src/HOL/Library/BigO.thy
author wenzelm
Wed Jun 13 18:30:11 2007 +0200 (2007-06-13)
changeset 23373 ead82c82da9e
parent 22665 cf152ff55d16
child 23413 5caa2710dd5b
permissions -rwxr-xr-x
tuned proofs: avoid implicit prems;
     1 (*  Title:      HOL/Library/BigO.thy
     2     ID:		$Id$
     3     Authors:    Jeremy Avigad and Kevin Donnelly
     4 *)
     5 
     6 header {* Big O notation *}
     7 
     8 theory BigO
     9 imports SetsAndFunctions
    10 begin
    11 
    12 text {*
    13 This library is designed to support asymptotic ``big O'' calculations,
    14 i.e.~reasoning with expressions of the form $f = O(g)$ and $f = g +
    15 O(h)$.  An earlier version of this library is described in detail in
    16 \cite{Avigad-Donnelly}.
    17 
    18 The main changes in this version are as follows:
    19 \begin{itemize}
    20 \item We have eliminated the @{text O} operator on sets. (Most uses of this seem
    21   to be inessential.)
    22 \item We no longer use @{text "+"} as output syntax for @{text "+o"}
    23 \item Lemmas involving @{text "sumr"} have been replaced by more general lemmas 
    24   involving `@{text "setsum"}.
    25 \item The library has been expanded, with e.g.~support for expressions of
    26   the form @{text "f < g + O(h)"}.
    27 \end{itemize}
    28 
    29 See \verb,Complex/ex/BigO_Complex.thy, for additional lemmas that
    30 require the \verb,HOL-Complex, logic image.
    31 
    32 Note also since the Big O library includes rules that demonstrate set
    33 inclusion, to use the automated reasoners effectively with the library
    34 one should redeclare the theorem @{text "subsetI"} as an intro rule,
    35 rather than as an @{text "intro!"} rule, for example, using
    36 \isa{\isakeyword{declare}}~@{text "subsetI [del, intro]"}.
    37 *}
    38 
    39 subsection {* Definitions *}
    40 
    41 definition
    42   bigo :: "('a => 'b::ordered_idom) => ('a => 'b) set"  ("(1O'(_'))") where
    43   "O(f::('a => 'b)) =
    44       {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
    45 
    46 lemma bigo_pos_const: "(EX (c::'a::ordered_idom). 
    47     ALL x. (abs (h x)) <= (c * (abs (f x))))
    48       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
    49   apply auto
    50   apply (case_tac "c = 0")
    51   apply simp
    52   apply (rule_tac x = "1" in exI)
    53   apply simp
    54   apply (rule_tac x = "abs c" in exI)
    55   apply auto
    56   apply (subgoal_tac "c * abs(f x) <= abs c * abs (f x)")
    57   apply (erule_tac x = x in allE)
    58   apply force
    59   apply (rule mult_right_mono)
    60   apply (rule abs_ge_self)
    61   apply (rule abs_ge_zero)
    62   done
    63 
    64 lemma bigo_alt_def: "O(f) = 
    65     {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
    66   by (auto simp add: bigo_def bigo_pos_const)
    67 
    68 lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
    69   apply (auto simp add: bigo_alt_def)
    70   apply (rule_tac x = "ca * c" in exI)
    71   apply (rule conjI)
    72   apply (rule mult_pos_pos)
    73   apply (assumption)+
    74   apply (rule allI)
    75   apply (drule_tac x = "xa" in spec)+
    76   apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))")
    77   apply (erule order_trans)
    78   apply (simp add: mult_ac)
    79   apply (rule mult_left_mono, assumption)
    80   apply (rule order_less_imp_le, assumption)
    81   done
    82 
    83 lemma bigo_refl [intro]: "f : O(f)"
    84   apply(auto simp add: bigo_def)
    85   apply(rule_tac x = 1 in exI)
    86   apply simp
    87   done
    88 
    89 lemma bigo_zero: "0 : O(g)"
    90   apply (auto simp add: bigo_def func_zero)
    91   apply (rule_tac x = 0 in exI)
    92   apply auto
