src/HOL/MetisExamples/BigO.thy
author haftmann
Tue Nov 06 08:47:30 2007 +0100 (2007-11-06)
changeset 25304 7491c00f0915
parent 25087 5908591fb881
child 25592 e8ddaf6bf5df
permissions -rw-r--r--
removed subclass edge ordered_ring < lordered_ring
     1 (*  Title:      HOL/MetisExamples/BigO.thy
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4 
     5 Testing the metis method
     6 *)
     7 
     8 header {* Big O notation *}
     9 
    10 theory BigO
    11 imports SetsAndFunctions 
    12 begin
    13 
    14 subsection {* Definitions *}
    15 
    16 constdefs 
    17 
    18   bigo :: "('a => 'b::ordered_idom) => ('a => 'b) set"    ("(1O'(_'))")
    19   "O(f::('a => 'b)) ==   {h. EX c. ALL x. abs (h x) <= c * abs (f x)}"
    20 
    21 ML{*ResAtp.problem_name := "BigO__bigo_pos_const"*}
    22 lemma bigo_pos_const: "(EX (c::'a::ordered_idom). 
    23     ALL x. (abs (h x)) <= (c * (abs (f x))))
    24       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
    25   apply auto
    26   apply (case_tac "c = 0", simp)
    27   apply (rule_tac x = "1" in exI, simp)
    28   apply (rule_tac x = "abs c" in exI, auto)
    29   apply (metis abs_ge_minus_self abs_ge_zero abs_minus_cancel abs_of_nonneg equation_minus_iff Orderings.xt1(6) abs_mult)
    30   done
    31 
    32 (*** Now various verions with an increasing modulus ***)
    33 
    34 ML{*ResReconstruct.modulus := 1*}
    35 
    36 lemma bigo_pos_const: "(EX (c::'a::ordered_idom). 
    37     ALL x. (abs (h x)) <= (c * (abs (f x))))
    38       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
    39   apply auto
    40   apply (case_tac "c = 0", simp)
    41   apply (rule_tac x = "1" in exI, simp)
    42   apply (rule_tac x = "abs c" in exI, auto)
    43 proof (neg_clausify)
    44 fix c x
    45 have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * X2\<bar> = \<bar>X2 * X1\<bar>"
    46   by (metis abs_mult mult_commute)
    47 have 1: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
    48    X1 \<le> (0\<Colon>'a\<Colon>ordered_idom) \<or> \<bar>X2\<bar> * X1 = \<bar>X2 * X1\<bar>"
    49   by (metis abs_mult_pos linorder_linear)
    50 have 2: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
    51    \<not> (0\<Colon>'a\<Colon>ordered_idom) < X1 * X2 \<or>
    52    \<not> (0\<Colon>'a\<Colon>ordered_idom) \<le> X2 \<or> \<not> X1 \<le> (0\<Colon>'a\<Colon>ordered_idom)"
    53   by (metis linorder_not_less mult_nonneg_nonpos2)
    54 assume 3: "\<And>x\<Colon>'b\<Colon>type.
    55    \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
    56    \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
    57 assume 4: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
    58   \<le> \<bar>c\<Colon>'a\<Colon>ordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
    59 have 5: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
    60   \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
    61   by (metis 4 abs_mult)
    62 have 6: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
    63    \<not> X1 \<le> (0\<Colon>'a\<Colon>ordered_idom) \<or> X1 \<le> \<bar>X2\<bar>"
    64   by (metis abs_ge_zero xt1(6))
    65 have 7: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
    66    X1 \<le> \<bar>X2\<bar> \<or> (0\<Colon>'a\<Colon>ordered_idom) < X1"
    67   by (metis not_leE 6)
    68 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>"
    69   by (metis 5 7)
    70 have 9: "\<And>X1\<Colon>'a\<Colon>ordered_idom.
