src/HOL/MetisExamples/BigO.thy
 author wenzelm Tue Oct 14 16:01:36 2008 +0200 (2008-10-14) changeset 28592 824f8390aaa2 parent 28486 873726bdfd47 child 29511 7071b017cb35 permissions -rw-r--r--
1 (*  Title:      HOL/MetisExamples/BigO.thy
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
5 Testing the metis method
6 *)
8 header {* Big O notation *}
10 theory BigO
11 imports Dense_Linear_Order Main SetsAndFunctions
12 begin
14 subsection {* Definitions *}
16 constdefs
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)}"
21 ML_command{*AtpWrapper.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
32 (*** Now various verions with an increasing modulus ***)
34 declare [[sledgehammer_modulus = 1]]
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
103 declare [[sledgehammer_modulus = 2]]
105 lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom).
106     ALL x. (abs (h x)) <= (c * (abs (f x))))
107       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
108   apply auto
109   apply (case_tac "c = 0", simp)
110   apply (rule_tac x = "1" in exI, simp)
111   apply (rule_tac x = "abs c" in exI, auto);
112 proof (neg_clausify)
113 fix c x
114 have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom. \<bar>X1 * X2\<bar> = \<bar>X2 * X1\<bar>"
115   by (metis abs_mult mult_commute)
116 assume 1: "\<And>x\<Colon>'b\<Colon>type.
117    \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
118    \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
119 assume 2: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
120   \<le> \<bar>c\<Colon>'a\<Colon>ordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
121 have 3: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
122   \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
123   by (metis 2 abs_mult)
124 have 4: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
125    \<not> X1 \<le> (0\<Colon>'a\<Colon>ordered_idom) \<or> X1 \<le> \<bar>X2\<bar>"
126   by (metis abs_ge_zero xt1(6))
127 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>"
128   by (metis not_leE 4 3)
129 have 6: "(0\<Colon>'a\<Colon>ordered_idom)
130 < (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>"
131   by (metis 1 order_less_le_trans 5)
132 have 7: "\<And>X1\<Colon>'a\<Colon>ordered_idom. (c\<Colon>'a\<Colon>ordered_idom) * \<bar>X1\<bar> = \<bar>X1 * c\<bar>"
133   by (metis abs_ge_zero linorder_not_less mult_nonneg_nonpos2 6 linorder_linear abs_mult_pos mult_commute)
134 have 8: "\<And>X1\<Colon>'b\<Colon>type.
135    - (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
136    \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
137   by (metis 0 7 abs_le_D2 1)
138 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>"
139   by (metis abs_ge_self xt1(6) abs_le_D1)
140 show "False"
141   by (metis 8 abs_ge_zero abs_mult_pos abs_mult 1 9 3 abs_le_iff)
142 qed
144 declare [[sledgehammer_modulus = 3]]
146 lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom).
147     ALL x. (abs (h x)) <= (c * (abs (f x))))
148       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
149   apply auto
150   apply (case_tac "c = 0", simp)
151   apply (rule_tac x = "1" in exI, simp)
152   apply (rule_tac x = "abs c" in exI, auto);
153 proof (neg_clausify)
154 fix c x
155 assume 0: "\<And>x\<Colon>'b\<Colon>type.
156    \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>
157    \<le> (c\<Colon>'a\<Colon>ordered_idom) * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
158 assume 1: "\<not> \<bar>(h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) (x\<Colon>'b\<Colon>type)\<bar>
159   \<le> \<bar>c\<Colon>'a\<Colon>ordered_idom\<bar> * \<bar>(f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) x\<bar>"
160 have 2: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2\<Colon>'a\<Colon>ordered_idom.
161    X1 \<le> \<bar>X2\<bar> \<or> (0\<Colon>'a\<Colon>ordered_idom) < X1"
162   by (metis abs_ge_zero xt1(6) not_leE)
163 have 3: "\<not> (c\<Colon>'a\<Colon>ordered_idom) \<le> (0\<Colon>'a\<Colon>ordered_idom)"
164   by (metis abs_ge_zero mult_nonneg_nonpos2 linorder_not_less order_less_le_trans 1 abs_mult 2 0)
165 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>"
166   by (metis abs_ge_zero abs_mult_pos abs_mult)
167 have 5: "\<And>X1\<Colon>'b\<Colon>type.
168    (h\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1
169    \<le> \<bar>(c\<Colon>'a\<Colon>ordered_idom) * (f\<Colon>'b\<Colon>type \<Rightarrow> 'a\<Colon>ordered_idom) X1\<bar>"
170   by (metis 4 0 xt1(6) abs_ge_self abs_le_D1)
171 show "False"
172   by (metis abs_mult mult_commute 3 abs_mult_pos linorder_linear 0 abs_le_D2 5 1 abs_le_iff)
173 qed
176 declare [[sledgehammer_modulus = 1]]
178 lemma (*bigo_pos_const:*) "(EX (c::'a::ordered_idom).
