src/HOL/Library/BigO.thy
author hoelzl
Tue Jul 19 14:37:49 2011 +0200 (2011-07-19)
changeset 43922 c6f35921056e
parent 42285 8d91a85b6d91
child 45270 d5b5c9259afd
permissions -rw-r--r--
add nat => enat coercion
     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    22 \item Lemmas involving @{text "sumr"} have been replaced by more general lemmas 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) \<oplus> 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) \<oplus> 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) \<oplus> 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 assumption
   133   apply (simp add: order_less_le)
   134   apply (rule mult_left_mono)
   135   apply (simp add: abs_triangle_ineq)
   136   apply (simp add: order_less_le)
   137   apply (rule mult_nonneg_nonneg)
   138   apply (rule add_nonneg_nonneg)
   139   apply auto
   140   apply (rule_tac x = "%n. if (abs (f n)) <  abs (g n) then x n else 0" 
   141      in exI)
   142   apply (rule conjI)
   143   apply (rule_tac x = "c + c" in exI)
   144   apply auto
   145   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
   146   apply (erule_tac x = xa in allE)
   147   apply (erule order_trans)
   148   apply (simp)
   149   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   150   apply (erule order_trans)
   151   apply (simp add: ring_distribs)
   152   apply (rule mult_left_mono)
   153   apply (simp add: order_less_le)
   154   apply (simp add: order_less_le)
   155   apply (rule mult_left_mono)
   156   apply (rule abs_triangle_ineq)
   157   apply (simp add: order_less_le)
   158   apply (rule mult_nonneg_nonneg)
   159   apply (rule add_nonneg_nonneg)
   160   apply (erule order_less_imp_le)+
   161   apply simp
   162   apply (rule ext)
   163   apply (auto simp add: if_splits linorder_not_le)
   164   done
   165 
   166 lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A \<oplus> B <= O(f)"
   167   apply (subgoal_tac "A \<oplus> B <= O(f) \<oplus> O(f)")
   168   apply (erule order_trans)
   169   apply simp
   170   apply (auto del: subsetI simp del: bigo_plus_idemp)
   171   done
   172 
   173 lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==> 
   174     O(f + g) = O(f) \<oplus> O(g)"
   175   apply (rule equalityI)
   176   apply (rule bigo_plus_subset)
   177   apply (simp add: bigo_alt_def set_plus_def func_plus)
   178   apply clarify
   179   apply (rule_tac x = "max c ca" in exI)
   180   apply (rule conjI)
   181   apply (subgoal_tac "c <= max c ca")
   182   apply (erule order_less_le_trans)
   183   apply assumption
   184   apply (rule le_maxI1)
   185   apply clarify
   186   apply (drule_tac x = "xa" in spec)+
   187   apply (subgoal_tac "0 <= f xa + g xa")
   188   apply (simp add: ring_distribs)
   189   apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
   190   apply (subgoal_tac "abs(a xa) + abs(b xa) <= 
   191       max c ca * f xa + max c ca * g xa")
   192   apply (force)
   193   apply (rule add_mono)
   194   apply (subgoal_tac "c * f xa <= max c ca * f xa")
   195   apply (force)
   196   apply (rule mult_right_mono)
   197   apply (rule le_maxI1)
   198   apply assumption
   199   apply (subgoal_tac "ca * g xa <= max c ca * g xa")
   200   apply (force)
   201   apply (rule mult_right_mono)
   202   apply (rule le_maxI2)
   203   apply assumption
   204   apply (rule abs_triangle_ineq)
   205   apply (rule add_nonneg_nonneg)
   206   apply assumption+
   207   done
   208 
   209 lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
   210     f : O(g)" 
   211   apply (auto simp add: bigo_def)
   212   apply (rule_tac x = "abs c" in exI)
   213   apply auto
   214   apply (drule_tac x = x in spec)+
   215   apply (simp add: abs_mult [symmetric])
   216   done
   217 
   218 lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> 
   219     f : O(g)" 
   220   apply (erule bigo_bounded_alt [of f 1 g])
   221   apply simp
   222   done
   223 
   224 lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
   225     f : lb +o O(g)"
   226   apply (rule set_minus_imp_plus)
   227   apply (rule bigo_bounded)
   228   apply (auto simp add: diff_minus fun_Compl_def func_plus)
   229   apply (drule_tac x = x in spec)+
   230   apply force
   231   apply (drule_tac x = x in spec)+
   232   apply force
   233   done
   234 
   235 lemma bigo_abs: "(%x. abs(f x)) =o O(f)" 
   236   apply (unfold bigo_def)