    71    \<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar> \<le> X1 \<or>
    72    (0\<Colon>'a\<Colon>ordered_idom) < X1"
    73   by (metis 8 order_less_le_trans)
    74 have 10: "(0\<Colon>'a\<Colon>ordered_idom)
    75 < (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
    76   by (metis 3 9)
    77 have 11: "\<not> (c\<Colon>'a\<Colon>ordered_idom) \<le> (0\<Colon>'a\<Colon>ordered_idom)"
    78   by (metis abs_ge_zero 2 10)
    79 have 12: "\<And>X1\<Colon>'a\<Colon>ordered_idom. (c\<Colon>'a\<Colon>ordered_idom) * \<bar>X1\<bar> = \<bar>X1 * c\<bar>"
    80   by (metis mult_commute 1 11)
    81 have 13: "\<And>X1\<Colon>'b\<Colon>type.
    82    - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
    83    \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
    84   by (metis 3 abs_le_D2)
    85 have 14: "\<And>X1\<Colon>'b\<Colon>type.
    86    - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
    87    \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
    88   by (metis 0 12 13)
    89 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>"
    90   by (metis abs_mult abs_mult_pos abs_ge_zero)
    91 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"
    92   by (metis xt1(6) abs_ge_self)
    93 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>"
    94   by (metis 16 abs_le_D1)
    95 have 18: "\<And>X1\<Colon>'b\<Colon>type.
    96    (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
    97    \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
    98   by (metis 17 3 15)
    99 show "False"
   100   by (metis abs_le_iff 5 18 14)
   101 qed
   102 
   103 lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom). 
   104     ALL x. (abs (h x)) <= (c * (abs (f x))))
   105       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
   106   apply auto
   107   apply (case_tac "c = 0", simp)
   108   apply (rule_tac x = "1" in exI, simp)
   109   apply (rule_tac x = "abs c" in exI, auto);
   110 ML{*ResReconstruct.modulus:=2*}
   111 proof (neg_clausify)
   112 fix c x
   113 have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * X2\<bar> = \<bar>X2 * X1\<bar>"
   114   by (metis abs_mult mult_commute)
   115 assume 1: "\<And>x\<Colon>'b\<Colon>type.
   116    \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
   117    \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
   118 assume 2: "\<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\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
   120 have 3: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
   121   \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
   122   by (metis 2 abs_mult)
   123 have 4: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
   124    \<not> X1 \<le> (0\<Colon>'a\<Colon>ordered_idom) \<or> X1 \<le> \<bar>X2\<bar>"
   125   by (metis abs_ge_zero xt1(6))
   126 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>"
   127   by (metis not_leE 4 3)
   128 have 6: "(0\<Colon>'a\<Colon>ordered_idom)
   129 < (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
   130   by (metis 1 order_less_le_trans 5)
   131 have 7: "\<And>X1\<Colon>'a\<Colon>ordered_idom. (c\<Colon>'a\<Colon>ordered_idom) * \<bar>X1\<bar> = \<bar>X1 * c\<bar>"
   132   by (metis abs_ge_zero linorder_not_less mult_nonneg_nonpos2 6 linorder_linear abs_mult_pos mult_commute)
   133 have 8: "\<And>X1\<Colon>'b\<Colon>type.
   134    - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
   135    \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
   136   by (metis 0 7 abs_le_D2 1)
   137 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>"
   138   by (metis abs_ge_self xt1(6) abs_le_D1)
   139 show "False"
   140   by (metis 8 abs_ge_zero abs_mult_pos abs_mult 1 9 3 abs_le_iff)
   141 qed
   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 ML{*ResReconstruct.modulus:=3*}
   151 proof (neg_clausify)
   152 fix c x
   153 assume 0: "\<And>x\<Colon>'b\<Colon>type.
   154    \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
   155    \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
   156 assume 1: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
   157   \<le> \<bar>c\<Colon>'a\<Colon>ordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
   158 have 2: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
   159    X1 \<le> \<bar>X2\<bar> \<or> (0\<Colon>'a\<Colon>ordered_idom) < X1"
   160   by (metis abs_ge_zero xt1(6) not_leE)
   161 have 3: "\<not> (c\<Colon>'a\<Colon>ordered_idom) \<le> (0\<Colon>'a\<Colon>ordered_idom)"
   162   by (metis abs_ge_zero mult_nonneg_nonpos2 linorder_not_less order_less_le_trans 1 abs_mult 2 0)
   163 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>"
   164   by (metis abs_ge_zero abs_mult_pos abs_mult)
   165 have 5: "\<And>X1\<Colon>'b\<Colon>type.
