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
```