179     ALL x. (abs (h x)) <= (c * (abs (f x))))
180       = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))"
181   apply auto
182   apply (case_tac "c = 0", simp)
183   apply (rule_tac x = "1" in exI, simp)
184   apply (rule_tac x = "abs c" in exI, auto);
185 proof (neg_clausify)
186 fix c x  (*sort/type constraint inserted by hand!*)
187 have 0: "\<And>(X1\<Colon>'a\<Colon>ordered_idom) X2. \<bar>X1 * \<bar>X2\<bar>\<bar> = \<bar>X1 * X2\<bar>"
188   by (metis abs_ge_zero abs_mult_pos abs_mult)
189 assume 1: "\<And>A. \<bar>h A\<bar> \<le> c * \<bar>f A\<bar>"
190 have 2: "\<And>X1 X2. \<not> \<bar>X1\<bar> \<le> X2 \<or> (0\<Colon>'a) \<le> X2"
191   by (metis abs_ge_zero order_trans)
192 have 3: "\<And>X1. (0\<Colon>'a) \<le> c * \<bar>f X1\<bar>"
193   by (metis 1 2)
194 have 4: "\<And>X1. c * \<bar>f X1\<bar> = \<bar>c * f X1\<bar>"
195   by (metis 0 abs_of_nonneg 3)
196 have 5: "\<And>X1. - h X1 \<le> c * \<bar>f X1\<bar>"
197   by (metis 1 abs_le_D2)
198 have 6: "\<And>X1. - h X1 \<le> \<bar>c * f X1\<bar>"
199   by (metis 4 5)
200 have 7: "\<And>X1. h X1 \<le> c * \<bar>f X1\<bar>"
201   by (metis 1 abs_le_D1)
202 have 8: "\<And>X1. h X1 \<le> \<bar>c * f X1\<bar>"
203   by (metis 4 7)
204 assume 9: "\<not> \<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>"
205 have 10: "\<not> \<bar>h x\<bar> \<le> \<bar>c * f x\<bar>"
206   by (metis abs_mult 9)
207 show "False"
208   by (metis 6 8 10 abs_leI)
209 qed
212 declare [[sledgehammer_sorts = true]]
214 lemma bigo_alt_def: "O(f) =
215     {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
216 by (auto simp add: bigo_def bigo_pos_const)
218 ML_command{*AtpWrapper.problem_name := "BigO__bigo_elt_subset"*}
219 lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
220   apply (auto simp add: bigo_alt_def)
221   apply (rule_tac x = "ca * c" in exI)
222   apply (rule conjI)
223   apply (rule mult_pos_pos)
224   apply (assumption)+
225 (*sledgehammer*);
226   apply (rule allI)
227   apply (drule_tac x = "xa" in spec)+
228   apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))");
229   apply (erule order_trans)
231   apply (rule mult_left_mono, assumption)
232   apply (rule order_less_imp_le, assumption);
233 done
236 ML_command{*AtpWrapper.problem_name := "BigO__bigo_refl"*}
237 lemma bigo_refl [intro]: "f : O(f)"
239 proof (neg_clausify)
240 fix x
241 assume 0: "\<And>xa. \<not> \<bar>f (x xa)\<bar> \<le> xa * \<bar>f (x xa)\<bar>"
242 have 1: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2 \<or> \<not> (1\<Colon>'b) \<le> (1\<Colon>'b)"
243   by (metis mult_le_cancel_right1 order_eq_iff)
244 have 2: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2"
245   by (metis order_eq_iff 1)
246 show "False"
247   by (metis 0 2)
248 qed
250 ML_command{*AtpWrapper.problem_name := "BigO__bigo_zero"*}
251 lemma bigo_zero: "0 : O(g)"
252   apply (auto simp add: bigo_def func_zero)
253 proof (neg_clausify)
254 fix x
255 assume 0: "\<And>xa. \<not> (0\<Colon>'b) \<le> xa * \<bar>g (x xa)\<bar>"
256 have 1: "\<not> (0\<Colon>'b) \<le> (0\<Colon>'b)"
257   by (metis 0 mult_eq_0_iff)
258 show "False"
259   by (metis 1 linorder_neq_iff linorder_antisym_conv1)
260 qed
262 lemma bigo_zero2: "O(%x.0) = {%x.0}"
263   apply (auto simp add: bigo_def)
264   apply (rule ext)
265   apply auto
266 done
268 lemma bigo_plus_self_subset [intro]:
269   "O(f) \<oplus> O(f) <= O(f)"
270   apply (auto simp add: bigo_alt_def set_plus_def)
271   apply (rule_tac x = "c + ca" in exI)
272   apply auto
273   apply (simp add: ring_distribs func_plus)
274   apply (blast intro:order_trans abs_triangle_ineq add_mono elim:)
275 done
277 lemma bigo_plus_idemp [simp]: "O(f) \<oplus> O(f) = O(f)"
278   apply (rule equalityI)
279   apply (rule bigo_plus_self_subset)
280   apply (rule set_zero_plus2)
281   apply (rule bigo_zero)
282 done
284 lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) \<oplus> O(g)"
285   apply (rule subsetI)
286   apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def)
287   apply (subst bigo_pos_const [symmetric])+
288   apply (rule_tac x =
289     "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI)
290   apply (rule conjI)
291   apply (rule_tac x = "c + c" in exI)
292   apply (clarsimp)
293   apply (auto)
294   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)")
295   apply (erule_tac x = xa in allE)
296   apply (erule order_trans)
297   apply (simp)
298   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
299   apply (erule order_trans)
301   apply (rule mult_left_mono)
302   apply assumption
304   apply (rule mult_left_mono)
307   apply (rule mult_nonneg_nonneg)
309   apply auto
310   apply (rule_tac x = "%n. if (abs (f n)) <  abs (g n) then x n else 0"
311      in exI)
312   apply (rule conjI)
313   apply (rule_tac x = "c + c" in exI)
314   apply auto
315   apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)")
316   apply (erule_tac x = xa in allE)
317   apply (erule order_trans)
318   apply (simp)
319   apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))")
320   apply (erule order_trans)
322   apply (rule mult_left_mono)
325   apply (rule mult_left_mono)
326   apply (rule abs_triangle_ineq)
328 apply (metis abs_not_less_zero double_less_0_iff less_not_permute linorder_not_less mult_less_0_iff)
329   apply (rule ext)
330   apply (auto simp add: if_splits linorder_not_le)
331 done
333 lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A \<oplus> B <= O(f)"
334   apply (subgoal_tac "A \<oplus> B <= O(f) \<oplus> O(f)")
335   apply (erule order_trans)
336   apply simp
337   apply (auto del: subsetI simp del: bigo_plus_idemp)
338 done
340 ML_command{*AtpWrapper.problem_name := "BigO__bigo_plus_eq"*}
341 lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==>