    93   done
    94 
    95 lemma bigo_zero2: "O(%x.0) = {%x.0}"
    96   apply (auto simp add: bigo_def) 
    97   apply (rule ext)
    98   apply auto
    99   done
   100 
   101 lemma bigo_plus_self_subset [intro]: 
   102   "O(f) + O(f) <= O(f)"
   103   apply (auto simp add: bigo_alt_def set_plus)
   104   apply (rule_tac x = "c + ca" in exI)
   105   apply auto
   106   apply (simp add: ring_distrib func_plus)
   107   apply (rule order_trans)
   108   apply (rule abs_triangle_ineq)
   109   apply (rule add_mono)
   110   apply force
   111   apply force
   112 done
   113 
   114 lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
   115   apply (rule equalityI)
   116   apply (rule bigo_plus_self_subset)
   117   apply (rule set_zero_plus2) 
   118   apply (rule bigo_zero)
   119   done
   120 
   121 lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
   122   apply (rule subsetI)
   123   apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus)
   124   apply (subst bigo_pos_const [symmetric])+
   125   apply (rule_tac x = 
   126     "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
   127   apply (rule conjI)
   128   apply (rule_tac x = "c + c" in exI)
   129   apply (clarsimp)
   130   apply (auto)
   131   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
   132   apply (erule_tac x = xa in allE)
   133   apply (erule order_trans)
   134   apply (simp)
   135   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   136   apply (erule order_trans)
   137   apply (simp add: ring_distrib)
   138   apply (rule mult_left_mono)
   139   apply assumption
   140   apply (simp add: order_less_le)
   141   apply (rule mult_left_mono)
   142   apply (simp add: abs_triangle_ineq)
   143   apply (simp add: order_less_le)
   144   apply (rule mult_nonneg_nonneg)
   145   apply (rule add_nonneg_nonneg)
   146   apply auto
   147   apply (rule_tac x = "%n. if (abs (f n)) <  abs (g n) then x n else 0" 
   148      in exI)
   149   apply (rule conjI)
   150   apply (rule_tac x = "c + c" in exI)
   151   apply auto
   152   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
   153   apply (erule_tac x = xa in allE)
   154   apply (erule order_trans)
   155   apply (simp)
   156   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   157   apply (erule order_trans)
   158   apply (simp add: ring_distrib)
   159   apply (rule mult_left_mono)
   160   apply (simp add: order_less_le)
   161   apply (simp add: order_less_le)
   162   apply (rule mult_left_mono)
   163   apply (rule abs_triangle_ineq)
   164   apply (simp add: order_less_le)
   165   apply (rule mult_nonneg_nonneg)
   166   apply (rule add_nonneg_nonneg)
   167   apply (erule order_less_imp_le)+
   168   apply simp
   169   apply (rule ext)
   170   apply (auto simp add: if_splits linorder_not_le)
   171   done
   172 
   173 lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A + B <= O(f)"
   174   apply (subgoal_tac "A + B <= O(f) + O(f)")
   175   apply (erule order_trans)
   176   apply simp
   177   apply (auto del: subsetI simp del: bigo_plus_idemp)
   178   done
   179 
   180 lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==> 
   181     O(f + g) = O(f) + O(g)"
   182   apply (rule equalityI)
   183   apply (rule bigo_plus_subset)
   184   apply (simp add: bigo_alt_def set_plus func_plus)
   185   apply clarify
   186   apply (rule_tac x = "max c ca" in exI)
   187   apply (rule conjI)
   188   apply (subgoal_tac "c <= max c ca")
   189   apply (erule order_less_le_trans)
   190   apply assumption
   191   apply (rule le_maxI1)
   192   apply clarify
   193   apply (drule_tac x = "xa" in spec)+
   194   apply (subgoal_tac "0 <= f xa + g xa")
   195   apply (simp add: ring_distrib)
