src/HOL/MetisExamples/BigO.thy
author wenzelm
Mon Mar 16 18:24:30 2009 +0100 (2009-03-16)
changeset 30549 d2d7874648bd
parent 29823 0ab754d13ccd
child 32864 a226f29d4bdc
permissions -rw-r--r--
simplified method setup;
     1 (*  Title:      HOL/MetisExamples/BigO.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3 
     4 Testing the metis method
     5 *)
     6 
     7 header {* Big O notation *}
     8 
     9 theory BigO
    10 imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" Main SetsAndFunctions 
    11 begin
    12 
    13 subsection {* Definitions *}
    14 
    15 definition bigo :: "('a => 'b::ordered_idom) => ('a => 'b) set"    ("(1O'(_'))") where
    16   "O(f::('a => 'b)) ==   {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
    17 
    18 ML_command{*AtpWrapper.problem_name := "BigO__bigo_pos_const"*}
    19 lemma bigo_pos_const: "(EX (c::'a::ordered_idom). 
    20     ALL x. (abs (h x)) <= (c * (abs (f x))))
    21       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
    22   apply auto
    23   apply (case_tac "c = 0", simp)
    24   apply (rule_tac x = "1" in exI, simp)
    25   apply (rule_tac x = "abs c" in exI, auto)
    26   apply (metis abs_ge_minus_self abs_ge_zero abs_minus_cancel abs_of_nonneg equation_minus_iff Orderings.xt1(6) abs_mult)
    27   done
    28 
    29 (*** Now various verions with an increasing modulus ***)
    30 
    31 declare [[sledgehammer_modulus = 1]]
    32 
    33 lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom). 
    34     ALL x. (abs (h x)) <= (c * (abs (f x))))
    35       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
    36   apply auto
    37   apply (case_tac "c = 0", simp)
    38   apply (rule_tac x = "1" in exI, simp)
    39   apply (rule_tac x = "abs c" in exI, auto)
    40 proof (neg_clausify)
    41 fix c x
    42 have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * X2\<bar> = \<bar>X2 * X1\<bar>"
    43   by (metis abs_mult mult_commute)
    44 have 1: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
    45    X1 \<le> (0\<Colon>'a\<Colon>ordered_idom) \<or> \<bar>X2\<bar> * X1 = \<bar>X2 * X1\<bar>"
    46   by (metis abs_mult_pos linorder_linear)
    47 have 2: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
    48    \<not> (0\<Colon>'a\<Colon>ordered_idom) < X1 * X2 \<or>
    49    \<not> (0\<Colon>'a\<Colon>ordered_idom) \<le> X2 \<or> \<not> X1 \<le> (0\<Colon>'a\<Colon>ordered_idom)"
    50   by (metis linorder_not_less mult_nonneg_nonpos2)
    51 assume 3: "\<And>x\<Colon>'b\<Colon>type.
    52    \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
    53    \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
    54 assume 4: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
    55   \<le> \<bar>c\<Colon>'a\<Colon>ordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
    56 have 5: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
    57   \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
    58   by (metis 4 abs_mult)
    59 have 6: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
    60    \<not> X1 \<le> (0\<Colon>'a\<Colon>ordered_idom) \<or> X1 \<le> \<bar>X2\<bar>"
    61   by (metis abs_ge_zero xt1(6))
    62 have 7: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
    63    X1 \<le> \<bar>X2\<bar> \<or> (0\<Colon>'a\<Colon>ordered_idom) < X1"
    64   by (metis not_leE 6)
    65 have 8: "(0\<Colon>'a\<Colon>ordered_idom) < \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
    66   by (metis 5 7)
    67 have 9: "\<And>X1\<Colon>'a\<Colon>ordered_idom.
    68    \<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar> \<le> X1 \<or>
    69    (0\<Colon>'a\<Colon>ordered_idom) < X1"
    70   by (metis 8 order_less_le_trans)
    71 have 10: "(0\<Colon>'a\<Colon>ordered_idom)
    72 < (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
    73   by (metis 3 9)
    74 have 11: "\<not> (c\<Colon>'a\<Colon>ordered_idom) \<le> (0\<Colon>'a\<Colon>ordered_idom)"
    75   by (metis abs_ge_zero 2 10)
    76 have 12: "\<And>X1\<Colon>'a\<Colon>ordered_idom. (c\<Colon>'a\<Colon>ordered_idom) * \<bar>X1\<bar> = \<bar>X1 * c\<bar>"
    77   by (metis mult_commute 1 11)
    78 have 13: "\<And>X1\<Colon>'b\<Colon>type.
    79    - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
    80    \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
    81   by (metis 3 abs_le_D2)
    82 have 14: "\<And>X1\<Colon>'b\<Colon>type.
    83    - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
    84    \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
    85   by (metis 0 12 13)
    86 have 15: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
    87   by (metis abs_mult abs_mult_pos abs_ge_zero)
    88 have 16: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. X1 \<le> \<bar>X2\<bar> \<or> \<not> X1 \<le> X2"
    89   by (metis xt1(6) abs_ge_self)
    90 have 17: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<not> \<bar>X1\<bar> \<le> X2 \<or> X1 \<le> \<bar>X2\<bar>"
    91   by (metis 16 abs_le_D1)
    92 have 18: "\<And>X1\<Colon>'b\<Colon>type.
    93    (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
    94    \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
    95   by (metis 17 3 15)
    96 show "False"
    97   by (metis abs_le_iff 5 18 14)
    98 qed
    99 
   100 declare [[sledgehammer_modulus = 2]]
   101 
   102 lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom). 
   103     ALL x. (abs (h x)) <= (c * (abs (f x))))
   104       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
   105   apply auto
   106   apply (case_tac "c = 0", simp)
   107   apply (rule_tac x = "1" in exI, simp)
   108   apply (rule_tac x = "abs c" in exI, auto);
   109 proof (neg_clausify)
   110 fix c x
   111 have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * X2\<bar> = \<bar>X2 * X1\<bar>"
   112   by (metis abs_mult mult_commute)
   113 assume 1: "\<And>x\<Colon>'b\<Colon>type.
   114    \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
   115    \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
   116 assume 2: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
   117   \<le> \<bar>c\<Colon>'a\<Colon>ordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
   118 have 3: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
   119   \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
   120   by (metis 2 abs_mult)
   121 have 4: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
   122    \<not> X1 \<le> (0\<Colon>'a\<Colon>ordered_idom) \<or> X1 \<le> \<bar>X2\<bar>"
   123   by (metis abs_ge_zero xt1(6))
   124 have 5: "(0\<Colon>'a\<Colon>ordered_idom) < \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
   125   by (metis not_leE 4 3)
   126 have 6: "(0\<Colon>'a\<Colon>ordered_idom)
   127 < (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
   128   by (metis 1 order_less_le_trans 5)
   129 have 7: "\<And>X1\<Colon>'a\<Colon>ordered_idom. (c\<Colon>'a\<Colon>ordered_idom) * \<bar>X1\<bar> = \<bar>X1 * c\<bar>"
   130   by (metis abs_ge_zero linorder_not_less mult_nonneg_nonpos2 6 linorder_linear abs_mult_pos mult_commute)
   131 have 8: "\<And>X1\<Colon>'b\<Colon>type.
   132    - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
   133    \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
   134   by (metis 0 7 abs_le_D2 1)
   135 have 9: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<not> \<bar>X1\<bar> \<le> X2 \<or> X1 \<le> \<bar>X2\<bar>"
   136   by (metis abs_ge_self xt1(6) abs_le_D1)
   137 show "False"
   138   by (metis 8 abs_ge_zero abs_mult_pos abs_mult 1 9 3 abs_le_iff)
   139 qed
   140 
   141 declare [[sledgehammer_modulus = 3]]
   142 
   143 lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom). 