342   O(f + g) = O(f) \<oplus> O(g)"
343   apply (rule equalityI)
344   apply (rule bigo_plus_subset)
345   apply (simp add: bigo_alt_def set_plus_def func_plus)
346   apply clarify
347 (*sledgehammer*);
348   apply (rule_tac x = "max c ca" in exI)
349   apply (rule conjI)
350    apply (metis Orderings.less_max_iff_disj)
351   apply clarify
352   apply (drule_tac x = "xa" in spec)+
353   apply (subgoal_tac "0 <= f xa + g xa")
355   apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)")
356   apply (subgoal_tac "abs(a xa) + abs(b xa) <=
357       max c ca * f xa + max c ca * g xa")
358   apply (blast intro: order_trans)
359   defer 1
360   apply (rule abs_triangle_ineq)
363 ML_command{*AtpWrapper.problem_name := "BigO__bigo_plus_eq_simpler"*}
364 (*Found by SPASS; SLOW*)
365 apply (metis le_maxI2 linorder_linear linorder_not_le min_max.less_eq_less_sup.sup_absorb1 mult_le_cancel_right order_trans)
366 apply (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans)
367 done
369 ML_command{*AtpWrapper.problem_name := "BigO__bigo_bounded_alt"*}
370 lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==>
371     f : O(g)"
372   apply (auto simp add: bigo_def)
373 (*Version 1: one-shot proof*)
374   apply (metis OrderedGroup.abs_le_D1 linorder_class.not_less  order_less_le  Orderings.xt1(12)  Ring_and_Field.abs_mult)
375   done
377 lemma (*bigo_bounded_alt:*) "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==>
378     f : O(g)"
379   apply (auto simp add: bigo_def)
380 (*Version 2: single-step proof*)
381 proof (neg_clausify)
382 fix x
383 assume 0: "\<And>x. f x \<le> c * g x"
384 assume 1: "\<And>xa. \<not> f (x xa) \<le> xa * \<bar>g (x xa)\<bar>"
385 have 2: "\<And>X3. c * g X3 = f X3 \<or> \<not> c * g X3 \<le> f X3"
386   by (metis 0 order_antisym_conv)
387 have 3: "\<And>X3. \<not> f (x \<bar>X3\<bar>) \<le> \<bar>X3 * g (x \<bar>X3\<bar>)\<bar>"
388   by (metis 1 abs_mult)
389 have 4: "\<And>X1 X3\<Colon>'b\<Colon>ordered_idom. X3 \<le> X1 \<or> X1 \<le> \<bar>X3\<bar>"
390   by (metis linorder_linear abs_le_D1)
391 have 5: "\<And>X3::'b. \<bar>X3\<bar> * \<bar>X3\<bar> = X3 * X3"
392   by (metis abs_mult_self)
393 have 6: "\<And>X3. \<not> X3 * X3 < (0\<Colon>'b\<Colon>ordered_idom)"
394   by (metis not_square_less_zero)
395 have 7: "\<And>X1 X3::'b. \<bar>X1\<bar> * \<bar>X3\<bar> = \<bar>X3 * X1\<bar>"
396   by (metis abs_mult mult_commute)
397 have 8: "\<And>X3::'b. X3 * X3 = \<bar>X3 * X3\<bar>"
398   by (metis abs_mult 5)
399 have 9: "\<And>X3. X3 * g (x \<bar>X3\<bar>) \<le> f (x \<bar>X3\<bar>)"
400   by (metis 3 4)
401 have 10: "c * g (x \<bar>c\<bar>) = f (x \<bar>c\<bar>)"
402   by (metis 2 9)
403 have 11: "\<And>X3::'b. \<bar>X3\<bar> * \<bar>\<bar>X3\<bar>\<bar> = \<bar>X3\<bar> * \<bar>X3\<bar>"
404   by (metis abs_idempotent abs_mult 8)
405 have 12: "\<And>X3::'b. \<bar>X3 * \<bar>X3\<bar>\<bar> = \<bar>X3\<bar> * \<bar>X3\<bar>"
406   by (metis mult_commute 7 11)
407 have 13: "\<And>X3::'b. \<bar>X3 * \<bar>X3\<bar>\<bar> = X3 * X3"
408   by (metis 8 7 12)
409 have 14: "\<And>X3. X3 \<le> \<bar>X3\<bar> \<or> X3 < (0\<Colon>'b)"
410   by (metis abs_ge_self abs_le_D1 abs_if)
411 have 15: "\<And>X3. X3 \<le> \<bar>X3\<bar> \<or> \<bar>X3\<bar> < (0\<Colon>'b)"
412   by (metis abs_ge_self abs_le_D1 abs_if)
413 have 16: "\<And>X3. X3 * X3 < (0\<Colon>'b) \<or> X3 * \<bar>X3\<bar> \<le> X3 * X3"
414   by (metis 15 13)
415 have 17: "\<And>X3::'b. X3 * \<bar>X3\<bar> \<le> X3 * X3"
416   by (metis 16 6)
417 have 18: "\<And>X3. X3 \<le> \<bar>X3\<bar> \<or> \<not> X3 < (0\<Colon>'b)"
418   by (metis mult_le_cancel_left 17)
419 have 19: "\<And>X3::'b. X3 \<le> \<bar>X3\<bar>"
420   by (metis 18 14)
421 have 20: "\<not> f (x \<bar>c\<bar>) \<le> \<bar>f (x \<bar>c\<bar>)\<bar>"
422   by (metis 3 10)
423 show "False"
424   by (metis 20 19)
425 qed
428 text{*So here is the easier (and more natural) problem using transitivity*}
429 ML_command{*AtpWrapper.problem_name := "BigO__bigo_bounded_alt_trans"*}
430 lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)"
431   apply (auto simp add: bigo_def)
432   (*Version 1: one-shot proof*)
433   apply (metis Orderings.leD Orderings.leI abs_ge_self abs_le_D1 abs_mult abs_of_nonneg order_le_less)
434   done
436 text{*So here is the easier (and more natural) problem using transitivity*}
437 ML_command{*AtpWrapper.problem_name := "BigO__bigo_bounded_alt_trans"*}
438 lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)"
439   apply (auto simp add: bigo_def)
440 (*Version 2: single-step proof*)
441 proof (neg_clausify)
442 fix x
443 assume 0: "\<And>A\<Colon>'a\<Colon>type.
444    (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) A
445    \<le> (c\<Colon>'b\<Colon>ordered_idom) * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) A"
446 assume 1: "\<And>A\<Colon>'b\<Colon>ordered_idom.
447    \<not> (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) A)
448      \<le> A * \<bar>(g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x A)\<bar>"
449 have 2: "\<And>X2\<Colon>'a\<Colon>type.
450    \<not> (c\<Colon>'b\<Colon>ordered_idom) * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) X2
451      < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) X2"
452   by (metis 0 linorder_not_le)
453 have 3: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
454    \<not> (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)
455      \<le> \<bar>X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)\<bar>"
456   by (metis abs_mult 1)
457 have 4: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
458    \<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>
459    < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)"
460   by (metis 3 linorder_not_less)
461 have 5: "\<And>X2\<Colon>'b\<Colon>ordered_idom.