   237   apply auto
   238   apply (rule_tac x = 1 in exI)
   239   apply auto
   240   done
   241 
   242 lemma bigo_abs2: "f =o O(%x. abs(f x))"
   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_abs3: "O(f) = O(%x. abs(f x))"
   250   apply (rule equalityI)
   251   apply (rule bigo_elt_subset)
   252   apply (rule bigo_abs2)
   253   apply (rule bigo_elt_subset)
   254   apply (rule bigo_abs)
   255   done
   256 
   257 lemma bigo_abs4: "f =o g +o O(h) ==> 
   258     (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
   259   apply (drule set_plus_imp_minus)
   260   apply (rule set_minus_imp_plus)
   261   apply (subst fun_diff_def)
   262 proof -
   263   assume a: "f - g : O(h)"
   264   have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
   265     by (rule bigo_abs2)
   266   also have "... <= O(%x. abs (f x - g x))"
   267     apply (rule bigo_elt_subset)
   268     apply (rule bigo_bounded)
   269     apply force
   270     apply (rule allI)
   271     apply (rule abs_triangle_ineq3)
   272     done
   273   also have "... <= O(f - g)"
   274     apply (rule bigo_elt_subset)
   275     apply (subst fun_diff_def)
   276     apply (rule bigo_abs)
   277     done
   278   also from a have "... <= O(h)"
   279     by (rule bigo_elt_subset)
   280   finally show "(%x. abs (f x) - abs (g x)) : O(h)".
   281 qed
   282 
   283 lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" 
   284   by (unfold bigo_def, auto)
   285 
   286 lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) \<oplus> O(h)"
   287 proof -
   288   assume "f : g +o O(h)"
   289   also have "... <= O(g) \<oplus> O(h)"
   290     by (auto del: subsetI)
   291   also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
   292     apply (subst bigo_abs3 [symmetric])+
   293     apply (rule refl)
   294     done
   295   also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
   296     by (rule bigo_plus_eq [symmetric], auto)
   297   finally have "f : ...".
   298   then have "O(f) <= ..."
   299     by (elim bigo_elt_subset)
   300   also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
   301     by (rule bigo_plus_eq, auto)
   302   finally show ?thesis
   303     by (simp add: bigo_abs3 [symmetric])
   304 qed
   305 
   306 lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)"
   307   apply (rule subsetI)
   308   apply (subst bigo_def)
   309   apply (auto simp add: bigo_alt_def set_times_def func_times)
   310   apply (rule_tac x = "c * ca" in exI)
   311   apply(rule allI)
   312   apply(erule_tac x = x in allE)+
   313   apply(subgoal_tac "c * ca * abs(f x * g x) = 
   314       (c * abs(f x)) * (ca * abs(g x))")
   315   apply(erule ssubst)
   316   apply (subst abs_mult)
   317   apply (rule mult_mono)
   318   apply assumption+
   319   apply (rule mult_nonneg_nonneg)
   320   apply auto
   321   apply (simp add: mult_ac abs_mult)
   322   done
   323 
   324 lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
   325   apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
   326   apply (rule_tac x = c in exI)
   327   apply auto
   328   apply (drule_tac x = x in spec)
   329   apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
   330   apply (force simp add: mult_ac)
   331   apply (rule mult_left_mono, assumption)
   332   apply (rule abs_ge_zero)
   333   done
   334 
   335 lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
   336   apply (rule subsetD)
   337   apply (rule bigo_mult)
   338   apply (erule set_times_intro, assumption)
   339   done
   340 
   341 lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
   342   apply (drule set_plus_imp_minus)
   343   apply (rule set_minus_imp_plus)
   344   apply (drule bigo_mult3 [where g = g and j = g])
   345   apply (auto simp add: algebra_simps)
   346   done
   347 
   348 lemma bigo_mult5:
   349   assumes "ALL x. f x ~= 0"
   350   shows "O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
   351 proof
   352   fix h
   353   assume "h : O(f * g)"
   354   then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
   355     by auto
   356   also have "... <= O((%x. 1 / f x) * (f * g))"
   357     by (rule bigo_mult2)
   358   also have "(%x. 1 / f x) * (f * g) = g"
   359     apply (simp add: func_times) 
   360     apply (rule ext)
   361     apply (simp add: assms nonzero_divide_eq_eq mult_ac)
   362     done
   363   finally have "(%x. (1::'b) / f x) * h : O(g)" .