   144     ALL x. (abs (h x)) <= (c * (abs (f x))))
   145       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
   146   apply auto
   147   apply (case_tac "c = 0", simp)
   148   apply (rule_tac x = "1" in exI, simp)
   149   apply (rule_tac x = "abs c" in exI, auto);
   150 proof (neg_clausify)
   151 fix c x
   152 assume 0: "\<And>x\<Colon>'b\<Colon>type.
   153    \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
   154    \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
   155 assume 1: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
   156   \<le> \<bar>c\<Colon>'a\<Colon>ordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
   157 have 2: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
   158    X1 \<le> \<bar>X2\<bar> \<or> (0\<Colon>'a\<Colon>ordered_idom) < X1"
   159   by (metis abs_ge_zero xt1(6) not_leE)
   160 have 3: "\<not> (c\<Colon>'a\<Colon>ordered_idom) \<le> (0\<Colon>'a\<Colon>ordered_idom)"
   161   by (metis abs_ge_zero mult_nonneg_nonpos2 linorder_not_less order_less_le_trans 1 abs_mult 2 0)
   162 have 4: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
   163   by (metis abs_ge_zero abs_mult_pos abs_mult)
   164 have 5: "\<And>X1\<Colon>'b\<Colon>type.
   165    (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
   166    \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
   167   by (metis 4 0 xt1(6) abs_ge_self abs_le_D1)
   168 show "False"
   169   by (metis abs_mult mult_commute 3 abs_mult_pos linorder_linear 0 abs_le_D2 5 1 abs_le_iff)
   170 qed
   171 
   172 
   173 declare [[sledgehammer_modulus = 1]]
   174 
   175 lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom). 
   176     ALL x. (abs (h x)) <= (c * (abs (f x))))
   177       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
   178   apply auto
   179   apply (case_tac "c = 0", simp)
   180   apply (rule_tac x = "1" in exI, simp)
   181   apply (rule_tac x = "abs c" in exI, auto);
   182 proof (neg_clausify)
   183 fix c x  (*sort/type constraint inserted by hand!*)
   184 have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
   185   by (metis abs_ge_zero abs_mult_pos abs_mult)
   186 assume 1: "\<And>A. \<bar>h A\<bar> \<le> c * \<bar>f A\<bar>"
   187 have 2: "\<And>X1 X2. \<not> \<bar>X1\<bar> \<le> X2 \<or> (0\<Colon>'a) \<le> X2"
   188   by (metis abs_ge_zero order_trans)
   189 have 3: "\<And>X1. (0\<Colon>'a) \<le> c * \<bar>f X1\<bar>"
   190   by (metis 1 2)
   191 have 4: "\<And>X1. c * \<bar>f X1\<bar> = \<bar>c * f X1\<bar>"
   192   by (metis 0 abs_of_nonneg 3)
   193 have 5: "\<And>X1. - h X1 \<le> c * \<bar>f X1\<bar>"
   194   by (metis 1 abs_le_D2)
   195 have 6: "\<And>X1. - h X1 \<le> \<bar>c * f X1\<bar>"
   196   by (metis 4 5)
   197 have 7: "\<And>X1. h X1 \<le> c * \<bar>f X1\<bar>"
   198   by (metis 1 abs_le_D1)
   199 have 8: "\<And>X1. h X1 \<le> \<bar>c * f X1\<bar>"
   200   by (metis 4 7)
   201 assume 9: "\<not> \<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>"
   202 have 10: "\<not> \<bar>h x\<bar> \<le> \<bar>c * f x\<bar>"
   203   by (metis abs_mult 9)
   204 show "False"
   205   by (metis 6 8 10 abs_leI)
   206 qed
   207 
   208 
   209 declare [[sledgehammer_sorts = true]]
   210 
   211 lemma bigo_alt_def: "O(f) = 
   212     {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
   213 by (auto simp add: bigo_def bigo_pos_const)
   214 
   215 ML_command{*AtpWrapper.problem_name := "BigO__bigo_elt_subset"*}
   216 lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
   217   apply (auto simp add: bigo_alt_def)
   218   apply (rule_tac x = "ca * c" in exI)
   219   apply (rule conjI)
   220   apply (rule mult_pos_pos)
   221   apply (assumption)+ 
   222 (*sledgehammer*);
   223   apply (rule allI)
   224   apply (drule_tac x = "xa" in spec)+
   225   apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))");
   226   apply (erule order_trans)
   227   apply (simp add: mult_ac)
   228   apply (rule mult_left_mono, assumption)
   229   apply (rule order_less_imp_le, assumption);
   230 done
   231 
   232 
   233 ML_command{*AtpWrapper.problem_name := "BigO__bigo_refl"*}
   234 lemma bigo_refl [intro]: "f : O(f)"
   235   apply(auto simp add: bigo_def)
   236 proof (neg_clausify)
   237 fix x
   238 assume 0: "\<And>xa. \<not> \<bar>f (x xa)\<bar> \<le> xa * \<bar>f (x xa)\<bar>"
   239 have 1: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2 \<or> \<not> (1\<Colon>'b) \<le> (1\<Colon>'b)"
   240   by (metis mult_le_cancel_right1 order_eq_iff)
   241 have 2: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2"
   242   by (metis order_eq_iff 1)
   243 show "False"
   244   by (metis 0 2)
   245 qed
   246 
   247 ML_command{*AtpWrapper.problem_name := "BigO__bigo_zero"*}
   248 lemma bigo_zero: "0 : O(g)"
   249   apply (auto simp add: bigo_def func_zero)
   250 proof (neg_clausify)
   251 fix x
   252 assume 0: "\<And>xa. \<not> (0\<Colon>'b) \<le> xa * \<bar>g (x xa)\<bar>"
   253 have 1: "\<not> (0\<Colon>'b) \<le> (0\<Colon>'b)"
   254   by (metis 0 mult_eq_0_iff)
   255 show "False"
   256   by (metis 1 linorder_neq_iff linorder_antisym_conv1)
   257 qed
   258 
   259 lemma bigo_zero2: "O(%x.0) = {%x.0}"
   260   apply (auto simp add: bigo_def) 
   261   apply (rule ext)
   262   apply auto
   263 done
   264 
   265 lemma bigo_plus_self_subset [intro]: 
   266   "O(f) \<oplus> O(f) <= O(f)"
   267   apply (auto simp add: bigo_alt_def set_plus_def)
   268   apply (rule_tac x = "c + ca" in exI)
   269   apply auto
   270   apply (simp add: ring_distribs func_plus)
   271   apply (blast intro:order_trans abs_triangle_ineq add_mono elim:) 
   272 done
   273 
   274 lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)"
   275   apply (rule equalityI)
   276   apply (rule bigo_plus_self_subset)
   277   apply (rule set_zero_plus2) 
   278   apply (rule bigo_zero)
   279 done
   280 
   281 lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) \<oplus> O(g)"
   282   apply (rule subsetI)
   283   apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
   284   apply (subst bigo_pos_const [symmetric])+
   285   apply (rule_tac x = 
   286     "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
   287   apply (rule conjI)
   288   apply (rule_tac x = "c + c" in exI)
   289   apply (clarsimp)
   290   apply (auto)
   291   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
   292   apply (erule_tac x = xa in allE)
   293   apply (erule order_trans)
   294   apply (simp)
   295   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   296   apply (erule order_trans)
   297   apply (simp add: ring_distribs)
   298   apply (rule mult_left_mono)