   196   apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
   197   apply (subgoal_tac "abs(a xa) + abs(b xa) <= 
   198       max c ca * f xa + max c ca * g xa")
   199   apply (force)
   200   apply (rule add_mono)
   201   apply (subgoal_tac "c * f xa <= max c ca * f xa")
   202   apply (force)
   203   apply (rule mult_right_mono)
   204   apply (rule le_maxI1)
   205   apply assumption
   206   apply (subgoal_tac "ca * g xa <= max c ca * g xa")
   207   apply (force)
   208   apply (rule mult_right_mono)
   209   apply (rule le_maxI2)
   210   apply assumption
   211   apply (rule abs_triangle_ineq)
   212   apply (rule add_nonneg_nonneg)
   213   apply assumption+
   214   done
   215 
   216 lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
   217     f : O(g)" 
   218   apply (auto simp add: bigo_def)
   219   apply (rule_tac x = "abs c" in exI)
   220   apply auto
   221   apply (drule_tac x = x in spec)+
   222   apply (simp add: abs_mult [symmetric])
   223   done
   224 
   225 lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> 
   226     f : O(g)" 
   227   apply (erule bigo_bounded_alt [of f 1 g])
   228   apply simp
   229   done
   230 
   231 lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
   232     f : lb +o O(g)"
   233   apply (rule set_minus_imp_plus)
   234   apply (rule bigo_bounded)
   235   apply (auto simp add: diff_minus func_minus func_plus)
   236   apply (drule_tac x = x in spec)+
   237   apply force
   238   apply (drule_tac x = x in spec)+
   239   apply force
   240   done
   241 
   242 lemma bigo_abs: "(%x. abs(f x)) =o O(f)" 
   243   apply (unfold bigo_def)
   244   apply auto
   245   apply (rule_tac x = 1 in exI)
   246   apply auto
   247   done
   248 
   249 lemma bigo_abs2: "f =o O(%x. abs(f x))"
   250   apply (unfold bigo_def)
   251   apply auto
   252   apply (rule_tac x = 1 in exI)
   253   apply auto
   254   done
   255 
   256 lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
   257   apply (rule equalityI)
   258   apply (rule bigo_elt_subset)
   259   apply (rule bigo_abs2)
   260   apply (rule bigo_elt_subset)
   261   apply (rule bigo_abs)
   262   done
   263 
   264 lemma bigo_abs4: "f =o g +o O(h) ==> 
   265     (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
   266   apply (drule set_plus_imp_minus)
   267   apply (rule set_minus_imp_plus)
   268   apply (subst func_diff)
   269 proof -
   270   assume a: "f - g : O(h)"
   271   have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
   272     by (rule bigo_abs2)
   273   also have "... <= O(%x. abs (f x - g x))"
   274     apply (rule bigo_elt_subset)
   275     apply (rule bigo_bounded)
   276     apply force
   277     apply (rule allI)
   278     apply (rule abs_triangle_ineq3)
   279     done
   280   also have "... <= O(f - g)"
   281     apply (rule bigo_elt_subset)
   282     apply (subst func_diff)
   283     apply (rule bigo_abs)
   284     done
   285   also from a have "... <= O(h)"
   286     by (rule bigo_elt_subset)
   287   finally show "(%x. abs (f x) - abs (g x)) : O(h)".
   288 qed
   289 
   290 lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" 
   291   by (unfold bigo_def, auto)
   292 
   293 lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) + O(h)"
   294 proof -
   295   assume "f : g +o O(h)"
   296   also have "... <= O(g) + O(h)"
   297     by (auto del: subsetI)
   298   also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
   299     apply (subst bigo_abs3 [symmetric])+
   300     apply (rule refl)
   301     done
   302   also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
   303     by (rule bigo_plus_eq [symmetric], auto)
   304   finally have "f : ...".
   305   then have "O(f) <= ..."