   166    (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
   167    \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
   168   by (metis 4 0 xt1(6) abs_ge_self abs_le_D1)
   169 show "False"
   170   by (metis abs_mult mult_commute 3 abs_mult_pos linorder_linear 0 abs_le_D2 5 1 abs_le_iff)
   171 qed
   172 
   173 
   174 ML{*ResReconstruct.modulus:=1*}
   175 
   176 lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom). 
   177     ALL x. (abs (h x)) <= (c * (abs (f x))))
   178       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
   179   apply auto
   180   apply (case_tac "c = 0", simp)
   181   apply (rule_tac x = "1" in exI, simp)
   182   apply (rule_tac x = "abs c" in exI, auto);
   183 proof (neg_clausify)
   184 fix c x  (*sort/type constraint inserted by hand!*)
   185 have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
   186   by (metis abs_ge_zero abs_mult_pos abs_mult)
   187 assume 1: "\<And>A. \<bar>h A\<bar> \<le> c * \<bar>f A\<bar>"
   188 have 2: "\<And>X1 X2. \<not> \<bar>X1\<bar> \<le> X2 \<or> (0\<Colon>'a) \<le> X2"
   189   by (metis abs_ge_zero order_trans)
   190 have 3: "\<And>X1. (0\<Colon>'a) \<le> c * \<bar>f X1\<bar>"
   191   by (metis 1 2)
   192 have 4: "\<And>X1. c * \<bar>f X1\<bar> = \<bar>c * f X1\<bar>"
   193   by (metis 0 abs_of_nonneg 3)
   194 have 5: "\<And>X1. - h X1 \<le> c * \<bar>f X1\<bar>"
   195   by (metis 1 abs_le_D2)
   196 have 6: "\<And>X1. - h X1 \<le> \<bar>c * f X1\<bar>"
   197   by (metis 4 5)
   198 have 7: "\<And>X1. h X1 \<le> c * \<bar>f X1\<bar>"
   199   by (metis 1 abs_le_D1)
   200 have 8: "\<And>X1. h X1 \<le> \<bar>c * f X1\<bar>"
   201   by (metis 4 7)
   202 assume 9: "\<not> \<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>"
   203 have 10: "\<not> \<bar>h x\<bar> \<le> \<bar>c * f x\<bar>"
   204   by (metis abs_mult 9)
   205 show "False"
   206   by (metis 6 8 10 abs_leI)
   207 qed
   208 
   209 
   210 ML{*ResReconstruct.recon_sorts:=true*}
   211 
   212 
   213 lemma bigo_alt_def: "O(f) = 
   214     {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
   215 by (auto simp add: bigo_def bigo_pos_const)
   216 
   217 ML{*ResAtp.problem_name := "BigO__bigo_elt_subset"*}
   218 lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
   219   apply (auto simp add: bigo_alt_def)
   220   apply (rule_tac x = "ca * c" in exI)
   221   apply (rule conjI)
   222   apply (rule mult_pos_pos)
   223   apply (assumption)+ 
   224 (*sledgehammer*);
   225   apply (rule allI)
   226   apply (drule_tac x = "xa" in spec)+
   227   apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))");
   228   apply (erule order_trans)
   229   apply (simp add: mult_ac)
   230   apply (rule mult_left_mono, assumption)
   231   apply (rule order_less_imp_le, assumption);
   232 done
   233 
   234 
   235 ML{*ResAtp.problem_name := "BigO__bigo_refl"*}
   236 lemma bigo_refl [intro]: "f : O(f)"
   237   apply(auto simp add: bigo_def)
   238 proof (neg_clausify)
   239 fix x
   240 assume 0: "\<And>xa. \<not> \<bar>f (x xa)\<bar> \<le> xa * \<bar>f (x xa)\<bar>"
   241 have 1: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2 \<or> \<not> (1\<Colon>'b) \<le> (1\<Colon>'b)"
   242   by (metis mult_le_cancel_right1 order_eq_iff)
   243 have 2: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2"
   244   by (metis order_eq_iff 1)
   245 show "False"
   246   by (metis 0 2)
   247 qed
   248 
   249 ML{*ResAtp.problem_name := "BigO__bigo_zero"*}
   250 lemma bigo_zero: "0 : O(g)"
