598   apply (rule bigo_setsum1)
   599   apply (rule allI)+
   600   apply (rule abs_ge_zero)
   601   apply (unfold bigo_def)
   602   apply auto
   603   apply (rule_tac x = c in exI)
   604   apply (rule allI)+
   605   apply (subst abs_mult)+
   606   apply (subst mult_left_commute)
   607   apply (rule mult_left_mono)
   608   apply (erule spec)
   609   apply (rule abs_ge_zero)
   610   done
   611 
   612 lemma bigo_setsum4: "f =o g +o O(h) ==>
   613     (%x. SUM y : A x. l x y * f(k x y)) =o
   614       (%x. SUM y : A x. l x y * g(k x y)) +o
   615         O(%x. SUM y : A x. abs(l x y * h(k x y)))"
   616   apply (rule set_minus_imp_plus)
   617   apply (subst fun_diff_def)
   618   apply (subst setsum_subtractf [symmetric])
   619   apply (subst right_diff_distrib [symmetric])
   620   apply (rule bigo_setsum3)
   621   apply (subst fun_diff_def [symmetric])
   622   apply (erule set_plus_imp_minus)
   623   done
   624 
   625 lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==> 
   626     ALL x. 0 <= h x ==>
   627       (%x. SUM y : A x. (l x y) * f(k x y)) =o
   628         O(%x. SUM y : A x. (l x y) * h(k x y))" 
   629   apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) = 
   630       (%x. SUM y : A x. abs((l x y) * h(k x y)))")
   631   apply (erule ssubst)
   632   apply (erule bigo_setsum3)
   633   apply (rule ext)
   634   apply (rule setsum_cong2)
   635   apply (subst abs_of_nonneg)
   636   apply (rule mult_nonneg_nonneg)
   637   apply auto
   638   done
   639 
   640 lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
   641     ALL x. 0 <= h x ==>
   642       (%x. SUM y : A x. (l x y) * f(k x y)) =o
   643         (%x. SUM y : A x. (l x y) * g(k x y)) +o
   644           O(%x. SUM y : A x. (l x y) * h(k x y))" 
   645   apply (rule set_minus_imp_plus)
   646   apply (subst fun_diff_def)
   647   apply (subst setsum_subtractf [symmetric])
   648   apply (subst right_diff_distrib [symmetric])
   649   apply (rule bigo_setsum5)
   650   apply (subst fun_diff_def [symmetric])
   651   apply (drule set_plus_imp_minus)
   652   apply auto
   653   done
   654 
   655 
   656 subsection {* Misc useful stuff *}
   657 
   658 lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
   659   A + B <= O(f)"
   660   apply (subst bigo_plus_idemp [symmetric])
   661   apply (rule set_plus_mono2)
   662   apply assumption+
   663   done
   664 
   665 lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
   666   apply (subst bigo_plus_idemp [symmetric])
   667   apply (rule set_plus_intro)
   668   apply assumption+
   669   done
   670   
   671 lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 ==> 
   672     (%x. c) * f =o O(h) ==> f =o O(h)"
   673   apply (rule subsetD)
   674   apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
   675   apply assumption
   676   apply (rule bigo_const_mult6)
   677   apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
   678   apply (erule ssubst)
   679   apply (erule set_times_intro2)
   680   apply (simp add: func_times)
   681   done
   682 
   683 lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
   684     f =o O(h)"
   685   apply (simp add: bigo_alt_def)
   686   apply auto
   687   apply (rule_tac x = c in exI)
   688   apply auto
   689   apply (case_tac "x = 0")
   690   apply simp
   691   apply (rule mult_nonneg_nonneg)
   692   apply force
   693   apply force
   694   apply (subgoal_tac "x = Suc (x - 1)")
   695   apply (erule ssubst) back
   696   apply (erule spec)
   697   apply simp
   698   done
   699 
   700 lemma bigo_fix2: 
   701     "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==> 
   702        f 0 = g 0 ==> f =o g +o O(h)"
   703   apply (rule set_minus_imp_plus)
   704   apply (rule bigo_fix)
   705   apply (subst fun_diff_def)
   706   apply (subst fun_diff_def [symmetric])
   707   apply (rule set_plus_imp_minus)
   708   apply simp
   709   apply (simp add: fun_diff_def)
   710   done
   711 
   712 
   713 subsection {* Less than or equal to *}
   714 
   715 definition
   716   lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)"
   717     (infixl "<o" 70) where
   718   "f <o g = (%x. max (f x - g x) 0)"
   719 
   720 lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
   721     g =o O(h)"
   722   apply (unfold bigo_def)
   723   apply clarsimp
   724   apply (rule_tac x = c in exI)
   725   apply (rule allI)
   726   apply (rule order_trans)
   727   apply (erule spec)+
   728   done
   729 
   730 lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
   731       g =o O(h)"
   732   apply (erule bigo_lesseq1)
   733   apply (rule allI)
   734   apply (drule_tac x = x in spec)
   735   apply (rule order_trans)
   736   apply assumption
   737   apply (rule abs_ge_self)
   738   done
   739 
   740 lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
   741     g =o O(h)"
   742   apply (erule bigo_lesseq2)