462    X2 * (g\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) \<bar>X2\<bar>)
463    < (f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) (x \<bar>X2\<bar>)"
464   by (metis abs_less_iff 4)
465 show "False"
466   by (metis 2 5)
467 qed
470 lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==>
471     f : O(g)"
472   apply (erule bigo_bounded_alt [of f 1 g])
473   apply simp
474 done
476 ML_command{*AtpWrapper.problem_name := "BigO__bigo_bounded2"*}
477 lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
478     f : lb +o O(g)"
479   apply (rule set_minus_imp_plus)
480   apply (rule bigo_bounded)
481   apply (auto simp add: diff_minus fun_Compl_def func_plus)
482   prefer 2
483   apply (drule_tac x = x in spec)+
484   apply arith (*not clear that it's provable otherwise*)
485 proof (neg_clausify)
486 fix x
487 assume 0: "\<And>y. lb y \<le> f y"
488 assume 1: "\<not> (0\<Colon>'b) \<le> f x + - lb x"
489 have 2: "\<And>X3. (0\<Colon>'b) + X3 = X3"
490   by (metis diff_eq_eq right_minus_eq)
491 have 3: "\<not> (0\<Colon>'b) \<le> f x - lb x"
492   by (metis 1 compare_rls(1))
493 have 4: "\<not> (0\<Colon>'b) + lb x \<le> f x"
494   by (metis 3 le_diff_eq)
495 show "False"
496   by (metis 4 2 0)
497 qed
499 ML_command{*AtpWrapper.problem_name := "BigO__bigo_abs"*}
500 lemma bigo_abs: "(%x. abs(f x)) =o O(f)"
501   apply (unfold bigo_def)
502   apply auto
503 proof (neg_clausify)
504 fix x
505 assume 0: "\<And>xa. \<not> \<bar>f (x xa)\<bar> \<le> xa * \<bar>f (x xa)\<bar>"
506 have 1: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2 \<or> \<not> (1\<Colon>'b) \<le> (1\<Colon>'b)"
507   by (metis mult_le_cancel_right1 order_eq_iff)
508 have 2: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2"
509   by (metis order_eq_iff 1)
510 show "False"
511   by (metis 0 2)
512 qed
514 ML_command{*AtpWrapper.problem_name := "BigO__bigo_abs2"*}
515 lemma bigo_abs2: "f =o O(%x. abs(f x))"
516   apply (unfold bigo_def)
517   apply auto
518 proof (neg_clausify)
519 fix x
520 assume 0: "\<And>xa. \<not> \<bar>f (x xa)\<bar> \<le> xa * \<bar>f (x xa)\<bar>"
521 have 1: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2 \<or> \<not> (1\<Colon>'b) \<le> (1\<Colon>'b)"
522   by (metis mult_le_cancel_right1 order_eq_iff)
523 have 2: "\<And>X2. X2 \<le> (1\<Colon>'b) * X2"
524   by (metis order_eq_iff 1)
525 show "False"
526   by (metis 0 2)
527 qed
529 lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
530   apply (rule equalityI)
531   apply (rule bigo_elt_subset)
532   apply (rule bigo_abs2)
533   apply (rule bigo_elt_subset)
534   apply (rule bigo_abs)
535 done
537 lemma bigo_abs4: "f =o g +o O(h) ==>
538     (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
539   apply (drule set_plus_imp_minus)
540   apply (rule set_minus_imp_plus)
541   apply (subst fun_diff_def)
542 proof -
543   assume a: "f - g : O(h)"
544   have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))"
545     by (rule bigo_abs2)
546   also have "... <= O(%x. abs (f x - g x))"
547     apply (rule bigo_elt_subset)
548     apply (rule bigo_bounded)
549     apply force
550     apply (rule allI)
551     apply (rule abs_triangle_ineq3)
552     done
553   also have "... <= O(f - g)"
554     apply (rule bigo_elt_subset)
555     apply (subst fun_diff_def)
556     apply (rule bigo_abs)
557     done
558   also have "... <= O(h)"
559     using a by (rule bigo_elt_subset)
560   finally show "(%x. abs (f x) - abs (g x)) : O(h)".
561 qed
563 lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)"
564 by (unfold bigo_def, auto)
566 lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) \<oplus> O(h)"
567 proof -
568   assume "f : g +o O(h)"
569   also have "... <= O(g) \<oplus> O(h)"
570     by (auto del: subsetI)
571   also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
572     apply (subst bigo_abs3 [symmetric])+
573     apply (rule refl)
574     done
575   also have "... = O((%x. abs(g x)) + (%x. abs(h x)))"
576     by (rule bigo_plus_eq [symmetric], auto)
577   finally have "f : ...".
578   then have "O(f) <= ..."
579     by (elim bigo_elt_subset)
580   also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
581     by (rule bigo_plus_eq, auto)
582   finally show ?thesis
583     by (simp add: bigo_abs3 [symmetric])
584 qed
586 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult"*}
587 lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)"
588   apply (rule subsetI)
589   apply (subst bigo_def)
590   apply (auto simp del: abs_mult mult_ac
591               simp add: bigo_alt_def set_times_def func_times)
592 (*sledgehammer*);
593   apply (rule_tac x = "c * ca" in exI)
594   apply(rule allI)
595   apply(erule_tac x = x in allE)+
596   apply(subgoal_tac "c * ca * abs(f x * g x) =
597       (c * abs(f x)) * (ca * abs(g x))")
598 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult_simpler"*}
599 prefer 2
600 apply (metis mult_assoc mult_left_commute
601   OrderedGroup.abs_of_pos OrderedGroup.mult_left_commute
602   Ring_and_Field.abs_mult Ring_and_Field.mult_pos_pos)
603   apply (erule ssubst)
604   apply (subst abs_mult)
605 (*not qute BigO__bigo_mult_simpler_1 (a hard problem!) as abs_mult has
606   just been done*)
607 proof (neg_clausify)
608 fix a c b ca x
609 assume 0: "(0\<Colon>'b\<Colon>ordered_idom) < (c\<Colon>'b\<Colon>ordered_idom)"
610 assume 1: "\<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
611 \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
612 assume 2: "\<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
613 \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
614 assume 3: "\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar> *
615   \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>
616   \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> *
617     ((ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>)"
618 have 4: "\<bar>c\<Colon>'b\<Colon>ordered_idom\<bar> = c"
619   by (metis OrderedGroup.abs_of_pos 0)
620 have 5: "\<And>X1\<Colon>'b\<Colon>ordered_idom. (c\<Colon>'b\<Colon>ordered_idom) * \<bar>X1\<bar> = \<bar>c * X1\<bar>"
621   by (metis Ring_and_Field.abs_mult 4)
622 have 6: "(0\<Colon>'b\<Colon>ordered_idom) = (1\<Colon>'b\<Colon>ordered_idom) \<or>