   299   apply assumption
   300   apply (simp add: order_less_le)
   301   apply (rule mult_left_mono)
   302   apply (simp add: abs_triangle_ineq)
   303   apply (simp add: order_less_le)
   304   apply (rule mult_nonneg_nonneg)
   305   apply (rule add_nonneg_nonneg)
   306   apply auto
   307   apply (rule_tac x = "%n. if (abs (f n)) <  abs (g n) then x n else 0" 
   308      in exI)
   309   apply (rule conjI)
   310   apply (rule_tac x = "c + c" in exI)
   311   apply auto
   312   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
   313   apply (erule_tac x = xa in allE)
   314   apply (erule order_trans)
   315   apply (simp)
   316   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   317   apply (erule order_trans)
   318   apply (simp add: ring_distribs)
   319   apply (rule mult_left_mono)
   320   apply (simp add: order_less_le)
   321   apply (simp add: order_less_le)
   322   apply (rule mult_left_mono)
   323   apply (rule abs_triangle_ineq)
   324   apply (simp add: order_less_le)
   325 apply (metis abs_not_less_zero double_less_0_iff less_not_permute linorder_not_less mult_less_0_iff)
   326   apply (rule ext)
   327   apply (auto simp add: if_splits linorder_not_le)
   328 done
   329 
   330 lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A \<oplus> B <= O(f)"
   331   apply (subgoal_tac "A \<oplus> B <= O(f) \<oplus> O(f)")
   332   apply (erule order_trans)
   333   apply simp
   334   apply (auto del: subsetI simp del: bigo_plus_idemp)
   335 done
   336 
   337 ML_command{*AtpWrapper.problem_name := "BigO__bigo_plus_eq"*}
   338 lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==> 
   339   O(f + g) = O(f) \<oplus> O(g)"
   340   apply (rule equalityI)
   341   apply (rule bigo_plus_subset)
   342   apply (simp add: bigo_alt_def set_plus_def func_plus)
   343   apply clarify 
   344 (*sledgehammer*); 
   345   apply (rule_tac x = "max c ca" in exI)
   346   apply (rule conjI)
   347    apply (metis Orderings.less_max_iff_disj)
   348   apply clarify
   349   apply (drule_tac x = "xa" in spec)+
   350   apply (subgoal_tac "0 <= f xa + g xa")
   351   apply (simp add: ring_distribs)
   352   apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
   353   apply (subgoal_tac "abs(a xa) + abs(b xa) <= 
   354       max c ca * f xa + max c ca * g xa")
   355   apply (blast intro: order_trans)
   356   defer 1
   357   apply (rule abs_triangle_ineq)
   358   apply (metis add_nonneg_nonneg)
   359   apply (rule add_mono)
   360 ML_command{*AtpWrapper.problem_name := "BigO__bigo_plus_eq_simpler"*} 
   361 (*Found by SPASS; SLOW*)
   362 apply (metis le_maxI2 linorder_linear linorder_not_le min_max.sup_absorb1 mult_le_cancel_right order_trans)
   363 apply (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans)
   364 done
   365 
   366 ML_command{*AtpWrapper.problem_name := "BigO__bigo_bounded_alt"*}
   367 lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
   368     f : O(g)" 
   369   apply (auto simp add: bigo_def)
   370 (*Version 1: one-shot proof*)
   371   apply (metis OrderedGroup.abs_le_D1 linorder_class.not_less  order_less_le  Orderings.xt1(12)  Ring_and_Field.abs_mult)
   372   done
   373 
   374 lemma (*bigo_bounded_alt:*) "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
   375     f : O(g)" 
   376   apply (auto simp add: bigo_def)
   377 (*Version 2: single-step proof*)
   378 proof (neg_clausify)
   379 fix x
   380 assume 0: "\<And>x. f x \<le> c * g x"
   381 assume 1: "\<And>xa. \<not> f (x xa) \<le> xa * \<bar>g (x xa)\<bar>"
   382 have 2: "\<And>X3. c * g X3 = f X3 \<or> \<not> c * g X3 \<le> f X3"
   383   by (metis 0 order_antisym_conv)
   384 have 3: "\<And>X3. \<not> f (x \<bar>X3\<bar>) \<le> \<bar>X3 * g (x \<bar>X3\<bar>)\<bar>"
   385   by (metis 1 abs_mult)
   386 have 4: "\<And>X1 X3\<Colon>'b\<Colon>ordered_idom. X3 \<le> X1 \<or> X1 \<le> \<bar>X3\<bar>"
   387   by (metis linorder_linear abs_le_D1)
   388 have 5: "\<And>X3::'b. \<bar>X3\<bar> * \<bar>X3\<bar> = X3 * X3"
   389   by (metis abs_mult_self)
   390 have 6: "\<And>X3. \<not> X3 * X3 < (0\<Colon>'b\<Colon>ordered_idom)"
   391   by (metis not_square_less_zero)
   392 have 7: "\<And>X1 X3::'b. \<bar>X1\<bar> * \<bar>X3\<bar> = \<bar>X3 * X1\<bar>"
   393   by (metis abs_mult mult_commute)
   394 have 8: "\<And>X3::'b. X3 * X3 = \<bar>X3 * X3\<bar>"
   395   by (metis abs_mult 5)
   396 have 9: "\<And>X3. X3 * g (x \<bar>X3\<bar>) \<le> f (x \<bar>X3\<bar>)"
   397   by (metis 3 4)
   398 have 10: "c * g (x \<bar>c\<bar>) = f (x \<bar>c\<bar>)"
   399   by (metis 2 9)
   400 have 11: "\<And>X3::'b. \<bar>X3\<bar> * \<bar>\<bar>X3\<bar>\<bar> = \<bar>X3\<bar> * \<bar>X3\<bar>"
   401   by (metis abs_idempotent abs_mult 8)
   402 have 12: "\<And>X3::'b. \<bar>X3 * \<bar>X3\<bar>\<bar> = \<bar>X3\<bar> * \<bar>X3\<bar>"
   403   by (metis mult_commute 7 11)
   404 have 13: "\<And>X3::'b. \<bar>X3 * \<bar>X3\<bar>\<bar> = X3 * X3"
   405   by (metis 8 7 12)
   406 have 14: "\<And>X3. X3 \<le> \<bar>X3\<bar> \<or> X3 < (0\<Colon>'b)"
   407   by (metis abs_ge_self abs_le_D1 abs_if)
   408 have 15: "\<And>X3. X3 \<le> \<bar>X3\<bar> \<or> \<bar>X3\<bar> < (0\<Colon>'b)"
   409   by (metis abs_ge_self abs_le_D1 abs_if)
   410 have 16: "\<And>X3. X3 * X3 < (0\<Colon>'b) \<or> X3 * \<bar>X3\<bar> \<le> X3 * X3"
   411   by (metis 15 13)
   412 have 17: "\<And>X3::'b. X3 * \<bar>X3\<bar> \<le> X3 * X3"
   413   by (metis 16 6)
   414 have 18: "\<And>X3. X3 \<le> \<bar>X3\<bar> \<or> \<not> X3 < (0\<Colon>'b)"
   415   by (metis mult_le_cancel_left 17)
   416 have 19: "\<And>X3::'b. X3 \<le> \<bar>X3\<bar>"
   417   by (metis 18 14)
   418 have 20: "\<not> f (x \<bar>c\<bar>) \<le> \<bar>f (x \<bar>c\<bar>)\<bar>"
   419   by (metis 3 10)
   420 show "False"
   421   by (metis 20 19)
   422 qed
   423 
   424 
   425 text{*So here is the easier (and more natural) problem using transitivity*}
   426 ML_command{*AtpWrapper.problem_name := "BigO__bigo_bounded_alt_trans"*}
   427 lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" 
   428   apply (auto simp add: bigo_def)
   429   (*Version 1: one-shot proof*) 
   430   apply (metis Orderings.leD Orderings.leI abs_ge_self abs_le_D1 abs_mult abs_of_nonneg order_le_less)
   431   done
   432 
   433 text{*So here is the easier (and more natural) problem using transitivity*}
   434 ML_command{*AtpWrapper.problem_name := "BigO__bigo_bounded_alt_trans"*}
   435 lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" 
   436   apply (auto simp add: bigo_def)
   437 (*Version 2: single-step proof*)
   438 proof (neg_clausify)
   439 fix x
   440 assume 0: "\<And>A\<Colon>'a\<Colon>type.