   364   then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
   365     by auto
   366   also have "f * ((%x. (1::'b) / f x) * h) = h"
   367     apply (simp add: func_times) 
   368     apply (rule ext)
   369     apply (simp add: assms nonzero_divide_eq_eq mult_ac)
   370     done
   371   finally show "h : f *o O(g)" .
   372 qed
   373 
   374 lemma bigo_mult6: "ALL x. f x ~= 0 ==>
   375     O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)"
   376   apply (rule equalityI)
   377   apply (erule bigo_mult5)
   378   apply (rule bigo_mult2)
   379   done
   380 
   381 lemma bigo_mult7: "ALL x. f x ~= 0 ==>
   382     O(f * g) <= O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
   383   apply (subst bigo_mult6)
   384   apply assumption
   385   apply (rule set_times_mono3)
   386   apply (rule bigo_refl)
   387   done
   388 
   389 lemma bigo_mult8: "ALL x. f x ~= 0 ==>
   390     O(f * g) = O(f::'a => ('b::linordered_field)) \<otimes> O(g)"
   391   apply (rule equalityI)
   392   apply (erule bigo_mult7)
   393   apply (rule bigo_mult)
   394   done
   395 
   396 lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
   397   by (auto simp add: bigo_def fun_Compl_def)
   398 
   399 lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
   400   apply (rule set_minus_imp_plus)
   401   apply (drule set_plus_imp_minus)
   402   apply (drule bigo_minus)
   403   apply (simp add: diff_minus)
   404   done
   405 
   406 lemma bigo_minus3: "O(-f) = O(f)"
   407   by (auto simp add: bigo_def fun_Compl_def)
   408 
   409 lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
   410 proof -
   411   assume a: "f : O(g)"
   412   show "f +o O(g) <= O(g)"
   413   proof -
   414     have "f : O(f)" by auto
   415     then have "f +o O(g) <= O(f) \<oplus> O(g)"
   416       by (auto del: subsetI)
   417     also have "... <= O(g) \<oplus> O(g)"
   418     proof -
   419       from a have "O(f) <= O(g)" by (auto del: subsetI)
   420       thus ?thesis by (auto del: subsetI)
   421     qed
   422     also have "... <= O(g)" by simp
   423     finally show ?thesis .
   424   qed
   425 qed
   426 
   427 lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
   428 proof -
   429   assume a: "f : O(g)"
   430   show "O(g) <= f +o O(g)"
   431   proof -
   432     from a have "-f : O(g)" by auto
   433     then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1)
   434     then have "f +o (-f +o O(g)) <= f +o O(g)" by auto
   435     also have "f +o (-f +o O(g)) = O(g)"
   436       by (simp add: set_plus_rearranges)
   437     finally show ?thesis .
   438   qed
   439 qed
   440 
   441 lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
   442   apply (rule equalityI)
   443   apply (erule bigo_plus_absorb_lemma1)
   444   apply (erule bigo_plus_absorb_lemma2)
   445   done
   446 
   447 lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
   448   apply (subgoal_tac "f +o A <= f +o O(g)")
   449   apply force+
   450   done
   451 
   452 lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
   453   apply (subst set_minus_plus [symmetric])
   454   apply (subgoal_tac "g - f = - (f - g)")
   455   apply (erule ssubst)
   456   apply (rule bigo_minus)
   457   apply (subst set_minus_plus)
   458   apply assumption
   459   apply  (simp add: diff_minus add_ac)
   460   done
   461 
   462 lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
   463   apply (rule iffI)
   464   apply (erule bigo_add_commute_imp)+
   465   done
   466 
   467 lemma bigo_const1: "(%x. c) : O(%x. 1)"
   468   by (auto simp add: bigo_def mult_ac)
   469 
   470 lemma bigo_const2 [intro]: "O(%x. c) <= O(%x. 1)"
   471   apply (rule bigo_elt_subset)
   472   apply (rule bigo_const1)
   473   done
   474 
   475 lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
   476   apply (simp add: bigo_def)
   477   apply (rule_tac x = "abs(inverse c)" in exI)
   478   apply (simp add: abs_mult [symmetric])
   479   done
   480 
   481 lemma bigo_const4: "(c::'a::linordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
   482   by (rule bigo_elt_subset, rule bigo_const3, assumption)
   483 
   484 lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ==> 
   485     O(%x. c) = O(%x. 1)"
   486   by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
   487 
   488 lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
   489   apply (simp add: bigo_def)