623 (0\<Colon>'b\<Colon>ordered_idom) < (1\<Colon>'b\<Colon>ordered_idom)"
624   by (metis OrderedGroup.abs_not_less_zero Ring_and_Field.abs_one Ring_and_Field.linorder_neqE_ordered_idom)
625 have 7: "(0\<Colon>'b\<Colon>ordered_idom) < (1\<Colon>'b\<Colon>ordered_idom)"
626   by (metis 6 Ring_and_Field.one_neq_zero)
627 have 8: "\<bar>1\<Colon>'b\<Colon>ordered_idom\<bar> = (1\<Colon>'b\<Colon>ordered_idom)"
628   by (metis OrderedGroup.abs_of_pos 7)
629 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>"
630   by (metis OrderedGroup.abs_ge_zero 5)
631 have 10: "\<And>X1\<Colon>'b\<Colon>ordered_idom. X1 * (1\<Colon>'b\<Colon>ordered_idom) = X1"
632   by (metis Ring_and_Field.mult_cancel_right2 mult_commute)
633 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>"
634   by (metis Ring_and_Field.abs_mult OrderedGroup.abs_idempotent 10)
635 have 12: "\<And>X1\<Colon>'b\<Colon>ordered_idom. \<bar>\<bar>X1\<bar>\<bar> = \<bar>X1\<bar>"
636   by (metis 11 8 10)
637 have 13: "\<And>X1\<Colon>'b\<Colon>ordered_idom. (0\<Colon>'b\<Colon>ordered_idom) \<le> \<bar>X1\<bar>"
638   by (metis OrderedGroup.abs_ge_zero 12)
639 have 14: "\<not> (0\<Colon>'b\<Colon>ordered_idom)
640   \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar> \<or>
641 \<not> (0\<Colon>'b\<Colon>ordered_idom) \<le> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
642 \<not> \<bar>b x\<bar> \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
643 \<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<le> c * \<bar>f x\<bar>"
644   by (metis 3 Ring_and_Field.mult_mono)
645 have 15: "\<not> (0\<Colon>'b\<Colon>ordered_idom) \<le> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar> \<or>
646 \<not> \<bar>b x\<bar> \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
647 \<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>
648   \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
649   by (metis 14 9)
650 have 16: "\<not> \<bar>(b\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
651   \<le> (ca\<Colon>'b\<Colon>ordered_idom) * \<bar>(g\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar> \<or>
652 \<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>
653   \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
654   by (metis 15 13)
655 have 17: "\<not> \<bar>(a\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) (x\<Colon>'a)\<bar>
656   \<le> (c\<Colon>'b\<Colon>ordered_idom) * \<bar>(f\<Colon>'a \<Rightarrow> 'b\<Colon>ordered_idom) x\<bar>"
657   by (metis 16 2)
658 show 18: "False"
659   by (metis 17 1)
660 qed
663 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult2"*}
664 lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
665   apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
666 (*sledgehammer*);
667   apply (rule_tac x = c in exI)
668   apply clarify
669   apply (drule_tac x = x in spec)
670 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult2_simpler"*}
671 (*sledgehammer [no luck]*);
672   apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))")
674   apply (rule mult_left_mono, assumption)
675   apply (rule abs_ge_zero)
676 done
678 ML_command{*AtpWrapper.problem_name:="BigO__bigo_mult3"*}
679 lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
680 by (metis bigo_mult set_times_intro subset_iff)
682 ML_command{*AtpWrapper.problem_name:="BigO__bigo_mult4"*}
683 lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
684 by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib)
687 lemma bigo_mult5: "ALL x. f x ~= 0 ==>
688     O(f * g) <= (f::'a => ('b::ordered_field)) *o O(g)"
689 proof -
690   assume "ALL x. f x ~= 0"
691   show "O(f * g) <= f *o O(g)"
692   proof
693     fix h
694     assume "h : O(f * g)"
695     then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)"
696       by auto
697     also have "... <= O((%x. 1 / f x) * (f * g))"
698       by (rule bigo_mult2)
699     also have "(%x. 1 / f x) * (f * g) = g"
701       apply (rule ext)
702       apply (simp add: prems nonzero_divide_eq_eq mult_ac)
703       done
704     finally have "(%x. (1::'b) / f x) * h : O(g)".
705     then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)"
706       by auto
707     also have "f * ((%x. (1::'b) / f x) * h) = h"
709       apply (rule ext)
710       apply (simp add: prems nonzero_divide_eq_eq mult_ac)
711       done
712     finally show "h : f *o O(g)".
713   qed
714 qed
716 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult6"*}
717 lemma bigo_mult6: "ALL x. f x ~= 0 ==>
718     O(f * g) = (f::'a => ('b::ordered_field)) *o O(g)"
719 by (metis bigo_mult2 bigo_mult5 order_antisym)
721 (*proof requires relaxing relevance: 2007-01-25*)
722 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult7"*}
723   declare bigo_mult6 [simp]
724 lemma bigo_mult7: "ALL x. f x ~= 0 ==>
725     O(f * g) <= O(f::'a => ('b::ordered_field)) \<otimes> O(g)"
726 (*sledgehammer*)
727   apply (subst bigo_mult6)
728   apply assumption
729   apply (rule set_times_mono3)
730   apply (rule bigo_refl)
731 done
732   declare bigo_mult6 [simp del]
734 ML_command{*AtpWrapper.problem_name := "BigO__bigo_mult8"*}
735   declare bigo_mult7[intro!]
736 lemma bigo_mult8: "ALL x. f x ~= 0 ==>
737     O(f * g) = O(f::'a => ('b::ordered_field)) \<otimes> O(g)"
738 by (metis bigo_mult bigo_mult7 order_antisym_conv)
740 lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
741   by (auto simp add: bigo_def fun_Compl_def)
743 lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)"
744   apply (rule set_minus_imp_plus)
745   apply (drule set_plus_imp_minus)
746   apply (drule bigo_minus)
748 done
750 lemma bigo_minus3: "O(-f) = O(f)"
751   by (auto simp add: bigo_def fun_Compl_def abs_minus_cancel)
753 lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)"
754 proof -
755   assume a: "f : O(g)"
756   show "f +o O(g) <= O(g)"
757   proof -
758     have "f : O(f)" by auto
759     then have "f +o O(g) <= O(f) \<oplus> O(g)"
760       by (auto del: subsetI)
761     also have "... <= O(g) \<oplus> O(g)"
762     proof -
763       from a have "O(f) <= O(g)" by (auto del: subsetI)
764       thus ?thesis by (auto del: subsetI)
765     qed
766     also have "... <= O(g)" by (simp add: bigo_plus_idemp)
767     finally show ?thesis .
