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
```