   490   apply (rule_tac x = "abs(c)" in exI)
   491   apply (auto simp add: abs_mult [symmetric])
   492   done
   493 
   494 lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
   495   by (rule bigo_elt_subset, rule bigo_const_mult1)
   496 
   497 lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%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: "(c::'a::linordered_field) ~= 0 ==> 
   504     O(f) <= O(%x. c * f x)"
   505   by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
   506 
   507 lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ==> 
   508     O(%x. c * f x) = O(f)"
   509   by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
   510 
   511 lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==> 
   512     (%x. c) *o O(f) = O(f)"
   513   apply (auto del: subsetI)
   514   apply (rule order_trans)
   515   apply (rule bigo_mult2)
   516   apply (simp add: func_times)
   517   apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
   518   apply (rule_tac x = "%y. inverse c * x y" in exI)
   519   apply (simp add: mult_assoc [symmetric] abs_mult)
   520   apply (rule_tac x = "abs (inverse c) * ca" in exI)
   521   apply (rule allI)
   522   apply (subst mult_assoc)
   523   apply (rule mult_left_mono)
   524   apply (erule spec)
   525   apply force
   526   done
   527 
   528 lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
   529   apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
   530   apply (rule_tac x = "ca * (abs c)" in exI)
   531   apply (rule allI)
   532   apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
   533   apply (erule ssubst)
   534   apply (subst abs_mult)
   535   apply (rule mult_left_mono)
   536   apply (erule spec)
   537   apply simp
   538   apply(simp add: mult_ac)
   539   done
   540 
   541 lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
   542 proof -
   543   assume "f =o O(g)"
   544   then have "(%x. c) * f =o (%x. c) *o O(g)"
   545     by auto
   546   also have "(%x. c) * f = (%x. c * f x)"
   547     by (simp add: func_times)
   548   also have "(%x. c) *o O(g) <= O(g)"
   549     by (auto del: subsetI)
   550   finally show ?thesis .
   551 qed
   552 
   553 lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
   554 by (unfold bigo_def, auto)
   555 
   556 lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o 
   557     O(%x. h(k x))"
   558   apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def
   559       func_plus)
   560   apply (erule bigo_compose1)
   561 done
   562 
   563 
   564 subsection {* Setsum *}
   565 
   566 lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==> 
   567     EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
   568       (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"  
   569   apply (auto simp add: bigo_def)
   570   apply (rule_tac x = "abs c" in exI)
   571   apply (subst abs_of_nonneg) back back
   572   apply (rule setsum_nonneg)
   573   apply force
   574   apply (subst setsum_right_distrib)
   575   apply (rule allI)
   576   apply (rule order_trans)
   577   apply (rule setsum_abs)
   578   apply (rule setsum_mono)
   579   apply (rule order_trans)
   580   apply (drule spec)+
   581   apply (drule bspec)+
   582   apply assumption+
   583   apply (drule bspec)
   584   apply assumption+
   585   apply (rule mult_right_mono) 
   586   apply (rule abs_ge_self)
   587   apply force
   588   done
   589 
   590 lemma bigo_setsum1: "ALL x y. 0 <= h x y ==> 
   591     EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
   592       (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
   593   apply (rule bigo_setsum_main)
   594   apply force
   595   apply clarsimp
   596   apply (rule_tac x = c in exI)
   597   apply force
   598   done
   599 
   600 lemma bigo_setsum2: "ALL y. 0 <= h y ==> 
   601     EX c. ALL y. abs(f y) <= c * (h y) ==>
   602       (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
   603   by (rule bigo_setsum1, auto)  
   604 
   605 lemma bigo_setsum3: "f =o O(h) ==>
   606     (%x. SUM y : A x. (l x y) * f(k x y)) =o
   607       O(%x. SUM y : A x. abs(l x y * h(k x y)))"
   608   apply (rule bigo_setsum1)
   609   apply (rule allI)+
   610   apply (rule abs_ge_zero)
   611   apply (unfold bigo_def)
   612   apply auto
   613   apply (rule_tac x = c in exI)
   614   apply (rule allI)+
   615   apply (subst abs_mult)+