   441    (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) A
   442    \<le> (c\<Colon>'b\<Colon>ordered_idom) * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) A"
   443 assume 1: "\<And>A\<Colon>'b\<Colon>ordered_idom.
   444    \<not> (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) A)
   445      \<le> A * \<bar>(g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x A)\<bar>"
   446 have 2: "\<And>X2\<Colon>'a\<Colon>type.
   447    \<not> (c\<Colon>'b\<Colon>ordered_idom) * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) X2
   448      < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) X2"
   449   by (metis 0 linorder_not_le)
   450 have 3: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
   451    \<not> (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)
   452      \<le> \<bar>X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)\<bar>"
   453   by (metis abs_mult 1)
   454 have 4: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
   455    \<bar>X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)\<bar>
   456    < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)"
   457   by (metis 3 linorder_not_less)
   458 have 5: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
   459    X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)
   460    < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)"
   461   by (metis abs_less_iff 4)
   462 show "False"
   463   by (metis 2 5)
   464 qed
   465 
   466 
   467 lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> 
   468     f : O(g)" 
   469   apply (erule bigo_bounded_alt [of f 1 g])
   470   apply simp
   471 done
   472 
   473 ML_command{*AtpWrapper.problem_name := "BigO__bigo_bounded2"*}
   474 lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
   475     f : lb +o O(g)"
   476   apply (rule set_minus_imp_plus)
   477   apply (rule bigo_bounded)
   478   apply (auto simp add: diff_minus fun_Compl_def func_plus)
   479   prefer 2
   480   apply (drule_tac x = x in spec)+ 
   481   apply arith (*not clear that it's provable otherwise*) 
   482 proof (neg_clausify)
   483 fix x
   484 assume 0: "\<And>y. lb y \<le> f y"
   485 assume 1: "\<not> (0\<Colon>'b) \<le> f x + - lb x"
   486 have 2: "\<And>X3. (0\<Colon>'b) + X3 = X3"
   487   by (metis diff_eq_eq right_minus_eq)
   488 have 3: "\<not> (0\<Colon>'b) \<le> f x - lb x"
   489   by (metis 1 diff_minus)
   490 have 4: "\<not> (0\<Colon>'b) + lb x \<le> f x"
   491   by (metis 3 le_diff_eq)
   492 show "False"
   493   by (metis 4 2 0)
   494 qed
   495 
   496 ML_command{*AtpWrapper.problem_name := "BigO__bigo_abs"*}
   497 lemma bigo_abs: "(%x. abs(f x)) =o O(f)" 
   498   apply (unfold bigo_def)
   499   apply auto
   500 proof (neg_clausify)
   501 fix x
   502 assume 0: "\<And>xa. \<not> \<bar>f (x xa)\<bar> \<le> xa * \<bar>f (x xa)\<bar>"
   503 have 1: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2 \<or> \<not> (1\<Colon>'b) \<le> (1\<Colon>'b)"
   504   by (metis mult_le_cancel_right1 order_eq_iff)
   505 have 2: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2"
   506   by (metis order_eq_iff 1)
   507 show "False"
   508   by (metis 0 2)
   509 qed
   510 
   511 ML_command{*AtpWrapper.problem_name := "BigO__bigo_abs2"*}
   512 lemma bigo_abs2: "f =o O(%x. abs(f x))"
   513   apply (unfold bigo_def)
   514   apply auto
   515 proof (neg_clausify)
   516 fix x
   517 assume 0: "\<And>xa. \<not> \<bar>f (x xa)\<bar> \<le> xa * \<bar>f (x xa)\<bar>"
   518 have 1: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2 \<or> \<not> (1\<Colon>'b) \<le> (1\<Colon>'b)"
   519   by (metis mult_le_cancel_right1 order_eq_iff)
   520 have 2: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2"
   521   by (metis order_eq_iff 1)
   522 show "False"
   523   by (metis 0 2)
   524 qed
   525  
   526 lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
   527   apply (rule equalityI)
   528   apply (rule bigo_elt_subset)
   529   apply (rule bigo_abs2)
   530   apply (rule bigo_elt_subset)
   531   apply (rule bigo_abs)
   532 done
   533 
   534 lemma bigo_abs4: "f =o g +o O(h) ==> 
   535     (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
   536   apply (drule set_plus_imp_minus)
   537   apply (rule set_minus_imp_plus)
   538   apply (subst fun_diff_def)
   539 proof -
   540   assume a: "f - g : O(h)"
   541   have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
   542     by (rule bigo_abs2)
   543   also have "... <= O(%x. abs (f x - g x))"
   544     apply (rule bigo_elt_subset)
   545     apply (rule bigo_bounded)
   546     apply force
   547     apply (rule allI)
   548     apply (rule abs_triangle_ineq3)
   549     done
   550   also have "... <= O(f - g)"
   551     apply (rule bigo_elt_subset)
   552     apply (subst fun_diff_def)
   553     apply (rule bigo_abs)
   554     done
   555   also have "... <= O(h)"
   556     using a by (rule bigo_elt_subset)
   557   finally show "(%x. abs (f x) - abs (g x)) : O(h)".
   558 qed
   559 
   560 lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" 
   561 by (unfold bigo_def, auto)
   562 
   563 lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) \<oplus> O(h)"
   564 proof -
   565   assume "f : g +o O(h)"
   566   also have "... <= O(g) \<oplus> O(h)"
   567     by (auto del: subsetI)
   568   also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
   569     apply (subst bigo_abs3 [symmetric])+
   570     apply (rule refl)
   571     done
   572   also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
   573     by (rule bigo_plus_eq [symmetric], auto)
   574   finally have "f : ...".
   575   then have "O(f) <= ..."