   251   apply (auto simp add: bigo_def func_zero)
   252 proof (neg_clausify)
   253 fix x
   254 assume 0: "\<And>xa. \<not> (0\<Colon>'b) \<le> xa * \<bar>g (x xa)\<bar>"
   255 have 1: "\<not> (0\<Colon>'b) \<le> (0\<Colon>'b)"
   256   by (metis 0 mult_eq_0_iff)
   257 show "False"
   258   by (metis 1 linorder_neq_iff linorder_antisym_conv1)
   259 qed
   260 
   261 lemma bigo_zero2: "O(%x.0) = {%x.0}"
   262   apply (auto simp add: bigo_def) 
   263   apply (rule ext)
   264   apply auto
   265 done
   266 
   267 lemma bigo_plus_self_subset [intro]: 
   268   "O(f) + O(f) <= O(f)"
   269   apply (auto simp add: bigo_alt_def set_plus)
   270   apply (rule_tac x = "c + ca" in exI)
   271   apply auto
   272   apply (simp add: ring_distribs func_plus)
   273   apply (blast intro:order_trans abs_triangle_ineq add_mono elim:) 
   274 done
   275 
   276 lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
   277   apply (rule equalityI)
   278   apply (rule bigo_plus_self_subset)
   279   apply (rule set_zero_plus2) 
   280   apply (rule bigo_zero)
   281 done
   282 
   283 lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
   284   apply (rule subsetI)
   285   apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus)
   286   apply (subst bigo_pos_const [symmetric])+
   287   apply (rule_tac x = 
   288     "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
   289   apply (rule conjI)
   290   apply (rule_tac x = "c + c" in exI)
   291   apply (clarsimp)
   292   apply (auto)
   293   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
   294   apply (erule_tac x = xa in allE)
   295   apply (erule order_trans)
   296   apply (simp)
   297   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   298   apply (erule order_trans)
   299   apply (simp add: ring_distribs)
   300   apply (rule mult_left_mono)
   301   apply assumption
   302   apply (simp add: order_less_le)
   303   apply (rule mult_left_mono)
   304   apply (simp add: abs_triangle_ineq)
   305   apply (simp add: order_less_le)
   306   apply (rule mult_nonneg_nonneg)
   307   apply (rule add_nonneg_nonneg)
   308   apply auto
   309   apply (rule_tac x = "%n. if (abs (f n)) <  abs (g n) then x n else 0" 
   310      in exI)
   311   apply (rule conjI)
   312   apply (rule_tac x = "c + c" in exI)
   313   apply auto
   314   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
   315   apply (erule_tac x = xa in allE)
   316   apply (erule order_trans)
   317   apply (simp)
   318   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
   319   apply (erule order_trans)
   320   apply (simp add: ring_distribs)
   321   apply (rule mult_left_mono)
   322   apply (simp add: order_less_le)
   323   apply (simp add: order_less_le)
   324   apply (rule mult_left_mono)
   325   apply (rule abs_triangle_ineq)
   326   apply (simp add: order_less_le)
   327 apply (metis abs_not_less_zero double_less_0_iff less_not_permute linorder_not_less mult_less_0_iff)
   328   apply (rule ext)
   329   apply (auto simp add: if_splits linorder_not_le)
   330 done
   331 
   332 lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A + B <= O(f)"
   333   apply (subgoal_tac "A + B <= O(f) + O(f)")
   334   apply (erule order_trans)
   335   apply simp
   336   apply (auto del: subsetI simp del: bigo_plus_idemp)
   337 done
   338 
   339 ML{*ResAtp.problem_name := "BigO__bigo_plus_eq"*}
   340 lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==> 
   341   O(f + g) = O(f) + O(g)"
   342   apply (rule equalityI)