768   qed
769 qed
771 lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)"
772 proof -
773   assume a: "f : O(g)"
774   show "O(g) <= f +o O(g)"
775   proof -
776     from a have "-f : O(g)" by auto
777     then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1)
778     then have "f +o (-f +o O(g)) <= f +o O(g)" by auto
779     also have "f +o (-f +o O(g)) = O(g)"
781     finally show ?thesis .
782   qed
783 qed
785 ML_command{*AtpWrapper.problem_name:="BigO__bigo_plus_absorb"*}
786 lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)"
787 by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff);
789 lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
790   apply (subgoal_tac "f +o A <= f +o O(g)")
791   apply force+
792 done
794 lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
795   apply (subst set_minus_plus [symmetric])
796   apply (subgoal_tac "g - f = - (f - g)")
797   apply (erule ssubst)
798   apply (rule bigo_minus)
799   apply (subst set_minus_plus)
800   apply assumption
802 done
804 lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
805   apply (rule iffI)
807 done
809 lemma bigo_const1: "(%x. c) : O(%x. 1)"
810 by (auto simp add: bigo_def mult_ac)
812 ML_command{*AtpWrapper.problem_name:="BigO__bigo_const2"*}
813 lemma (*bigo_const2 [intro]:*) "O(%x. c) <= O(%x. 1)"
814 by (metis bigo_const1 bigo_elt_subset);
816 lemma bigo_const2 [intro]: "O(%x. c::'b::ordered_idom) <= O(%x. 1)";
817 (*??FAILS because the two occurrences of COMBK have different polymorphic types
818 proof (neg_clausify)
819 assume 0: "\<not> O(COMBK (c\<Colon>'b\<Colon>ordered_idom)) \<subseteq> O(COMBK (1\<Colon>'b\<Colon>ordered_idom))"
820 have 1: "COMBK (c\<Colon>'b\<Colon>ordered_idom) \<notin> O(COMBK (1\<Colon>'b\<Colon>ordered_idom))"
821 apply (rule notI)
822 apply (rule 0 [THEN notE])
823 apply (rule bigo_elt_subset)
824 apply assumption;
825 sorry
826   by (metis 0 bigo_elt_subset)  loops??
827 show "False"
828   by (metis 1 bigo_const1)
829 qed
830 *)
831   apply (rule bigo_elt_subset)
832   apply (rule bigo_const1)
833 done
835 ML_command{*AtpWrapper.problem_name := "BigO__bigo_const3"*}
836 lemma bigo_const3: "(c::'a::ordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
838 proof (neg_clausify)
839 assume 0: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> (0\<Colon>'a\<Colon>ordered_field)"
840 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>"
841 have 2: "(0\<Colon>'a\<Colon>ordered_field) = \<bar>c\<Colon>'a\<Colon>ordered_field\<bar> \<or>
842 \<not> (1\<Colon>'a\<Colon>ordered_field) \<le> (1\<Colon>'a\<Colon>ordered_field)"
843   by (metis 1 field_inverse)
844 have 3: "\<bar>c\<Colon>'a\<Colon>ordered_field\<bar> = (0\<Colon>'a\<Colon>ordered_field)"
845   by (metis linorder_neq_iff linorder_antisym_conv1 2)
846 have 4: "(0\<Colon>'a\<Colon>ordered_field) = (c\<Colon>'a\<Colon>ordered_field)"
847   by (metis 3 abs_eq_0)
848 show "False"
849   by (metis 0 4)
850 qed
852 lemma bigo_const4: "(c::'a::ordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
853 by (rule bigo_elt_subset, rule bigo_const3, assumption)
855 lemma bigo_const [simp]: "(c::'a::ordered_field) ~= 0 ==>
856     O(%x. c) = O(%x. 1)"
857 by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
859 ML_command{*AtpWrapper.problem_name := "BigO__bigo_const_mult1"*}
860 lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
861   apply (simp add: bigo_def abs_mult)
862 proof (neg_clausify)
863 fix x
864 assume 0: "\<And>xa\<Colon>'b\<Colon>ordered_idom.
865    \<not> \<bar>c\<Colon>'b\<Colon>ordered_idom\<bar> *
866      \<bar>(f\<Colon>'a\<Colon>type \<Rightarrow> 'b\<Colon>ordered_idom) ((x\<Colon>'b\<Colon>ordered_idom \<Rightarrow> 'a\<Colon>type) xa)\<bar>
867      \<le> xa * \<bar>f (x xa)\<bar>"
868 show "False"
869   by (metis linorder_neq_iff linorder_antisym_conv1 0)
870 qed
872 lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
873 by (rule bigo_elt_subset, rule bigo_const_mult1)
875 ML_command{*AtpWrapper.problem_name := "BigO__bigo_const_mult3"*}
876 lemma bigo_const_mult3: "(c::'a::ordered_field) ~= 0 ==> f : O(%x. c * f x)"
878 (*sledgehammer [no luck]*);
879   apply (rule_tac x = "abs(inverse c)" in exI)
880   apply (simp only: abs_mult [symmetric] mult_assoc [symmetric])
881 apply (subst left_inverse)
882 apply (auto );
883 done
885 lemma bigo_const_mult4: "(c::'a::ordered_field) ~= 0 ==>
886     O(f) <= O(%x. c * f x)"
887 by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
889 lemma bigo_const_mult [simp]: "(c::'a::ordered_field) ~= 0 ==>
890     O(%x. c * f x) = O(f)"
891 by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
893 ML_command{*AtpWrapper.problem_name := "BigO__bigo_const_mult5"*}
894 lemma bigo_const_mult5 [simp]: "(c::'a::ordered_field) ~= 0 ==>
895     (%x. c) *o O(f) = O(f)"
896   apply (auto del: subsetI)
897   apply (rule order_trans)
898   apply (rule bigo_mult2)
900   apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times)
901   apply (rule_tac x = "%y. inverse c * x y" in exI)
902   apply (rename_tac g d)
903   apply safe
904   apply (rule_tac [2] ext)
905    prefer 2
906    apply simp
907   apply (simp add: mult_assoc [symmetric] abs_mult)
908   (*couldn't get this proof without the step above; SLOW*)
909   apply (metis mult_assoc abs_ge_zero mult_left_mono)
910 done
913 ML_command{*AtpWrapper.problem_name := "BigO__bigo_const_mult6"*}
914 lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
915   apply (auto intro!: subsetI
916     simp add: bigo_def elt_set_times_def func_times
917     simp del: abs_mult mult_ac)
918 (*sledgehammer*);
919   apply (rule_tac x = "ca * (abs c)" in exI)
920   apply (rule allI)
921   apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))")
922   apply (erule ssubst)
923   apply (subst abs_mult)
924   apply (rule mult_left_mono)
925   apply (erule spec)
926   apply simp
928 done
930 lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
931 proof -
932   assume "f =o O(g)"
933   then have "(%x. c) * f =o (%x. c) *o O(g)"
934     by auto
935   also have "(%x. c) * f = (%x. c * f x)"
937   also have "(%x. c) *o O(g) <= O(g)"
938     by (auto del: subsetI)
939   finally show ?thesis .