   306     by (elim bigo_elt_subset)
   307   also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
   308     by (rule bigo_plus_eq, auto)
   309   finally show ?thesis
   310     by (simp add: bigo_abs3 [symmetric])
   311 qed
   312 
   313 lemma bigo_mult [intro]: "O(f)*O(g) <= O(f * g)"
   314   apply (rule subsetI)
   315   apply (subst bigo_def)
   316   apply (auto simp add: bigo_alt_def set_times func_times)
   317   apply (rule_tac x = "c * ca" in exI)
   318   apply(rule allI)
   319   apply(erule_tac x = x in allE)+
   320   apply(subgoal_tac "c * ca * abs(f x * g x) = 
   321       (c * abs(f x)) * (ca * abs(g x))")
   322   apply(erule ssubst)
   323   apply (subst abs_mult)
   324   apply (rule mult_mono)
   325   apply assumption+
   326   apply (rule mult_nonneg_nonneg)
   327   apply auto
   328   apply (simp add: mult_ac abs_mult)
   329   done
   330 
   331 lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
   332   apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
   333   apply (rule_tac x = c in exI)
   334   apply auto
   335   apply (drule_tac x = x in spec)
   336   apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
   337   apply (force simp add: mult_ac)
   338   apply (rule mult_left_mono, assumption)
   339   apply (rule abs_ge_zero)
   340   done
   341 
   342 lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
   343   apply (rule subsetD)
   344   apply (rule bigo_mult)
   345   apply (erule set_times_intro, assumption)
   346   done
   347 
   348 lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
   349   apply (drule set_plus_imp_minus)
   350   apply (rule set_minus_imp_plus)
   351   apply (drule bigo_mult3 [where g = g and j = g])
   352   apply (auto simp add: ring_eq_simps mult_ac)
   353   done
   354 
   355 lemma bigo_mult5: "ALL x. f x ~= 0 ==>
   356     O(f * g) <= (f::'a => ('b::ordered_field)) *o O(g)"
   357 proof -
   358   assume "ALL x. f x ~= 0"
   359   show "O(f * g) <= f *o O(g)"
   360   proof
   361     fix h
   362     assume "h : O(f * g)"
   363     then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
   364       by auto
   365     also have "... <= O((%x. 1 / f x) * (f * g))"
   366       by (rule bigo_mult2)
   367     also have "(%x. 1 / f x) * (f * g) = g"
   368       apply (simp add: func_times) 
   369       apply (rule ext)
   370       apply (simp add: prems nonzero_divide_eq_eq mult_ac)
   371       done
   372     finally have "(%x. (1::'b) / f x) * h : O(g)".
   373     then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
   374       by auto
   375     also have "f * ((%x. (1::'b) / f x) * h) = h"
   376       apply (simp add: func_times) 
   377       apply (rule ext)
   378       apply (simp add: prems nonzero_divide_eq_eq mult_ac)
   379       done
   380     finally show "h : f *o O(g)".
   381   qed
   382 qed
   383 
   384 lemma bigo_mult6: "ALL x. f x ~= 0 ==>
   385     O(f * g) = (f::'a => ('b::ordered_field)) *o O(g)"
   386   apply (rule equalityI)
   387   apply (erule bigo_mult5)
   388   apply (rule bigo_mult2)
   389   done
   390 
   391 lemma bigo_mult7: "ALL x. f x ~= 0 ==>
   392     O(f * g) <= O(f::'a => ('b::ordered_field)) * O(g)"
   393   apply (subst bigo_mult6)
   394   apply assumption
   395   apply (rule set_times_mono3)
   396   apply (rule bigo_refl)
   397   done
   398 
   399 lemma bigo_mult8: "ALL x. f x ~= 0 ==>
   400     O(f * g) = O(f::'a => ('b::ordered_field)) * O(g)"
   401   apply (rule equalityI)
   402   apply (erule bigo_mult7)
   403   apply (rule bigo_mult)
   404   done
   405 
   406 lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
   407   by (auto simp add: bigo_def func_minus)
   408 
   409 lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
   410   apply (rule set_minus_imp_plus)
   411   apply (drule set_plus_imp_minus)
   412   apply (drule bigo_minus)
   413   apply (simp add: diff_minus)
   414   done
   415 
   416 lemma bigo_minus3: "O(-f) = O(f)"
   417   by (auto simp add: bigo_def func_minus abs_minus_cancel)
   418 
   419 lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
   420 proof -
   421   assume a: "f : O(g)"
   422   show "f +o O(g) <= O(g)"
   423   proof -
   424     have "f : O(f)" by auto
   425     then have "f +o O(g) <= O(f) + O(g)"
   426       by (auto del: subsetI)
   427     also have "... <= O(g) + O(g)"
   428     proof -
   429       from a have "O(f) <= O(g)" by (auto del: subsetI)
   430       thus ?thesis by (auto del: subsetI)
   431     qed
   432     also have "... <= O(g)" by (simp add: bigo_plus_idemp)
   433     finally show ?thesis .