   576     by (elim bigo_elt_subset)
   577   also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
   578     by (rule bigo_plus_eq, auto)
   579   finally show ?thesis
   580     by (simp add: bigo_abs3 [symmetric])
   581 qed
   582 
   583 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult"*}
   584 lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)"
   585   apply (rule subsetI)
   586   apply (subst bigo_def)
   587   apply (auto simp del: abs_mult mult_ac
   588               simp add: bigo_alt_def set_times_def func_times)
   589 (*sledgehammer*); 
   590   apply (rule_tac x = "c * ca" in exI)
   591   apply(rule allI)
   592   apply(erule_tac x = x in allE)+
   593   apply(subgoal_tac "c * ca * abs(f x * g x) = 
   594       (c * abs(f x)) * (ca * abs(g x))")
   595 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult_simpler"*}
   596 prefer 2 
   597 apply (metis mult_assoc mult_left_commute
   598   OrderedGroup.abs_of_pos OrderedGroup.mult_left_commute
   599   Ring_and_Field.abs_mult Ring_and_Field.mult_pos_pos)
   600   apply (erule ssubst) 
   601   apply (subst abs_mult)
   602 (*not qute BigO__bigo_mult_simpler_1 (a hard problem!) as abs_mult has
   603   just been done*)
   604 proof (neg_clausify)
   605 fix a c b ca x
   606 assume 0: "(0\<Colon>'b\<Colon>ordered_idom) < (c\<Colon>'b\<Colon>ordered_idom)"
   607 assume 1: "\<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
   608 \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
   609 assume 2: "\<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
   610 \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
   611 assume 3: "\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar> *
   612   \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>
   613   \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> *
   614     ((ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>)"
   615 have 4: "\<bar>c\<Colon>'b\<Colon>ordered_idom\<bar> = c"
   616   by (metis OrderedGroup.abs_of_pos 0)
   617 have 5: "\<And>X1\<Colon>'b\<Colon>ordered_idom. (c\<Colon>'b\<Colon>ordered_idom) * \<bar>X1\<bar> = \<bar>c * X1\<bar>"
   618   by (metis Ring_and_Field.abs_mult 4)
   619 have 6: "(0\<Colon>'b\<Colon>ordered_idom) = (1\<Colon>'b\<Colon>ordered_idom) \<or>
   620 (0\<Colon>'b\<Colon>ordered_idom) < (1\<Colon>'b\<Colon>ordered_idom)"
   621   by (metis OrderedGroup.abs_not_less_zero Ring_and_Field.abs_one Ring_and_Field.linorder_neqE_ordered_idom)
   622 have 7: "(0\<Colon>'b\<Colon>ordered_idom) < (1\<Colon>'b\<Colon>ordered_idom)"
   623   by (metis 6 Ring_and_Field.one_neq_zero)
   624 have 8: "\<bar>1\<Colon>'b\<Colon>ordered_idom\<bar> = (1\<Colon>'b\<Colon>ordered_idom)"
   625   by (metis OrderedGroup.abs_of_pos 7)
   626 have 9: "\<And>X1\<Colon>'b\<Colon>ordered_idom. (0\<Colon>'b\<Colon>ordered_idom) \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>X1\<bar>"
   627   by (metis OrderedGroup.abs_ge_zero 5)
   628 have 10: "\<And>X1\<Colon>'b\<Colon>ordered_idom. X1 * (1\<Colon>'b\<Colon>ordered_idom) = X1"
   629   by (metis Ring_and_Field.mult_cancel_right2 mult_commute)
   630 have 11: "\<And>X1\<Colon>'b\<Colon>ordered_idom. \<bar>\<bar>X1\<bar>\<bar> = \<bar>X1\<bar> * \<bar>1\<Colon>'b\<Colon>ordered_idom\<bar>"
   631   by (metis Ring_and_Field.abs_mult OrderedGroup.abs_idempotent 10)
   632 have 12: "\<And>X1\<Colon>'b\<Colon>ordered_idom. \<bar>\<bar>X1\<bar>\<bar> = \<bar>X1\<bar>"
   633   by (metis 11 8 10)
   634 have 13: "\<And>X1\<Colon>'b\<Colon>ordered_idom. (0\<Colon>'b\<Colon>ordered_idom) \<le> \<bar>X1\<bar>"
   635   by (metis OrderedGroup.abs_ge_zero 12)
   636 have 14: "\<not> (0\<Colon>'b\<Colon>ordered_idom)
   637   \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar> \<or>
   638 \<not> (0\<Colon>'b\<Colon>ordered_idom) \<le> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
   639 \<not> \<bar>b x\<bar> \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
   640 \<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<le> c * \<bar>f x\<bar>"
   641   by (metis 3 Ring_and_Field.mult_mono)
   642 have 15: "\<not> (0\<Colon>'b\<Colon>ordered_idom) \<le> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar> \<or>
   643 \<not> \<bar>b x\<bar> \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
   644 \<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>
   645   \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
   646   by (metis 14 9)
   647 have 16: "\<not> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
   648   \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
   649 \<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>
   650   \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
   651   by (metis 15 13)
   652 have 17: "\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
   653   \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
   654   by (metis 16 2)
   655 show 18: "False"
   656   by (metis 17 1)
   657 qed
   658 
   659 
   660 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult2"*}
   661 lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
   662   apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
   663 (*sledgehammer*); 
   664   apply (rule_tac x = c in exI)
   665   apply clarify
   666   apply (drule_tac x = x in spec)
   667 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult2_simpler"*}
   668 (*sledgehammer [no luck]*); 
   669   apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
   670   apply (simp add: mult_ac)
   671   apply (rule mult_left_mono, assumption)
   672   apply (rule abs_ge_zero)
   673 done
   674 
   675 ML_command{*AtpWrapper.problem_name:="BigO__bigo_mult3"*}
   676 lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
   677 by (metis bigo_mult set_times_intro subset_iff)
   678 
   679 ML_command{*AtpWrapper.problem_name:="BigO__bigo_mult4"*}
   680 lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
   681 by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib)
   682 
   683 
   684 lemma bigo_mult5: "ALL x. f x ~= 0 ==>
   685     O(f * g) <= (f::'a => ('b::ordered_field)) *o O(g)"
   686 proof -
   687   assume "ALL x. f x ~= 0"
   688   show "O(f * g) <= f *o O(g)"
   689   proof
   690     fix h
   691     assume "h : O(f * g)"
   692     then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
   693       by auto
   694     also have "... <= O((%x. 1 / f x) * (f * g))"
   695       by (rule bigo_mult2)
   696     also have "(%x. 1 / f x) * (f * g) = g"
   697       apply (simp add: func_times) 
   698       apply (rule ext)
   699       apply (simp add: prems nonzero_divide_eq_eq mult_ac)
   700       done
   701     finally have "(%x. (1::'b) / f x) * h : O(g)".
   702     then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
   703       by auto
   704     also have "f * ((%x. (1::'b) / f x) * h) = h"
   705       apply (simp add: func_times) 
   706       apply (rule ext)
   707       apply (simp add: prems nonzero_divide_eq_eq mult_ac)
   708       done
   709     finally show "h : f *o O(g)".
   710   qed
   711 qed
   712 
   713 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult6"*}
   714 lemma bigo_mult6: "ALL x. f x ~= 0 ==>
   715     O(f * g) = (f::'a => ('b::ordered_field)) *o O(g)"
   716 by (metis bigo_mult2 bigo_mult5 order_antisym)
   717 
   718 (*proof requires relaxing relevance: 2007-01-25*)
   719 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult7"*}
   720   declare bigo_mult6 [simp]
   721 lemma bigo_mult7: "ALL x. f x ~= 0 ==>
   722     O(f * g) <= O(f::'a => ('b::ordered_field)) \<otimes> O(g)"
   723 (*sledgehammer*)
   724   apply (subst bigo_mult6)
   725   apply assumption
   726   apply (rule set_times_mono3) 
   727   apply (rule bigo_refl)
   728 done
   729   declare bigo_mult6 [simp del]
   730 
   731 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult8"*}
   732   declare bigo_mult7[intro!]