940 qed
942 lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))"
943 by (unfold bigo_def, auto)
945 lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o
946     O(%x. h(k x))"
947   apply (simp only: set_minus_plus [symmetric] diff_minus fun_Compl_def
948       func_plus)
949   apply (erule bigo_compose1)
950 done
952 subsection {* Setsum *}
954 lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==>
955     EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==>
956       (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
957   apply (auto simp add: bigo_def)
958   apply (rule_tac x = "abs c" in exI)
959   apply (subst abs_of_nonneg) back back
960   apply (rule setsum_nonneg)
961   apply force
962   apply (subst setsum_right_distrib)
963   apply (rule allI)
964   apply (rule order_trans)
965   apply (rule setsum_abs)
966   apply (rule setsum_mono)
967 apply (blast intro: order_trans mult_right_mono abs_ge_self)
968 done
970 ML_command{*AtpWrapper.problem_name := "BigO__bigo_setsum1"*}
971 lemma bigo_setsum1: "ALL x y. 0 <= h x y ==>
972     EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
973       (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)"
974   apply (rule bigo_setsum_main)
975 (*sledgehammer*);
976   apply force
977   apply clarsimp
978   apply (rule_tac x = c in exI)
979   apply force
980 done
982 lemma bigo_setsum2: "ALL y. 0 <= h y ==>
983     EX c. ALL y. abs(f y) <= c * (h y) ==>
984       (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
985 by (rule bigo_setsum1, auto)
987 ML_command{*AtpWrapper.problem_name := "BigO__bigo_setsum3"*}
988 lemma bigo_setsum3: "f =o O(h) ==>
989     (%x. SUM y : A x. (l x y) * f(k x y)) =o
990       O(%x. SUM y : A x. abs(l x y * h(k x y)))"
991   apply (rule bigo_setsum1)
992   apply (rule allI)+
993   apply (rule abs_ge_zero)
994   apply (unfold bigo_def)
995   apply (auto simp add: abs_mult);
996 (*sledgehammer*);
997   apply (rule_tac x = c in exI)
998   apply (rule allI)+
999   apply (subst mult_left_commute)
1000   apply (rule mult_left_mono)
1001   apply (erule spec)
1002   apply (rule abs_ge_zero)
1003 done
1005 lemma bigo_setsum4: "f =o g +o O(h) ==>
1006     (%x. SUM y : A x. l x y * f(k x y)) =o
1007       (%x. SUM y : A x. l x y * g(k x y)) +o
1008         O(%x. SUM y : A x. abs(l x y * h(k x y)))"
1009   apply (rule set_minus_imp_plus)
1010   apply (subst fun_diff_def)
1011   apply (subst setsum_subtractf [symmetric])
1012   apply (subst right_diff_distrib [symmetric])
1013   apply (rule bigo_setsum3)
1014   apply (subst fun_diff_def [symmetric])
1015   apply (erule set_plus_imp_minus)
1016 done
1018 ML_command{*AtpWrapper.problem_name := "BigO__bigo_setsum5"*}
1019 lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==>
1020     ALL x. 0 <= h x ==>
1021       (%x. SUM y : A x. (l x y) * f(k x y)) =o
1022         O(%x. SUM y : A x. (l x y) * h(k x y))"
1023   apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) =
1024       (%x. SUM y : A x. abs((l x y) * h(k x y)))")
1025   apply (erule ssubst)
1026   apply (erule bigo_setsum3)
1027   apply (rule ext)
1028   apply (rule setsum_cong2)
1029   apply (thin_tac "f \<in> O(h)")
1030 apply (metis abs_of_nonneg zero_le_mult_iff)
1031 done
1033 lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
1034     ALL x. 0 <= h x ==>
1035       (%x. SUM y : A x. (l x y) * f(k x y)) =o
1036         (%x. SUM y : A x. (l x y) * g(k x y)) +o
1037           O(%x. SUM y : A x. (l x y) * h(k x y))"
1038   apply (rule set_minus_imp_plus)
1039   apply (subst fun_diff_def)
1040   apply (subst setsum_subtractf [symmetric])
1041   apply (subst right_diff_distrib [symmetric])
1042   apply (rule bigo_setsum5)
1043   apply (subst fun_diff_def [symmetric])
1044   apply (drule set_plus_imp_minus)
1045   apply auto
1046 done
1048 subsection {* Misc useful stuff *}
1050 lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==>
1051   A \<oplus> B <= O(f)"
1052   apply (subst bigo_plus_idemp [symmetric])
1053   apply (rule set_plus_mono2)
1054   apply assumption+
1055 done
1057 lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
1058   apply (subst bigo_plus_idemp [symmetric])
1059   apply (rule set_plus_intro)
1060   apply assumption+
1061 done
1063 lemma bigo_useful_const_mult: "(c::'a::ordered_field) ~= 0 ==>
1064     (%x. c) * f =o O(h) ==> f =o O(h)"
1065   apply (rule subsetD)
1066   apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
1067   apply assumption
1068   apply (rule bigo_const_mult6)
1069   apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)")
1070   apply (erule ssubst)
1071   apply (erule set_times_intro2)
1073 done
1075 ML_command{*AtpWrapper.problem_name := "BigO__bigo_fix"*}
1076 lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
1077     f =o O(h)"
1079 (*sledgehammer*);
1080   apply clarify
1081   apply (rule_tac x = c in exI)
1082   apply safe
1083   apply (case_tac "x = 0")
1084 apply (metis OrderedGroup.abs_ge_zero  OrderedGroup.abs_zero  order_less_le  Ring_and_Field.split_mult_pos_le)
1085   apply (subgoal_tac "x = Suc (x - 1)")
1086   apply metis
1087   apply simp
1088   done
1091 lemma bigo_fix2:
1092     "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==>
1093        f 0 = g 0 ==> f =o g +o O(h)"
1094   apply (rule set_minus_imp_plus)
1095   apply (rule bigo_fix)
1096   apply (subst fun_diff_def)
1097   apply (subst fun_diff_def [symmetric])
1098   apply (rule set_plus_imp_minus)
1099   apply simp
1101 done
1103 subsection {* Less than or equal to *}
1105 constdefs
1106   lesso :: "('a => 'b::ordered_idom) => ('a => 'b) => ('a => 'b)"
1107       (infixl "<o" 70)
1108   "f <o g == (%x. max (f x - g x) 0)"
1110 lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==>
1111     g =o O(h)"
1112   apply (unfold bigo_def)
1113   apply clarsimp
1114 apply (blast intro: order_trans)
1115 done
1117 lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
1118       g =o O(h)"
1119   apply (erule bigo_lesseq1)
1120 apply (blast intro: abs_ge_self order_trans)
1121 done
1123 lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
1124       g =o O(h)"
1125   apply (erule bigo_lesseq2)
1126   apply (rule allI)
1127   apply (subst abs_of_nonneg)
1128   apply (erule spec)+
1129 done
1131 lemma bigo_lesseq4: "f =o O(h) ==>
1132     ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
1133       g =o O(h)"
1134   apply (erule bigo_lesseq1)
1135   apply (rule allI)
1136   apply (subst abs_of_nonneg)
1137   apply (erule spec)+
1138 done
1140 ML_command{*AtpWrapper.problem_name:="BigO__bigo_lesso1"*}
1141 lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
1142   apply (unfold lesso_def)
1143   apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
1144 (*??Translation of TSTP raised an exception: Type unification failed: Variable ?'X2.0::type not of sort ord*)
1145 apply (metis bigo_zero)
1146   apply (unfold func_zero)
1147   apply (rule ext)
1148   apply (simp split: split_max)
1149 done
1152 ML_command{*AtpWrapper.problem_name := "BigO__bigo_lesso2"*}
1153 lemma bigo_lesso2: "f =o g +o O(h) ==>
1154     ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
1155       k <o g =o O(h)"
1156   apply (unfold lesso_def)
1157   apply (rule bigo_lesseq4)
1158   apply (erule set_plus_imp_minus)
1159   apply (rule allI)
1160   apply (rule le_maxI2)
1161   apply (rule allI)
1162   apply (subst fun_diff_def)
1163 apply (erule thin_rl)
1164 (*sledgehammer*);
1165   apply (case_tac "0 <= k x - g x")
1166   prefer 2 (*re-order subgoals because I don't know what to put after a structured proof*)
1167    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)
1168 proof (neg_clausify)
1169 fix x
1170 assume 0: "\<And>A. k A \<le> f A"
1171 have 1: "\<And>(X1\<Colon>'b\<Colon>ordered_idom) X2. \<not> max X1 X2 < X1"
1172   by (metis linorder_not_less le_maxI1)  (*sort inserted by hand*)
1173 assume 2: "(0\<Colon>'b) \<le> k x - g x"
1174 have 3: "\<not> k x - g x < (0\<Colon>'b)"
1175   by (metis 2 linorder_not_less)
1176 have 4: "\<And>X1 X2. min X1 (k X2) \<le> f X2"
1177   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)
1178 have 5: "\<bar>g x - f x\<bar> = f x - g x"
1179   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)
1180 have 6: "max (0\<Colon>'b) (k x - g x) = k x - g x"
1181   by (metis min_max.less_eq_less_sup.le_iff_sup 2)
1182 assume 7: "\<not> max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>"
1183 have 8: "\<not> k x - g x \<le> f x - g x"
1184   by (metis 5 abs_minus_commute 7 min_max.less_eq_less_sup.sup_commute 6)
1185 show "False"
1186   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)
1187 qed
1189 ML_command{*AtpWrapper.problem_name := "BigO__bigo_lesso3"*}
1190 lemma bigo_lesso3: "f =o g +o O(h) ==>
1191     ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
1192       f <o k =o O(h)"
1193   apply (unfold lesso_def)
1194   apply (rule bigo_lesseq4)
1195   apply (erule set_plus_imp_minus)
1196   apply (rule allI)
1197   apply (rule le_maxI2)
1198   apply (rule allI)
1199   apply (subst fun_diff_def)
1200 apply (erule thin_rl)
1201 (*sledgehammer*);
1202   apply (case_tac "0 <= f x - k x")
1203   apply (simp del: compare_rls diff_minus);
1204   apply (subst abs_of_nonneg)
1205   apply (drule_tac x = x in spec) back
1206 ML_command{*AtpWrapper.problem_name := "BigO__bigo_lesso3_simpler"*}
1207 apply (metis diff_less_0_iff_less linorder_not_le not_leE uminus_add_conv_diff xt1(12) xt1(6))
1209 apply (metis abs_ge_zero linorder_linear min_max.less_eq_less_sup.sup_absorb1 min_max.less_eq_less_sup.sup_commute)
1210 done
1212 lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::ordered_field) ==>
1213     g =o h +o O(k) ==> f <o h =o O(k)"
1214   apply (unfold lesso_def)
1215   apply (drule set_plus_imp_minus)
1216   apply (drule bigo_abs5) back
1219   apply assumption
1220   apply (erule bigo_lesseq2) back
1221   apply (rule allI)
1222   apply (auto simp add: func_plus fun_diff_def compare_rls
1223     split: split_max abs_split)
1224 done
1226 ML_command{*AtpWrapper.problem_name := "BigO__bigo_lesso5"*}
1227 lemma bigo_lesso5: "f <o g =o O(h) ==>
1228     EX C. ALL x. f x <= g x + C * abs(h x)"
1229   apply (simp only: lesso_def bigo_alt_def)
1230   apply clarsimp
1231   apply (metis abs_if abs_mult add_commute diff_le_eq less_not_permute)
1232 done
1234 end