   616   apply (subst mult_left_commute)
   617   apply (rule mult_left_mono)
   618   apply (erule spec)
   619   apply (rule abs_ge_zero)
   620   done
   621 
   622 lemma bigo_setsum4: "f =o g +o O(h) ==>
   623     (%x. SUM y : A x. l x y * f(k x y)) =o
   624       (%x. SUM y : A x. l x y * g(k x y)) +o
   625         O(%x. SUM y : A x. abs(l x y * h(k x y)))"
   626   apply (rule set_minus_imp_plus)
   627   apply (subst fun_diff_def)
   628   apply (subst setsum_subtractf [symmetric])
   629   apply (subst right_diff_distrib [symmetric])
   630   apply (rule bigo_setsum3)
   631   apply (subst fun_diff_def [symmetric])
   632   apply (erule set_plus_imp_minus)
   633   done
   634 
   635 lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==> 
   636     ALL x. 0 <= h x ==>
   637       (%x. SUM y : A x. (l x y) * f(k x y)) =o
   638         O(%x. SUM y : A x. (l x y) * h(k x y))" 
   639   apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) = 
   640       (%x. SUM y : A x. abs((l x y) * h(k x y)))")
   641   apply (erule ssubst)
   642   apply (erule bigo_setsum3)
   643   apply (rule ext)
   644   apply (rule setsum_cong2)
   645   apply (subst abs_of_nonneg)
   646   apply (rule mult_nonneg_nonneg)
   647   apply auto
   648   done
   649 
   650 lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
   651     ALL x. 0 <= h x ==>
   652       (%x. SUM y : A x. (l x y) * f(k x y)) =o
   653         (%x. SUM y : A x. (l x y) * g(k x y)) +o
   654           O(%x. SUM y : A x. (l x y) * h(k x y))" 
   655   apply (rule set_minus_imp_plus)
   656   apply (subst fun_diff_def)
   657   apply (subst setsum_subtractf [symmetric])
   658   apply (subst right_diff_distrib [symmetric])
   659   apply (rule bigo_setsum5)
   660   apply (subst fun_diff_def [symmetric])
   661   apply (drule set_plus_imp_minus)
   662   apply auto
   663   done
   664 
   665 
   666 subsection {* Misc useful stuff *}
   667 
   668 lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
   669   A \<oplus> B <= O(f)"
   670   apply (subst bigo_plus_idemp [symmetric])
   671   apply (rule set_plus_mono2)
   672   apply assumption+
   673   done
   674 
   675 lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
   676   apply (subst bigo_plus_idemp [symmetric])
   677   apply (rule set_plus_intro)
   678   apply assumption+
   679   done
   680   
   681 lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 ==> 
   682     (%x. c) * f =o O(h) ==> f =o O(h)"
   683   apply (rule subsetD)
   684   apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
   685   apply assumption
   686   apply (rule bigo_const_mult6)
   687   apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
   688   apply (erule ssubst)
   689   apply (erule set_times_intro2)
   690   apply (simp add: func_times)
   691   done
   692 
   693 lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
   694     f =o O(h)"
   695   apply (simp add: bigo_alt_def)
   696   apply auto
   697   apply (rule_tac x = c in exI)
   698   apply auto
   699   apply (case_tac "x = 0")
   700   apply simp
   701   apply (rule mult_nonneg_nonneg)
   702   apply force
   703   apply force
   704   apply (subgoal_tac "x = Suc (x - 1)")
   705   apply (erule ssubst) back
   706   apply (erule spec)
   707   apply simp
   708   done
   709 
   710 lemma bigo_fix2: 
   711     "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==> 
   712        f 0 = g 0 ==> f =o g +o O(h)"
   713   apply (rule set_minus_imp_plus)
   714   apply (rule bigo_fix)
   715   apply (subst fun_diff_def)
   716   apply (subst fun_diff_def [symmetric])
   717   apply (rule set_plus_imp_minus)
   718   apply simp
   719   apply (simp add: fun_diff_def)
   720   done
   721 
   722 
   723 subsection {* Less than or equal to *}
   724 
   725 definition
   726   lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)"
   727     (infixl "<o" 70) where
   728   "f <o g = (%x. max (f x - g x) 0)"
   729 
   730 lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
   731     g =o O(h)"
   732   apply (unfold bigo_def)
   733   apply clarsimp
   734   apply (rule_tac x = c in exI)
   735   apply (rule allI)
   736   apply (rule order_trans)
   737   apply (erule spec)+
   738   done
   739 
   740 lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
   741       g =o O(h)"
   742   apply (erule bigo_lesseq1)
   743   apply (rule allI)
   744   apply (drule_tac x = x in spec)