   733 lemma bigo_mult8: "ALL x. f x ~= 0 ==>
   734     O(f * g) = O(f::'a => ('b::ordered_field)) \<otimes> O(g)"
   735 by (metis bigo_mult bigo_mult7 order_antisym_conv)
   736 
   737 lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
   738   by (auto simp add: bigo_def fun_Compl_def)
   739 
   740 lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
   741   apply (rule set_minus_imp_plus)
   742   apply (drule set_plus_imp_minus)
   743   apply (drule bigo_minus)
   744   apply (simp add: diff_minus)
   745 done
   746 
   747 lemma bigo_minus3: "O(-f) = O(f)"
   748   by (auto simp add: bigo_def fun_Compl_def abs_minus_cancel)
   749 
   750 lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
   751 proof -
   752   assume a: "f : O(g)"
   753   show "f +o O(g) <= O(g)"
   754   proof -
   755     have "f : O(f)" by auto
   756     then have "f +o O(g) <= O(f) \<oplus> O(g)"
   757       by (auto del: subsetI)
   758     also have "... <= O(g) \<oplus> O(g)"
   759     proof -
   760       from a have "O(f) <= O(g)" by (auto del: subsetI)
   761       thus ?thesis by (auto del: subsetI)
   762     qed
   763     also have "... <= O(g)" by (simp add: bigo_plus_idemp)
   764     finally show ?thesis .
   765   qed
   766 qed
   767 
   768 lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
   769 proof -
   770   assume a: "f : O(g)"
   771   show "O(g) <= f +o O(g)"
   772   proof -
   773     from a have "-f : O(g)" by auto
   774     then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1)
   775     then have "f +o (-f +o O(g)) <= f +o O(g)" by auto
   776     also have "f +o (-f +o O(g)) = O(g)"
   777       by (simp add: set_plus_rearranges)
   778     finally show ?thesis .
   779   qed
   780 qed
   781 
   782 ML_command{*AtpWrapper.problem_name:="BigO__bigo_plus_absorb"*}
   783 lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
   784 by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff);
   785 
   786 lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
   787   apply (subgoal_tac "f +o A <= f +o O(g)")
   788   apply force+
   789 done
   790 
   791 lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
   792   apply (subst set_minus_plus [symmetric])
   793   apply (subgoal_tac "g - f = - (f - g)")
   794   apply (erule ssubst)
   795   apply (rule bigo_minus)
   796   apply (subst set_minus_plus)
   797   apply assumption
   798   apply  (simp add: diff_minus add_ac)
   799 done
   800 
   801 lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
   802   apply (rule iffI)
   803   apply (erule bigo_add_commute_imp)+
   804 done
   805 
   806 lemma bigo_const1: "(%x. c) : O(%x. 1)"
   807 by (auto simp add: bigo_def mult_ac)
   808 
   809 ML_command{*AtpWrapper.problem_name:="BigO__bigo_const2"*}
   810 lemma (*bigo_const2 [intro]:*) "O(%x. c) <= O(%x. 1)"
   811 by (metis bigo_const1 bigo_elt_subset);
   812 
   813 lemma bigo_const2 [intro]: "O(%x. c::'b::ordered_idom) <= O(%x. 1)";
   814 (*??FAILS because the two occurrences of COMBK have different polymorphic types
   815 proof (neg_clausify)
   816 assume 0: "\<not> O(COMBK (c\<Colon>'b\<Colon>ordered_idom)) \<subseteq> O(COMBK (1\<Colon>'b\<Colon>ordered_idom))"
   817 have 1: "COMBK (c\<Colon>'b\<Colon>ordered_idom) \<notin> O(COMBK (1\<Colon>'b\<Colon>ordered_idom))"
   818 apply (rule notI) 
   819 apply (rule 0 [THEN notE]) 
   820 apply (rule bigo_elt_subset) 
   821 apply assumption; 
   822 sorry
   823   by (metis 0 bigo_elt_subset)  loops??
   824 show "False"
   825   by (metis 1 bigo_const1)
   826 qed
   827 *)
   828   apply (rule bigo_elt_subset)
   829   apply (rule bigo_const1)
   830 done
   831 
   832 ML_command{*AtpWrapper.problem_name := "BigO__bigo_const3"*}
   833 lemma bigo_const3: "(c::'a::ordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
   834 apply (simp add: bigo_def)
   835 proof (neg_clausify)
   836 assume 0: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> (0\<Colon>'a\<Colon>ordered_field)"
   837 assume 1: "\<And>A\<Colon>'a\<Colon>ordered_field. \<not> (1\<Colon>'a\<Colon>ordered_field) \<le> A * \<bar>c\<Colon>'a\<Colon>ordered_field\<bar>"
   838 have 2: "(0\<Colon>'a\<Colon>ordered_field) = \<bar>c\<Colon>'a\<Colon>ordered_field\<bar> \<or>
   839 \<not> (1\<Colon>'a\<Colon>ordered_field) \<le> (1\<Colon>'a\<Colon>ordered_field)"
   840   by (metis 1 field_inverse)
   841 have 3: "\<bar>c\<Colon>'a\<Colon>ordered_field\<bar> = (0\<Colon>'a\<Colon>ordered_field)"
   842   by (metis linorder_neq_iff linorder_antisym_conv1 2)
   843 have 4: "(0\<Colon>'a\<Colon>ordered_field) = (c\<Colon>'a\<Colon>ordered_field)"
   844   by (metis 3 abs_eq_0)
   845 show "False"
   846   by (metis 0 4)
   847 qed
   848 
   849 lemma bigo_const4: "(c::'a::ordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
   850 by (rule bigo_elt_subset, rule bigo_const3, assumption)
   851 
   852 lemma bigo_const [simp]: "(c::'a::ordered_field) ~= 0 ==> 
   853     O(%x. c) = O(%x. 1)"
   854 by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
   855 
   856 ML_command{*AtpWrapper.problem_name := "BigO__bigo_const_mult1"*}
   857 lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
   858   apply (simp add: bigo_def abs_mult)
   859 proof (neg_clausify)
   860 fix x
   861 assume 0: "\<And>xa\<Colon>'b\<Colon>ordered_idom.
   862    \<not> \<bar>c\<Colon>'b\<Colon>ordered_idom\<bar> *
   863      \<bar>(f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) xa)\<bar>
   864      \<le> xa * \<bar>f (x xa)\<bar>"
   865 show "False"
   866   by (metis linorder_neq_iff linorder_antisym_conv1 0)
   867 qed
   868 
   869 lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
   870 by (rule bigo_elt_subset, rule bigo_const_mult1)
   871 
   872 ML_command{*AtpWrapper.problem_name := "BigO__bigo_const_mult3"*}
   873 lemma bigo_const_mult3: "(c::'a::ordered_field) ~= 0 ==> f : O(%x. c * f x)"
   874   apply (simp add: bigo_def)
   875 (*sledgehammer [no luck]*); 
   876   apply (rule_tac x = "abs(inverse c)" in exI)
   877   apply (simp only: abs_mult [symmetric] mult_assoc [symmetric])
   878 apply (subst left_inverse) 
   879 apply (auto ); 
   880 done
   881 
   882 lemma bigo_const_mult4: "(c::'a::ordered_field) ~= 0 ==> 
   883     O(f) <= O(%x. c * f x)"
   884 by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
   885 
   886 lemma bigo_const_mult [simp]: "(c::'a::ordered_field) ~= 0 ==> 
   887     O(%x. c * f x) = O(f)"
   888 by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
   889 
   890 ML_command{*AtpWrapper.problem_name := "BigO__bigo_const_mult5"*}
   891 lemma bigo_const_mult5 [simp]: "(c::'a::ordered_field) ~= 0 ==> 
   892     (%x. c) *o O(f) = O(f)"
   893   apply (auto del: subsetI)
   894   apply (rule order_trans)
   895   apply (rule bigo_mult2)
   896   apply (simp add: func_times)
   897   apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times)
   898   apply (rule_tac x = "%y. inverse c * x y" in exI)
   899   apply (rename_tac g d) 
   900   apply safe
   901   apply (rule_tac [2] ext) 
   902    prefer 2 
   903    apply simp
   904   apply (simp add: mult_assoc [symmetric] abs_mult)
   905   (*couldn't get this proof without the step above; SLOW*)
   906   apply (metis mult_assoc abs_ge_zero mult_left_mono)
   907 done
   908 
   909 
   910 ML_command{*AtpWrapper.problem_name := "BigO__bigo_const_mult6"*}
   911 lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
   912   apply (auto intro!: subsetI
   913     simp add: bigo_def elt_set_times_def func_times
   914     simp del: abs_mult mult_ac)
   915 (*sledgehammer*); 
   916   apply (rule_tac x = "ca * (abs c)" in exI)
   917   apply (rule allI)
   918   apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
   919   apply (erule ssubst)
   920   apply (subst abs_mult)
   921   apply (rule mult_left_mono)
   922   apply (erule spec)
   923   apply simp
   924   apply(simp add: mult_ac)
   925 done
   926 
   927 lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
   928 proof -
   929   assume "f =o O(g)"
   930   then have "(%x. c) * f =o (%x. c) *o O(g)"
   931     by auto
   932   also have "(%x. c) * f = (%x. c * f x)"
   933     by (simp add: func_times)
   934   also have "(%x. c) *o O(g) <= O(g)"
   935     by (auto del: subsetI)
   936   finally show ?thesis .