   434   qed
   435 qed
   436 
   437 lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
   438 proof -
   439   assume a: "f : O(g)"
   440   show "O(g) <= f +o O(g)"
   441   proof -
   442     from a have "-f : O(g)" by auto
   443     then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1)
   444     then have "f +o (-f +o O(g)) <= f +o O(g)" by auto
   445     also have "f +o (-f +o O(g)) = O(g)"
   446       by (simp add: set_plus_rearranges)
   447     finally show ?thesis .
   448   qed
   449 qed
   450 
   451 lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
   452   apply (rule equalityI)
   453   apply (erule bigo_plus_absorb_lemma1)
   454   apply (erule bigo_plus_absorb_lemma2)
   455   done
   456 
   457 lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
   458   apply (subgoal_tac "f +o A <= f +o O(g)")
   459   apply force+
   460   done
   461 
   462 lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
   463   apply (subst set_minus_plus [symmetric])
   464   apply (subgoal_tac "g - f = - (f - g)")
   465   apply (erule ssubst)
   466   apply (rule bigo_minus)
   467   apply (subst set_minus_plus)
   468   apply assumption
   469   apply  (simp add: diff_minus add_ac)
   470   done
   471 
   472 lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
   473   apply (rule iffI)
   474   apply (erule bigo_add_commute_imp)+
   475   done
   476 
   477 lemma bigo_const1: "(%x. c) : O(%x. 1)"
   478   by (auto simp add: bigo_def mult_ac)
   479 
   480 lemma bigo_const2 [intro]: "O(%x. c) <= O(%x. 1)"
   481   apply (rule bigo_elt_subset)
   482   apply (rule bigo_const1)
   483   done
   484 
   485 lemma bigo_const3: "(c::'a::ordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
   486   apply (simp add: bigo_def)
   487   apply (rule_tac x = "abs(inverse c)" in exI)
   488   apply (simp add: abs_mult [symmetric])
   489   done
   490 
   491 lemma bigo_const4: "(c::'a::ordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
   492   by (rule bigo_elt_subset, rule bigo_const3, assumption)
   493 
   494 lemma bigo_const [simp]: "(c::'a::ordered_field) ~= 0 ==> 
   495     O(%x. c) = O(%x. 1)"
   496   by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
   497 
   498 lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
   499   apply (simp add: bigo_def)
   500   apply (rule_tac x = "abs(c)" in exI)
   501   apply (auto simp add: abs_mult [symmetric])
   502   done
   503 
   504 lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
   505   by (rule bigo_elt_subset, rule bigo_const_mult1)
   506 
   507 lemma bigo_const_mult3: "(c::'a::ordered_field) ~= 0 ==> f : O(%x. c * f x)"
   508   apply (simp add: bigo_def)
   509   apply (rule_tac x = "abs(inverse c)" in exI)
   510   apply (simp add: abs_mult [symmetric] mult_assoc [symmetric])
   511   done
   512 
   513 lemma bigo_const_mult4: "(c::'a::ordered_field) ~= 0 ==> 
   514     O(f) <= O(%x. c * f x)"
   515   by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
   516 
   517 lemma bigo_const_mult [simp]: "(c::'a::ordered_field) ~= 0 ==> 
   518     O(%x. c * f x) = O(f)"
   519   by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
   520 
   521 lemma bigo_const_mult5 [simp]: "(c::'a::ordered_field) ~= 0 ==> 
   522     (%x. c) *o O(f) = O(f)"
   523   apply (auto del: subsetI)
   524   apply (rule order_trans)
   525   apply (rule bigo_mult2)
   526   apply (simp add: func_times)
   527   apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times)
   528   apply (rule_tac x = "%y. inverse c * x y" in exI)
   529   apply (simp add: mult_assoc [symmetric] abs_mult)
   530   apply (rule_tac x = "abs (inverse c) * ca" in exI)
   531   apply (rule allI)
   532   apply (subst mult_assoc)
   533   apply (rule mult_left_mono)
   534   apply (erule spec)
   535   apply force
   536   done
   537 
   538 lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
   539   apply (auto intro!: subsetI
   540     simp add: bigo_def elt_set_times_def func_times)
   541   apply (rule_tac x = "ca * (abs c)" in exI)
   542   apply (rule allI)
   543   apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
   544   apply (erule ssubst)
   545   apply (subst abs_mult)
   546   apply (rule mult_left_mono)
   547   apply (erule spec)
   548   apply simp
   549   apply(simp add: mult_ac)
   550   done
   551 
   552 lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
   553 proof -
   554   assume "f =o O(g)"
   555   then have "(%x. c) * f =o (%x. c) *o O(g)"
   556     by auto
   557   also have "(%x. c) * f = (%x. c * f x)"
   558     by (simp add: func_times)
   559   also have "(%x. c) *o O(g) <= O(g)"
   560     by (auto del: subsetI)
   561   finally show ?thesis .