   343   apply (rule bigo_plus_subset)
   344   apply (simp add: bigo_alt_def set_plus func_plus)
   345   apply clarify 
   346 (*sledgehammer*); 
   347   apply (rule_tac x = "max c ca" in exI)
   348   apply (rule conjI)
   349    apply (metis Orderings.less_max_iff_disj)
   350   apply clarify
   351   apply (drule_tac x = "xa" in spec)+
   352   apply (subgoal_tac "0 <= f xa + g xa")
   353   apply (simp add: ring_distribs)
   354   apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
   355   apply (subgoal_tac "abs(a xa) + abs(b xa) <= 
   356       max c ca * f xa + max c ca * g xa")
   357   apply (blast intro: order_trans)
   358   defer 1
   359   apply (rule abs_triangle_ineq)
   360   apply (metis add_nonneg_nonneg)
   361   apply (rule add_mono)
   362 ML{*ResAtp.problem_name := "BigO__bigo_plus_eq_simpler"*} 
   363 (*Found by SPASS; SLOW*)
   364 apply (metis le_maxI2 linorder_linear linorder_not_le min_max.less_eq_less_sup.sup_absorb1 mult_le_cancel_right xt1(6))
   365 apply (metis le_maxI2 linorder_not_le mult_le_cancel_right xt1(6))
   366 done
   367 
   368 ML{*ResAtp.problem_name := "BigO__bigo_bounded_alt"*}
   369 lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
   370     f : O(g)" 
   371   apply (auto simp add: bigo_def)
   372 (*Version 1: one-shot proof*)
   373   apply (metis OrderedGroup.abs_le_D1 Orderings.linorder_class.not_less  order_less_le  Orderings.xt1(12)  Ring_and_Field.abs_mult)
   374   done
   375 
   376 lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> 
   377     f : O(g)" 
   378   apply (auto simp add: bigo_def)
   379 (*Version 2: single-step proof*)
   380 proof (neg_clausify)
   381 fix x
   382 assume 0: "\<And>x. f x \<le> c * g x"
   383 assume 1: "\<And>xa. \<not> f (x xa) \<le> xa * \<bar>g (x xa)\<bar>"
   384 have 2: "\<And>X3. c * g X3 = f X3 \<or> \<not> c * g X3 \<le> f X3"
   385   by (metis 0 order_antisym_conv)
   386 have 3: "\<And>X3. \<not> f (x \<bar>X3\<bar>) \<le> \<bar>X3 * g (x \<bar>X3\<bar>)\<bar>"
   387   by (metis 1 abs_mult)
   388 have 4: "\<And>X1 X3\<Colon>'b\<Colon>ordered_idom. X3 \<le> X1 \<or> X1 \<le> \<bar>X3\<bar>"
   389   by (metis linorder_linear abs_le_D1)
   390 have 5: "\<And>X3::'b. \<bar>X3\<bar> * \<bar>X3\<bar> = X3 * X3"
   391   by (metis abs_mult_self AC_mult.f.commute)
   392 have 6: "\<And>X3. \<not> X3 * X3 < (0\<Colon>'b\<Colon>ordered_idom)"
   393   by (metis not_square_less_zero AC_mult.f.commute)
   394 have 7: "\<And>X1 X3::'b. \<bar>X1\<bar> * \<bar>X3\<bar> = \<bar>X3 * X1\<bar>"
   395   by (metis abs_mult AC_mult.f.commute)
   396 have 8: "\<And>X3::'b. X3 * X3 = \<bar>X3 * X3\<bar>"
   397   by (metis abs_mult 5)
   398 have 9: "\<And>X3. X3 * g (x \<bar>X3\<bar>) \<le> f (x \<bar>X3\<bar>)"
   399   by (metis 3 4)
   400 have 10: "c * g (x \<bar>c\<bar>) = f (x \<bar>c\<bar>)"
   401   by (metis 2 9)
   402 have 11: "\<And>X3::'b. \<bar>X3\<bar> * \<bar>\<bar>X3\<bar>\<bar> = \<bar>X3\<bar> * \<bar>X3\<bar>"
   403   by (metis abs_idempotent abs_mult 8)
   404 have 12: "\<And>X3::'b. \<bar>X3 * \<bar>X3\<bar>\<bar> = \<bar>X3\<bar> * \<bar>X3\<bar>"
   405   by (metis AC_mult.f.commute 7 11)
   406 have 13: "\<And>X3::'b. \<bar>X3 * \<bar>X3\<bar>\<bar> = X3 * X3"
   407   by (metis 8 7 12)
   408 have 14: "\<And>X3. X3 \<le> \<bar>X3\<bar> \<or> X3 < (0\<Colon>'b)"
   409   by (metis abs_ge_self abs_le_D1 abs_if)
   410 have 15: "\<And>X3. X3 \<le> \<bar>X3\<bar> \<or> \<bar>X3\<bar> < (0\<Colon>'b)"
   411   by (metis abs_ge_self abs_le_D1 abs_if)