   743   apply (rule allI)
   744   apply (subst abs_of_nonneg)
   745   apply (erule spec)+
   746   done
   747 
   748 lemma bigo_lesseq4: "f =o O(h) ==>
   749     ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
   750       g =o O(h)"
   751   apply (erule bigo_lesseq1)
   752   apply (rule allI)
   753   apply (subst abs_of_nonneg)
   754   apply (erule spec)+
   755   done
   756 
   757 lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
   758   apply (unfold lesso_def)
   759   apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
   760   apply (erule ssubst)
   761   apply (rule bigo_zero)
   762   apply (unfold func_zero)
   763   apply (rule ext)
   764   apply (simp split: split_max)
   765   done
   766 
   767 lemma bigo_lesso2: "f =o g +o O(h) ==>
   768     ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
   769       k <o g =o O(h)"
   770   apply (unfold lesso_def)
   771   apply (rule bigo_lesseq4)
   772   apply (erule set_plus_imp_minus)
   773   apply (rule allI)
   774   apply (rule le_maxI2)
   775   apply (rule allI)
   776   apply (subst fun_diff_def)
   777   apply (case_tac "0 <= k x - g x")
   778   apply simp
   779   apply (subst abs_of_nonneg)
   780   apply (drule_tac x = x in spec) back
   781   apply (simp add: algebra_simps)
   782   apply (subst diff_minus)+
   783   apply (rule add_right_mono)
   784   apply (erule spec)
   785   apply (rule order_trans) 
   786   prefer 2
   787   apply (rule abs_ge_zero)
   788   apply (simp add: algebra_simps)
   789   done
   790 
   791 lemma bigo_lesso3: "f =o g +o O(h) ==>
   792     ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
   793       f <o k =o O(h)"
   794   apply (unfold lesso_def)
   795   apply (rule bigo_lesseq4)
   796   apply (erule set_plus_imp_minus)
   797   apply (rule allI)
   798   apply (rule le_maxI2)
   799   apply (rule allI)
   800   apply (subst fun_diff_def)
   801   apply (case_tac "0 <= f x - k x")
   802   apply simp
   803   apply (subst abs_of_nonneg)
   804   apply (drule_tac x = x in spec) back
   805   apply (simp add: algebra_simps)
   806   apply (subst diff_minus)+
   807   apply (rule add_left_mono)
   808   apply (rule le_imp_neg_le)
   809   apply (erule spec)
   810   apply (rule order_trans) 
   811   prefer 2
   812   apply (rule abs_ge_zero)
   813   apply (simp add: algebra_simps)
   814   done
   815 
   816 lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::linordered_field) ==>
   817     g =o h +o O(k) ==> f <o h =o O(k)"
   818   apply (unfold lesso_def)
   819   apply (drule set_plus_imp_minus)
   820   apply (drule bigo_abs5) back
   821   apply (simp add: fun_diff_def)
   822   apply (drule bigo_useful_add)
   823   apply assumption
   824   apply (erule bigo_lesseq2) back
   825   apply (rule allI)
   826   apply (auto simp add: func_plus fun_diff_def algebra_simps
   827     split: split_max abs_split)
   828   done
   829 
   830 lemma bigo_lesso5: "f <o g =o O(h) ==>
   831     EX C. ALL x. f x <= g x + C * abs(h x)"
   832   apply (simp only: lesso_def bigo_alt_def)
   833   apply clarsimp
   834   apply (rule_tac x = c in exI)
   835   apply (rule allI)
   836   apply (drule_tac x = x in spec)
   837   apply (subgoal_tac "abs(max (f x - g x) 0) = max (f x - g x) 0")
   838   apply (clarsimp simp add: algebra_simps) 
   839   apply (rule abs_of_nonneg)
   840   apply (rule le_maxI2)
   841   done
   842 
   843 lemma lesso_add: "f <o g =o O(h) ==>
   844       k <o l =o O(h) ==> (f + k) <o (g + l) =o O(h)"
   845   apply (unfold lesso_def)
   846   apply (rule bigo_lesseq3)
   847   apply (erule bigo_useful_add)
   848   apply assumption
   849   apply (force split: split_max)
   850   apply (auto split: split_max simp add: func_plus)
   851   done
   852 
   853 lemma bigo_LIMSEQ1: "f =o O(g) ==> g ----> 0 ==> f ----> (0::real)"
   854   apply (simp add: LIMSEQ_iff bigo_alt_def)
   855   apply clarify
   856   apply (drule_tac x = "r / c" in spec)
   857   apply (drule mp)
   858   apply (erule divide_pos_pos)
   859   apply assumption
   860   apply clarify
   861   apply (rule_tac x = no in exI)
   862   apply (rule allI)
   863   apply (drule_tac x = n in spec)+
   864   apply (rule impI)
   865   apply (drule mp)
   866   apply assumption
   867   apply (rule order_le_less_trans)
   868   apply assumption
   869   apply (rule order_less_le_trans)
   870   apply (subgoal_tac "c * abs(g n) < c * (r / c)")
   871   apply assumption
   872   apply (erule mult_strict_left_mono)
   873   apply assumption
   874   apply simp
   875 done
   876 
   877 lemma bigo_LIMSEQ2: "f =o g +o O(h) ==> h ----> 0 ==> f ----> a 
   878     ==> g ----> (a::real)"
   879   apply (drule set_plus_imp_minus)
   880   apply (drule bigo_LIMSEQ1)
   881   apply assumption
   882   apply (simp only: fun_diff_def)
   883   apply (erule LIMSEQ_diff_approach_zero2)
   884   apply assumption
   885 done
   886 
   887 end