   562 qed
   563 
   564 lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
   565 by (unfold bigo_def, auto)
   566 
   567 lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o 
   568     O(%x. h(k x))"
   569   apply (simp only: set_minus_plus [symmetric] diff_minus func_minus
   570       func_plus)
   571   apply (erule bigo_compose1)
   572 done
   573 
   574 
   575 subsection {* Setsum *}
   576 
   577 lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==> 
   578     EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
   579       (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"  
   580   apply (auto simp add: bigo_def)
   581   apply (rule_tac x = "abs c" in exI)
   582   apply (subst abs_of_nonneg) back back
   583   apply (rule setsum_nonneg)
   584   apply force
   585   apply (subst setsum_right_distrib)
   586   apply (rule allI)
   587   apply (rule order_trans)
   588   apply (rule setsum_abs)
   589   apply (rule setsum_mono)
   590   apply (rule order_trans)
   591   apply (drule spec)+
   592   apply (drule bspec)+
   593   apply assumption+
   594   apply (drule bspec)
   595   apply assumption+
   596   apply (rule mult_right_mono) 
   597   apply (rule abs_ge_self)
   598   apply force
   599   done
   600 
   601 lemma bigo_setsum1: "ALL x y. 0 <= h x y ==> 
   602     EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
   603       (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
   604   apply (rule bigo_setsum_main)
   605   apply force
   606   apply clarsimp
   607   apply (rule_tac x = c in exI)
   608   apply force
   609   done
   610 
   611 lemma bigo_setsum2: "ALL y. 0 <= h y ==> 
   612     EX c. ALL y. abs(f y) <= c * (h y) ==>
   613       (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
   614   by (rule bigo_setsum1, auto)  
   615 
   616 lemma bigo_setsum3: "f =o O(h) ==>
   617     (%x. SUM y : A x. (l x y) * f(k x y)) =o
   618       O(%x. SUM y : A x. abs(l x y * h(k x y)))"
   619   apply (rule bigo_setsum1)
   620   apply (rule allI)+
   621   apply (rule abs_ge_zero)
   622   apply (unfold bigo_def)
   623   apply auto
   624   apply (rule_tac x = c in exI)
   625   apply (rule allI)+
   626   apply (subst abs_mult)+
   627   apply (subst mult_left_commute)
   628   apply (rule mult_left_mono)
   629   apply (erule spec)
   630   apply (rule abs_ge_zero)
   631   done
   632 
   633 lemma bigo_setsum4: "f =o g +o O(h) ==>
   634     (%x. SUM y : A x. l x y * f(k x y)) =o
   635       (%x. SUM y : A x. l x y * g(k x y)) +o
   636         O(%x. SUM y : A x. abs(l x y * h(k x y)))"
   637   apply (rule set_minus_imp_plus)
   638   apply (subst func_diff)
   639   apply (subst setsum_subtractf [symmetric])
   640   apply (subst right_diff_distrib [symmetric])
   641   apply (rule bigo_setsum3)
   642   apply (subst func_diff [symmetric])
   643   apply (erule set_plus_imp_minus)
   644   done
   645 
   646 lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==> 
   647     ALL x. 0 <= h x ==>
   648       (%x. SUM y : A x. (l x y) * f(k x y)) =o
   649         O(%x. SUM y : A x. (l x y) * h(k x y))" 
   650   apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) = 
   651       (%x. SUM y : A x. abs((l x y) * h(k x y)))")
   652   apply (erule ssubst)
   653   apply (erule bigo_setsum3)
   654   apply (rule ext)
   655   apply (rule setsum_cong2)
   656   apply (subst abs_of_nonneg)
   657   apply (rule mult_nonneg_nonneg)
   658   apply auto
   659   done
   660 
   661 lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
   662     ALL x. 0 <= h x ==>