   412 have 16: "\<And>X3. X3 * X3 < (0\<Colon>'b) \<or> X3 * \<bar>X3\<bar> \<le> X3 * X3"
   413   by (metis 15 13)
   414 have 17: "\<And>X3::'b. X3 * \<bar>X3\<bar> \<le> X3 * X3"
   415   by (metis 16 6)
   416 have 18: "\<And>X3. X3 \<le> \<bar>X3\<bar> \<or> \<not> X3 < (0\<Colon>'b)"
   417   by (metis mult_le_cancel_left 17)
   418 have 19: "\<And>X3::'b. X3 \<le> \<bar>X3\<bar>"
   419   by (metis 18 14)
   420 have 20: "\<not> f (x \<bar>c\<bar>) \<le> \<bar>f (x \<bar>c\<bar>)\<bar>"
   421   by (metis 3 10)
   422 show "False"
   423   by (metis 20 19)
   424 qed
   425 
   426 
   427 text{*So here is the easier (and more natural) problem using transitivity*}
   428 ML{*ResAtp.problem_name := "BigO__bigo_bounded_alt_trans"*}
   429 lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" 
   430   apply (auto simp add: bigo_def)
   431   (*Version 1: one-shot proof*) 
   432 apply (metis Orderings.leD Orderings.leI abs_ge_self abs_le_D1 abs_mult abs_of_nonneg order_le_less xt1(12));
   433   done
   434 
   435 text{*So here is the easier (and more natural) problem using transitivity*}
   436 ML{*ResAtp.problem_name := "BigO__bigo_bounded_alt_trans"*}
   437 lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" 
   438   apply (auto simp add: bigo_def)
   439 (*Version 2: single-step proof*)
   440 proof (neg_clausify)
   441 fix x
   442 assume 0: "\<And>A\<Colon>'a\<Colon>type.
   443    (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) A
   444    \<le> (c\<Colon>'b\<Colon>ordered_idom) * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) A"
   445 assume 1: "\<And>A\<Colon>'b\<Colon>ordered_idom.
   446    \<not> (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) A)
   447      \<le> A * \<bar>(g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x A)\<bar>"
   448 have 2: "\<And>X2\<Colon>'a\<Colon>type.
   449    \<not> (c\<Colon>'b\<Colon>ordered_idom) * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) X2
   450      < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) X2"
   451   by (metis 0 linorder_not_le)
   452 have 3: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
   453    \<not> (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)
   454      \<le> \<bar>X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)\<bar>"
   455   by (metis abs_mult 1)
   456 have 4: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
   457    \<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>
   458    < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)"
   459   by (metis 3 linorder_not_less)
   460 have 5: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
   461    X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)
   462    < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)"
   463   by (metis abs_less_iff 4)
   464 show "False"
   465   by (metis 2 5)
   466 qed
   467 
   468 
   469 lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> 
   470     f : O(g)" 
   471   apply (erule bigo_bounded_alt [of f 1 g])
   472   apply simp
   473 done
   474 
   475 ML{*ResAtp.problem_name := "BigO__bigo_bounded2"*}
   476 lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
   477     f : lb +o O(g)"
   478   apply (rule set_minus_imp_plus)
   479   apply (rule bigo_bounded)
   480   apply (auto simp add: diff_minus func_minus func_plus)
   481   prefer 2
   482   apply (drule_tac x = x in spec)+ 
   483   apply arith (*not clear that it's provable otherwise*) 
   484 proof (neg_clausify)