   937 qed
   938 
   939 lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
   940 by (unfold bigo_def, auto)
   941 
   942 lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o 
   943     O(%x. h(k x))"
   944   apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def
   945       func_plus)
   946   apply (erule bigo_compose1)
   947 done
   948 
   949 subsection {* Setsum *}
   950 
   951 lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==> 
   952     EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
   953       (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"  
   954   apply (auto simp add: bigo_def)
   955   apply (rule_tac x = "abs c" in exI)
   956   apply (subst abs_of_nonneg) back back
   957   apply (rule setsum_nonneg)
   958   apply force
   959   apply (subst setsum_right_distrib)
   960   apply (rule allI)
   961   apply (rule order_trans)
   962   apply (rule setsum_abs)
   963   apply (rule setsum_mono)
   964 apply (blast intro: order_trans mult_right_mono abs_ge_self) 
   965 done
   966 
   967 ML_command{*AtpWrapper.problem_name := "BigO__bigo_setsum1"*}
   968 lemma bigo_setsum1: "ALL x y. 0 <= h x y ==> 
   969     EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
   970       (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
   971   apply (rule bigo_setsum_main)
   972 (*sledgehammer*); 
   973   apply force
   974   apply clarsimp
   975   apply (rule_tac x = c in exI)
   976   apply force
   977 done
   978 
   979 lemma bigo_setsum2: "ALL y. 0 <= h y ==> 
   980     EX c. ALL y. abs(f y) <= c * (h y) ==>
   981       (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
   982 by (rule bigo_setsum1, auto)  
   983 
   984 ML_command{*AtpWrapper.problem_name := "BigO__bigo_setsum3"*}
   985 lemma bigo_setsum3: "f =o O(h) ==>
   986     (%x. SUM y : A x. (l x y) * f(k x y)) =o
   987       O(%x. SUM y : A x. abs(l x y * h(k x y)))"
   988   apply (rule bigo_setsum1)
   989   apply (rule allI)+
   990   apply (rule abs_ge_zero)
   991   apply (unfold bigo_def)
   992   apply (auto simp add: abs_mult);
   993 (*sledgehammer*); 
   994   apply (rule_tac x = c in exI)
   995   apply (rule allI)+
   996   apply (subst mult_left_commute)
   997   apply (rule mult_left_mono)
   998   apply (erule spec)
   999   apply (rule abs_ge_zero)
  1000 done
  1001 
  1002 lemma bigo_setsum4: "f =o g +o O(h) ==>
  1003     (%x. SUM y : A x. l x y * f(k x y)) =o
  1004       (%x. SUM y : A x. l x y * g(k x y)) +o
  1005         O(%x. SUM y : A x. abs(l x y * h(k x y)))"
  1006   apply (rule set_minus_imp_plus)
  1007   apply (subst fun_diff_def)
  1008   apply (subst setsum_subtractf [symmetric])
  1009   apply (subst right_diff_distrib [symmetric])
  1010   apply (rule bigo_setsum3)
  1011   apply (subst fun_diff_def [symmetric])
  1012   apply (erule set_plus_imp_minus)
  1013 done
  1014 
  1015 ML_command{*AtpWrapper.problem_name := "BigO__bigo_setsum5"*}
  1016 lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==> 
  1017     ALL x. 0 <= h x ==>
  1018       (%x. SUM y : A x. (l x y) * f(k x y)) =o
  1019         O(%x. SUM y : A x. (l x y) * h(k x y))" 
  1020   apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) = 
  1021       (%x. SUM y : A x. abs((l x y) * h(k x y)))")
  1022   apply (erule ssubst)
  1023   apply (erule bigo_setsum3)
  1024   apply (rule ext)
  1025   apply (rule setsum_cong2)
  1026   apply (thin_tac "f \<in> O(h)") 
  1027 apply (metis abs_of_nonneg zero_le_mult_iff)
  1028 done
  1029 
  1030 lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
  1031     ALL x. 0 <= h x ==>
  1032       (%x. SUM y : A x. (l x y) * f(k x y)) =o
  1033         (%x. SUM y : A x. (l x y) * g(k x y)) +o
  1034           O(%x. SUM y : A x. (l x y) * h(k x y))" 
  1035   apply (rule set_minus_imp_plus)
  1036   apply (subst fun_diff_def)
  1037   apply (subst setsum_subtractf [symmetric])
  1038   apply (subst right_diff_distrib [symmetric])
  1039   apply (rule bigo_setsum5)
  1040   apply (subst fun_diff_def [symmetric])
  1041   apply (drule set_plus_imp_minus)
  1042   apply auto
  1043 done
  1044 
  1045 subsection {* Misc useful stuff *}
  1046 
  1047 lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
  1048   A \<oplus> B <= O(f)"
  1049   apply (subst bigo_plus_idemp [symmetric])
  1050   apply (rule set_plus_mono2)
  1051   apply assumption+
  1052 done
  1053 
  1054 lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
  1055   apply (subst bigo_plus_idemp [symmetric])
  1056   apply (rule set_plus_intro)
  1057   apply assumption+
  1058 done
  1059   
  1060 lemma bigo_useful_const_mult: "(c::'a::ordered_field) ~= 0 ==> 
  1061     (%x. c) * f =o O(h) ==> f =o O(h)"
  1062   apply (rule subsetD)
  1063   apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
  1064   apply assumption
  1065   apply (rule bigo_const_mult6)
  1066   apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
  1067   apply (erule ssubst)
  1068   apply (erule set_times_intro2)
  1069   apply (simp add: func_times) 
  1070 done
  1071 
  1072 ML_command{*AtpWrapper.problem_name := "BigO__bigo_fix"*}
  1073 lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
  1074     f =o O(h)"
  1075   apply (simp add: bigo_alt_def)
  1076 (*sledgehammer*); 
  1077   apply clarify
  1078   apply (rule_tac x = c in exI)
  1079   apply safe
  1080   apply (case_tac "x = 0")
  1081 apply (metis OrderedGroup.abs_ge_zero  OrderedGroup.abs_zero  order_less_le  Ring_and_Field.split_mult_pos_le) 
  1082   apply (subgoal_tac "x = Suc (x - 1)")
  1083   apply metis
  1084   apply simp
  1085   done
  1086 
  1087 
  1088 lemma bigo_fix2: 
  1089     "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==> 
  1090        f 0 = g 0 ==> f =o g +o O(h)"