   663       (%x. SUM y : A x. (l x y) * f(k x y)) =o
   664         (%x. SUM y : A x. (l x y) * g(k x y)) +o
   665           O(%x. SUM y : A x. (l x y) * h(k x y))" 
   666   apply (rule set_minus_imp_plus)
   667   apply (subst func_diff)
   668   apply (subst setsum_subtractf [symmetric])
   669   apply (subst right_diff_distrib [symmetric])
   670   apply (rule bigo_setsum5)
   671   apply (subst func_diff [symmetric])
   672   apply (drule set_plus_imp_minus)
   673   apply auto
   674   done
   675 
   676 
   677 subsection {* Misc useful stuff *}
   678 
   679 lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
   680   A + B <= O(f)"
   681   apply (subst bigo_plus_idemp [symmetric])
   682   apply (rule set_plus_mono2)
   683   apply assumption+
   684   done
   685 
   686 lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
   687   apply (subst bigo_plus_idemp [symmetric])
   688   apply (rule set_plus_intro)
   689   apply assumption+
   690   done
   691   
   692 lemma bigo_useful_const_mult: "(c::'a::ordered_field) ~= 0 ==> 
   693     (%x. c) * f =o O(h) ==> f =o O(h)"
   694   apply (rule subsetD)
   695   apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
   696   apply assumption
   697   apply (rule bigo_const_mult6)
   698   apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
   699   apply (erule ssubst)
   700   apply (erule set_times_intro2)
   701   apply (simp add: func_times) 
   702   apply (rule ext)
   703   apply (subst times_divide_eq_left [symmetric])
   704   apply (subst divide_self)
   705   apply (assumption, simp)
   706   done
   707 
   708 lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
   709     f =o O(h)"
   710   apply (simp add: bigo_alt_def)
   711   apply auto
   712   apply (rule_tac x = c in exI)
   713   apply auto
   714   apply (case_tac "x = 0")
   715   apply simp
   716   apply (rule mult_nonneg_nonneg)
   717   apply force
   718   apply force
   719   apply (subgoal_tac "x = Suc (x - 1)")
   720   apply (erule ssubst) back
   721   apply (erule spec)
   722   apply simp
   723   done
   724 
   725 lemma bigo_fix2: 
   726     "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==> 
   727        f 0 = g 0 ==> f =o g +o O(h)"
   728   apply (rule set_minus_imp_plus)
   729   apply (rule bigo_fix)
   730   apply (subst func_diff)
   731   apply (subst func_diff [symmetric])
   732   apply (rule set_plus_imp_minus)
   733   apply simp
   734   apply (simp add: func_diff)
   735   done
   736 
   737 
   738 subsection {* Less than or equal to *}
   739 
   740 definition
   741   lesso :: "('a => 'b::ordered_idom) => ('a => 'b) => ('a => 'b)"
   742     (infixl "<o" 70) where
   743   "f <o g = (%x. max (f x - g x) 0)"
   744 
   745 lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
   746     g =o O(h)"
   747   apply (unfold bigo_def)
   748   apply clarsimp
   749   apply (rule_tac x = c in exI)
   750   apply (rule allI)
   751   apply (rule order_trans)
   752   apply (erule spec)+
   753   done
   754 
   755 lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
   756       g =o O(h)"
   757   apply (erule bigo_lesseq1)
   758   apply (rule allI)
   759   apply (drule_tac x = x in spec)
   760   apply (rule order_trans)
   761   apply assumption
   762   apply (rule abs_ge_self)
   763   done
   764 
   765 lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
   766     g =o O(h)"
   767   apply (erule bigo_lesseq2)
   768   apply (rule allI)
   769   apply (subst abs_of_nonneg)
   770   apply (erule spec)+
   771   done
   772 
   773 lemma bigo_lesseq4: "f =o O(h) ==>