   485 fix x
   486 assume 0: "\<And>y. lb y \<le> f y"
   487 assume 1: "\<not> (0\<Colon>'b) \<le> f x + - lb x"
   488 have 2: "\<And>X3. (0\<Colon>'b) + X3 = X3"
   489   by (metis diff_eq_eq right_minus_eq)
   490 have 3: "\<not> (0\<Colon>'b) \<le> f x - lb x"
   491   by (metis 1 compare_rls(1))
   492 have 4: "\<not> (0\<Colon>'b) + lb x \<le> f x"
   493   by (metis 3 le_diff_eq)
   494 show "False"
   495   by (metis 4 2 0)
   496 qed
   497 
   498 ML{*ResAtp.problem_name := "BigO__bigo_abs"*}
   499 lemma bigo_abs: "(%x. abs(f x)) =o O(f)" 
   500   apply (unfold bigo_def)
   501   apply auto
   502 proof (neg_clausify)
   503 fix x
   504 assume 0: "\<And>xa. \<not> \<bar>f (x xa)\<bar> \<le> xa * \<bar>f (x xa)\<bar>"
   505 have 1: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2 \<or> \<not> (1\<Colon>'b) \<le> (1\<Colon>'b)"
   506   by (metis mult_le_cancel_right1 order_eq_iff)
   507 have 2: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2"
   508   by (metis order_eq_iff 1)
   509 show "False"
   510   by (metis 0 2)
   511 qed
   512 
   513 ML{*ResAtp.problem_name := "BigO__bigo_abs2"*}
   514 lemma bigo_abs2: "f =o O(%x. abs(f x))"
   515   apply (unfold bigo_def)
   516   apply auto
   517 proof (neg_clausify)
   518 fix x
   519 assume 0: "\<And>xa. \<not> \<bar>f (x xa)\<bar> \<le> xa * \<bar>f (x xa)\<bar>"
   520 have 1: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2 \<or> \<not> (1\<Colon>'b) \<le> (1\<Colon>'b)"
   521   by (metis mult_le_cancel_right1 order_eq_iff)
   522 have 2: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2"
   523   by (metis order_eq_iff 1)
   524 show "False"
   525   by (metis 0 2)
   526 qed
   527  
   528 lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
   529   apply (rule equalityI)
   530   apply (rule bigo_elt_subset)
   531   apply (rule bigo_abs2)
   532   apply (rule bigo_elt_subset)
   533   apply (rule bigo_abs)
   534 done
   535 
   536 lemma bigo_abs4: "f =o g +o O(h) ==> 
   537     (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
   538   apply (drule set_plus_imp_minus)
   539   apply (rule set_minus_imp_plus)
   540   apply (subst func_diff)
   541 proof -
   542   assume a: "f - g : O(h)"
   543   have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
   544     by (rule bigo_abs2)
   545   also have "... <= O(%x. abs (f x - g x))"
   546     apply (rule bigo_elt_subset)
   547     apply (rule bigo_bounded)
   548     apply force
   549     apply (rule allI)
   550     apply (rule abs_triangle_ineq3)
   551     done
   552   also have "... <= O(f - g)"
   553     apply (rule bigo_elt_subset)
   554     apply (subst func_diff)
   555     apply (rule bigo_abs)
   556     done
   557   also have "... <= O(h)"
   558     using a by (rule bigo_elt_subset)
   559   finally show "(%x. abs (f x) - abs (g x)) : O(h)".
   560 qed
   561 
   562 lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" 
   563 by (unfold bigo_def, auto)
   564 
   565 lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) + O(h)"
   566 proof -
   567   assume "f : g +o O(h)"
   568   also have "... <= O(g) + O(h)"
   569     by (auto del: subsetI)
   570   also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
   571     apply (subst bigo_abs3 [symmetric])+
   572     apply (rule refl)
   573     done
   574   also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
   575     by (rule bigo_plus_eq [symmetric], auto)
   576   finally have "f : ...".
   577   then have "O(f) <= ..."