   745   apply (rule order_trans)
   746   apply assumption
   747   apply (rule abs_ge_self)
   748   done
   749 
   750 lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
   751     g =o O(h)"
   752   apply (erule bigo_lesseq2)
   753   apply (rule allI)
   754   apply (subst abs_of_nonneg)
   755   apply (erule spec)+
   756   done
   757 
   758 lemma bigo_lesseq4: "f =o O(h) ==>
   759     ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
   760       g =o O(h)"
   761   apply (erule bigo_lesseq1)
   762   apply (rule allI)
   763   apply (subst abs_of_nonneg)
   764   apply (erule spec)+
   765   done
   766 
   767 lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
   768   apply (unfold lesso_def)
   769   apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
   770   apply (erule ssubst)
   771   apply (rule bigo_zero)
   772   apply (unfold func_zero)
   773   apply (rule ext)
   774   apply (simp split: split_max)
   775   done
   776 
   777 lemma bigo_lesso2: "f =o g +o O(h) ==>
   778     ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
   779       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 le_maxI2)
   785   apply (rule allI)
   786   apply (subst fun_diff_def)
   787   apply (case_tac "0 <= 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_minus)+
   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) ==>
   802     ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
   803       f <o k =o O(h)"
   804   apply (unfold lesso_def)
   805   apply (rule bigo_lesseq4)
   806   apply (erule set_plus_imp_minus)
   807   apply (rule allI)
   808   apply (rule le_maxI2)
   809   apply (rule allI)
   810   apply (subst fun_diff_def)
   811   apply (case_tac "0 <= f x - k x")
   812   apply simp
   813   apply (subst abs_of_nonneg)
   814   apply (drule_tac x = x in spec) back
   815   apply (simp add: algebra_simps)
   816   apply (subst diff_minus)+
   817   apply (rule add_left_mono)
   818   apply (rule le_imp_neg_le)
   819   apply (erule spec)
   820   apply (rule order_trans) 
   821   prefer 2
   822   apply (rule abs_ge_zero)
   823   apply (simp add: algebra_simps)
   824   done
   825 
   826 lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::linordered_field) ==>
   827     g =o h +o O(k) ==> 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
   837     split: split_max abs_split)
   838   done
   839 
   840 lemma bigo_lesso5: "f <o g =o O(h) ==>
   841     EX C. ALL x. f x <= g x + C * abs(h x)"
   842   apply (simp only: lesso_def bigo_alt_def)
   843   apply clarsimp
   844   apply (rule_tac x = c in exI)
   845   apply (rule allI)
   846   apply (drule_tac x = x in spec)
   847   apply (subgoal_tac "abs(max (f x - g x) 0) = max (f x - g x) 0")
   848   apply (clarsimp simp add: algebra_simps) 
   849   apply (rule abs_of_nonneg)
   850   apply (rule le_maxI2)
   851   done
   852 
   853 lemma lesso_add: "f <o g =o O(h) ==>
   854       k <o l =o O(h) ==> (f + k) <o (g + l) =o O(h)"
   855   apply (unfold lesso_def)
   856   apply (rule bigo_lesseq3)
   857   apply (erule bigo_useful_add)
   858   apply assumption
   859   apply (force split: split_max)
   860   apply (auto split: split_max simp add: func_plus)
   861   done
   862 
   863 lemma bigo_LIMSEQ1: "f =o O(g) ==> g ----> 0 ==> f ----> (0::real)"
   864   apply (simp add: LIMSEQ_iff bigo_alt_def)
   865   apply clarify
   866   apply (drule_tac x = "r / c" in spec)
   867   apply (drule mp)
   868   apply (erule divide_pos_pos)
   869   apply assumption
   870   apply clarify
   871   apply (rule_tac x = no in exI)
   872   apply (rule allI)
   873   apply (drule_tac x = n in spec)+
   874   apply (rule impI)
   875   apply (drule mp)
   876   apply assumption
   877   apply (rule order_le_less_trans)
   878   apply assumption
   879   apply (rule order_less_le_trans)
   880   apply (subgoal_tac "c * abs(g n) < c * (r / c)")
   881   apply assumption
   882   apply (erule mult_strict_left_mono)
   883   apply assumption
   884   apply simp
   885 done
   886 
   887 lemma bigo_LIMSEQ2: "f =o g +o O(h) ==> h ----> 0 ==> f ----> a 
   888     ==> g ----> (a::real)"
   889   apply (drule set_plus_imp_minus)
   890   apply (drule bigo_LIMSEQ1)
   891   apply assumption
   892   apply (simp only: fun_diff_def)
   893   apply (erule LIMSEQ_diff_approach_zero2)
   894   apply assumption
   895 done
   896 
   897 end