   774     ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
   775       g =o O(h)"
   776   apply (erule bigo_lesseq1)
   777   apply (rule allI)
   778   apply (subst abs_of_nonneg)
   779   apply (erule spec)+
   780   done
   781 
   782 lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
   783   apply (unfold lesso_def)
   784   apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
   785   apply (erule ssubst)
   786   apply (rule bigo_zero)
   787   apply (unfold func_zero)
   788   apply (rule ext)
   789   apply (simp split: split_max)
   790   done
   791 
   792 lemma bigo_lesso2: "f =o g +o O(h) ==>
   793     ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
   794       k <o g =o O(h)"
   795   apply (unfold lesso_def)
   796   apply (rule bigo_lesseq4)
   797   apply (erule set_plus_imp_minus)
   798   apply (rule allI)
   799   apply (rule le_maxI2)
   800   apply (rule allI)
   801   apply (subst func_diff)
   802   apply (case_tac "0 <= k x - g x")
   803   apply simp
   804   apply (subst abs_of_nonneg)
   805   apply (drule_tac x = x in spec) back
   806   apply (simp add: compare_rls)
   807   apply (subst diff_minus)+
   808   apply (rule add_right_mono)
   809   apply (erule spec)
   810   apply (rule order_trans) 
   811   prefer 2
   812   apply (rule abs_ge_zero)
   813   apply (simp add: compare_rls)
   814   done
   815 
   816 lemma bigo_lesso3: "f =o g +o O(h) ==>
   817     ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
   818       f <o k =o O(h)"
   819   apply (unfold lesso_def)
   820   apply (rule bigo_lesseq4)
   821   apply (erule set_plus_imp_minus)
   822   apply (rule allI)
   823   apply (rule le_maxI2)
   824   apply (rule allI)
   825   apply (subst func_diff)
   826   apply (case_tac "0 <= f x - k x")
   827   apply simp
   828   apply (subst abs_of_nonneg)
   829   apply (drule_tac x = x in spec) back
   830   apply (simp add: compare_rls)
   831   apply (subst diff_minus)+
   832   apply (rule add_left_mono)
   833   apply (rule le_imp_neg_le)
   834   apply (erule spec)
   835   apply (rule order_trans) 
   836   prefer 2
   837   apply (rule abs_ge_zero)
   838   apply (simp add: compare_rls)
   839   done
   840 
   841 lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::ordered_field) ==>
   842     g =o h +o O(k) ==> f <o h =o O(k)"
   843   apply (unfold lesso_def)
   844   apply (drule set_plus_imp_minus)
   845   apply (drule bigo_abs5) back
   846   apply (simp add: func_diff)
   847   apply (drule bigo_useful_add)
   848   apply assumption
   849   apply (erule bigo_lesseq2) back
   850   apply (rule allI)
   851   apply (auto simp add: func_plus func_diff compare_rls 
   852     split: split_max abs_split)
   853   done
   854 
   855 lemma bigo_lesso5: "f <o g =o O(h) ==>
   856     EX C. ALL x. f x <= g x + C * abs(h x)"
   857   apply (simp only: lesso_def bigo_alt_def)
   858   apply clarsimp
   859   apply (rule_tac x = c in exI)
   860   apply (rule allI)
   861   apply (drule_tac x = x in spec)
   862   apply (subgoal_tac "abs(max (f x - g x) 0) = max (f x - g x) 0")
   863   apply (clarsimp simp add: compare_rls add_ac) 
   864   apply (rule abs_of_nonneg)
   865   apply (rule le_maxI2)
   866   done
   867 
   868 lemma lesso_add: "f <o g =o O(h) ==>
   869       k <o l =o O(h) ==> (f + k) <o (g + l) =o O(h)"
   870   apply (unfold lesso_def)
   871   apply (rule bigo_lesseq3)
   872   apply (erule bigo_useful_add)
   873   apply assumption
   874   apply (force split: split_max)
   875   apply (auto split: split_max simp add: func_plus)
   876   done
   877 
   878 end