   578     by (elim bigo_elt_subset)
   579   also have "... = O(%x. abs(g x)) + O(%x. abs(h x))"
   580     by (rule bigo_plus_eq, auto)
   581   finally show ?thesis
   582     by (simp add: bigo_abs3 [symmetric])
   583 qed
   584 
   585 ML{*ResAtp.problem_name := "BigO__bigo_mult"*}
   586 lemma bigo_mult [intro]: "O(f)*O(g) <= O(f * g)"
   587   apply (rule subsetI)
   588   apply (subst bigo_def)
   589   apply (auto simp del: abs_mult mult_ac
   590               simp add: bigo_alt_def set_times func_times)
   591 (*sledgehammer*); 
   592   apply (rule_tac x = "c * ca" in exI)
   593   apply(rule allI)
   594   apply(erule_tac x = x in allE)+
   595   apply(subgoal_tac "c * ca * abs(f x * g x) = 
   596       (c * abs(f x)) * (ca * abs(g x))")
   597 ML{*ResAtp.problem_name := "BigO__bigo_mult_simpler"*}
   598 prefer 2 
   599 apply (metis  Finite_Set.AC_mult.f.assoc  Finite_Set.AC_mult.f.left_commute  OrderedGroup.abs_of_pos  OrderedGroup.mult_left_commute  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 Finite_Set.AC_mult.f.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{*ResAtp.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{*ResAtp.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{*ResAtp.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{*ResAtp.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{*ResAtp.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{*ResAtp.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)) * 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{*ResAtp.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)) * 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 func_minus)
   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 func_minus 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) + O(g)"
   757       by (auto del: subsetI)
   758     also have "... <= O(g) + 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{*ResAtp.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{*ResAtp.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{*ResAtp.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{*ResAtp.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{*ResAtp.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{*ResAtp.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 (metis AC_mult.f_e.left_ident mult_assoc right_inverse)
   904   apply (simp add: mult_assoc [symmetric] abs_mult)
   905   (*couldn't get this proof without the step above; SLOW*)
   906   apply (metis AC_mult.f.assoc abs_ge_zero mult_left_mono)
   907 done
   908 
   909 
   910 ML{*ResAtp.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 func_minus
   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{*ResAtp.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{*ResAtp.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 func_diff)
  1008   apply (subst setsum_subtractf [symmetric])
  1009   apply (subst right_diff_distrib [symmetric])
  1010   apply (rule bigo_setsum3)
  1011   apply (subst func_diff [symmetric])
  1012   apply (erule set_plus_imp_minus)
  1013 done
  1014 
  1015 ML{*ResAtp.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 func_diff)
  1037   apply (subst setsum_subtractf [symmetric])
  1038   apply (subst right_diff_distrib [symmetric])
  1039   apply (rule bigo_setsum5)
  1040   apply (subst func_diff [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 + 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{*ResAtp.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 func_diff)
  1094   apply (subst func_diff [symmetric])
  1095   apply (rule set_plus_imp_minus)
  1096   apply simp
  1097   apply (simp add: func_diff)
  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{*ResAtp.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{*ResAtp.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 func_diff)
  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.less_eq_less_sup.sup_absorb1 min_max.less_eq_less_sup.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.less_eq_less_inf.inf_le2 min_max.less_eq_less_inf.le_inf_iff min_max.less_eq_less_inf.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.less_eq_less_inf.inf_commute 4 linorder_not_le min_max.less_eq_less_inf.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.less_eq_less_sup.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.less_eq_less_sup.sup_commute 6)
  1182 show "False"
  1183   by (metis min_max.less_eq_less_sup.sup_commute min_max.less_eq_less_inf.inf_commute min_max.less_eq_less_inf_sup.sup_inf_absorb min_max.less_eq_less_inf.le_iff_inf 0 max_diff_distrib_left 1 linorder_not_le 8)
  1184 qed
  1185 
  1186 ML{*ResAtp.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 func_diff)
  1197 apply (erule thin_rl) 
  1198 (*sledgehammer*); 
  1199   apply (case_tac "0 <= f x - k x")
  1200   apply (simp del: compare_rls diff_minus);
  1201   apply (subst abs_of_nonneg)
  1202   apply (drule_tac x = x in spec) back
  1203 ML{*ResAtp.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.less_eq_less_sup.sup_absorb1 min_max.less_eq_less_sup.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: func_diff)
  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 func_diff compare_rls 
  1220     split: split_max abs_split)
  1221 done
  1222 
  1223 ML{*ResAtp.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