  1091   apply (rule set_minus_imp_plus)
  1092   apply (rule bigo_fix)
  1093   apply (subst fun_diff_def)
  1094   apply (subst fun_diff_def [symmetric])
  1095   apply (rule set_plus_imp_minus)
  1096   apply simp
  1097   apply (simp add: fun_diff_def)
  1098 done
  1099 
  1100 subsection {* Less than or equal to *}
  1101 
  1102 constdefs 
  1103   lesso :: "('a => 'b::ordered_idom) => ('a => 'b) => ('a => 'b)"
  1104       (infixl "<o" 70)
  1105   "f <o g == (%x. max (f x - g x) 0)"
  1106 
  1107 lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
  1108     g =o O(h)"
  1109   apply (unfold bigo_def)
  1110   apply clarsimp
  1111 apply (blast intro: order_trans) 
  1112 done
  1113 
  1114 lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
  1115       g =o O(h)"
  1116   apply (erule bigo_lesseq1)
  1117 apply (blast intro: abs_ge_self order_trans) 
  1118 done
  1119 
  1120 lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
  1121       g =o O(h)"
  1122   apply (erule bigo_lesseq2)
  1123   apply (rule allI)
  1124   apply (subst abs_of_nonneg)
  1125   apply (erule spec)+
  1126 done
  1127 
  1128 lemma bigo_lesseq4: "f =o O(h) ==>
  1129     ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
  1130       g =o O(h)"
  1131   apply (erule bigo_lesseq1)
  1132   apply (rule allI)
  1133   apply (subst abs_of_nonneg)
  1134   apply (erule spec)+
  1135 done
  1136 
  1137 ML_command{*AtpWrapper.problem_name:="BigO__bigo_lesso1"*}
  1138 lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
  1139   apply (unfold lesso_def)
  1140   apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
  1141 (*??Translation of TSTP raised an exception: Type unification failed: Variable ?'X2.0::type not of sort ord*)
  1142 apply (metis bigo_zero)
  1143   apply (unfold func_zero)
  1144   apply (rule ext)
  1145   apply (simp split: split_max)
  1146 done
  1147 
  1148 
  1149 ML_command{*AtpWrapper.problem_name := "BigO__bigo_lesso2"*}
  1150 lemma bigo_lesso2: "f =o g +o O(h) ==>
  1151     ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
  1152       k <o g =o O(h)"
  1153   apply (unfold lesso_def)
  1154   apply (rule bigo_lesseq4)
  1155   apply (erule set_plus_imp_minus)
  1156   apply (rule allI)
  1157   apply (rule le_maxI2)
  1158   apply (rule allI)
  1159   apply (subst fun_diff_def)
  1160 apply (erule thin_rl)
  1161 (*sledgehammer*);  
  1162   apply (case_tac "0 <= k x - g x")
  1163   prefer 2 (*re-order subgoals because I don't know what to put after a structured proof*)
  1164    apply (metis abs_ge_zero abs_minus_commute linorder_linear min_max.sup_absorb1 min_max.sup_commute)
  1165 proof (neg_clausify)
  1166 fix x
  1167 assume 0: "\<And>A. k A \<le> f A"
  1168 have 1: "\<And>(X1\<Colon>'b\<Colon>ordered_idom) X2. \<not> max X1 X2 < X1"
  1169   by (metis linorder_not_less le_maxI1)  (*sort inserted by hand*)
  1170 assume 2: "(0\<Colon>'b) \<le> k x - g x"
  1171 have 3: "\<not> k x - g x < (0\<Colon>'b)"
  1172   by (metis 2 linorder_not_less)
  1173 have 4: "\<And>X1 X2. min X1 (k X2) \<le> f X2"
  1174   by (metis min_max.inf_le2 min_max.le_inf_iff min_max.le_iff_inf 0)
  1175 have 5: "\<bar>g x - f x\<bar> = f x - g x"
  1176   by (metis abs_minus_commute combine_common_factor mult_zero_right minus_add_cancel minus_zero abs_if diff_less_eq min_max.inf_commute 4 linorder_not_le min_max.le_iff_inf 3 diff_less_0_iff_less linorder_not_less)
  1177 have 6: "max (0\<Colon>'b) (k x - g x) = k x - g x"
  1178   by (metis min_max.le_iff_sup 2)
  1179 assume 7: "\<not> max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>"
  1180 have 8: "\<not> k x - g x \<le> f x - g x"
  1181   by (metis 5 abs_minus_commute 7 min_max.sup_commute 6)
  1182 show "False"
  1183   by (metis min_max.sup_commute min_max.inf_commute min_max.sup_inf_absorb min_max.le_iff_inf 0 max_diff_distrib_left 1 linorder_not_le 8)
  1184 qed
  1185 
  1186 ML_command{*AtpWrapper.problem_name := "BigO__bigo_lesso3"*}
  1187 lemma bigo_lesso3: "f =o g +o O(h) ==>
  1188     ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
  1189       f <o k =o O(h)"
  1190   apply (unfold lesso_def)
  1191   apply (rule bigo_lesseq4)
  1192   apply (erule set_plus_imp_minus)
  1193   apply (rule allI)
  1194   apply (rule le_maxI2)
  1195   apply (rule allI)
  1196   apply (subst fun_diff_def)
  1197 apply (erule thin_rl) 
  1198 (*sledgehammer*); 
  1199   apply (case_tac "0 <= f x - k x")
  1200   apply (simp)
  1201   apply (subst abs_of_nonneg)
  1202   apply (drule_tac x = x in spec) back
  1203 ML_command{*AtpWrapper.problem_name := "BigO__bigo_lesso3_simpler"*}
  1204 apply (metis diff_less_0_iff_less linorder_not_le not_leE uminus_add_conv_diff xt1(12) xt1(6))
  1205 apply (metis add_minus_cancel diff_le_eq le_diff_eq uminus_add_conv_diff)
  1206 apply (metis abs_ge_zero linorder_linear min_max.sup_absorb1 min_max.sup_commute)
  1207 done
  1208 
  1209 lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::ordered_field) ==>
  1210     g =o h +o O(k) ==> f <o h =o O(k)"
  1211   apply (unfold lesso_def)
  1212   apply (drule set_plus_imp_minus)
  1213   apply (drule bigo_abs5) back
  1214   apply (simp add: fun_diff_def)
  1215   apply (drule bigo_useful_add)
  1216   apply assumption
  1217   apply (erule bigo_lesseq2) back
  1218   apply (rule allI)
  1219   apply (auto simp add: func_plus fun_diff_def algebra_simps
  1220     split: split_max abs_split)
  1221 done
  1222 
  1223 ML_command{*AtpWrapper.problem_name := "BigO__bigo_lesso5"*}
  1224 lemma bigo_lesso5: "f <o g =o O(h) ==>
  1225     EX C. ALL x. f x <= g x + C * abs(h x)"
  1226   apply (simp only: lesso_def bigo_alt_def)
  1227   apply clarsimp
  1228   apply (metis abs_if abs_mult add_commute diff_le_eq less_not_permute)  
  1229 done
  1230 
  1231 end