author | haftmann |
Fri, 04 Jul 2014 20:18:47 +0200 | |
changeset 57512 | cc97b347b301 |
parent 57418 | 6ab1c7cb0b8d |
child 57514 | bdc2c6b40bf2 |
permissions | -rw-r--r-- |
43197 | 1 |
(* Title: HOL/Metis_Examples/Big_O.thy |
2 |
Author: Lawrence C. Paulson, Cambridge University Computer Laboratory |
|
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Author: Jasmin Blanchette, TU Muenchen |
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|
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Metis example featuring the Big O notation. |
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*) |
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header {* Metis Example Featuring the Big O Notation *} |
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theory Big_O |
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explicit file specifications -- avoid secondary load path;
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imports |
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explicit file specifications -- avoid secondary load path;
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"~~/src/HOL/Decision_Procs/Dense_Linear_Order" |
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explicit file specifications -- avoid secondary load path;
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"~~/src/HOL/Library/Function_Algebras" |
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explicit file specifications -- avoid secondary load path;
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"~~/src/HOL/Library/Set_Algebras" |
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begin |
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subsection {* Definitions *} |
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||
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definition bigo :: "('a => 'b\<Colon>linordered_idom) => ('a => 'b) set" ("(1O'(_'))") where |
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"O(f\<Colon>('a => 'b)) == {h. \<exists>c. \<forall>x. abs (h x) <= c * abs (f x)}" |
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|
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lemma bigo_pos_const: |
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"(\<exists>c\<Colon>'a\<Colon>linordered_idom. |
24 |
\<forall>x. abs (h x) \<le> c * abs (f x)) |
|
25 |
\<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. abs(h x) \<le> c * abs (f x)))" |
|
26 |
by (metis (no_types) abs_ge_zero |
|
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comm_semiring_1_class.normalizing_semiring_rules(7) mult.comm_neutral |
28 |
mult_nonpos_nonneg not_leE order_trans zero_less_one) |
|
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|
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(*** Now various verions with an increasing shrink factor ***) |
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|
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sledgehammer_params [isar_proofs, compress = 1] |
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|
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lemma |
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"(\<exists>c\<Colon>'a\<Colon>linordered_idom. |
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\<forall>x. abs (h x) \<le> c * abs (f x)) |
|
37 |
\<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. abs(h x) \<le> c * abs (f x)))" |
|
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apply auto |
39 |
apply (case_tac "c = 0", simp) |
|
40 |
apply (rule_tac x = "1" in exI, simp) |
|
41 |
apply (rule_tac x = "abs c" in exI, auto) |
|
36561 | 42 |
proof - |
43 |
fix c :: 'a and x :: 'b |
|
44 |
assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
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diff
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|
45 |
have F1: "\<forall>x\<^sub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> \<bar>x\<^sub>1\<bar>" by (metis abs_ge_zero) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
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diff
changeset
|
46 |
have F2: "\<forall>x\<^sub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
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|
47 |
have F3: "\<forall>x\<^sub>1 x\<^sub>3. x\<^sub>3 \<le> \<bar>h x\<^sub>1\<bar> \<longrightarrow> x\<^sub>3 \<le> c * \<bar>f x\<^sub>1\<bar>" by (metis A1 order_trans) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
48 |
have F4: "\<forall>x\<^sub>2 x\<^sub>3\<Colon>'a\<Colon>linordered_idom. \<bar>x\<^sub>3\<bar> * \<bar>x\<^sub>2\<bar> = \<bar>x\<^sub>3 * x\<^sub>2\<bar>" |
36561 | 49 |
by (metis abs_mult) |
53015
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standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
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50 |
have F5: "\<forall>x\<^sub>3 x\<^sub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> x\<^sub>1 \<longrightarrow> \<bar>x\<^sub>3 * x\<^sub>1\<bar> = \<bar>x\<^sub>3\<bar> * x\<^sub>1" |
36561 | 51 |
by (metis abs_mult_pos) |
53015
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standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
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changeset
|
52 |
hence "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1\<Colon>'a\<Colon>linordered_idom\<bar> = \<bar>1\<bar> * x\<^sub>1" by (metis F2) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
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|
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hence "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^sub>1" by (metis F2 abs_one) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
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diff
changeset
|
54 |
hence "\<forall>x\<^sub>3. 0 \<le> \<bar>h x\<^sub>3\<bar> \<longrightarrow> \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>" by (metis F3) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
55 |
hence "\<forall>x\<^sub>3. \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>" by (metis F1) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
56 |
hence "\<forall>x\<^sub>3. (0\<Colon>'a) \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^sub>3\<bar>" by (metis F5) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
57 |
hence "\<forall>x\<^sub>3. (0\<Colon>'a) \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c * f x\<^sub>3\<bar>" by (metis F4) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
58 |
hence "\<forall>x\<^sub>3. c * \<bar>f x\<^sub>3\<bar> = \<bar>c * f x\<^sub>3\<bar>" by (metis F1) |
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hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1) |
60 |
thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis F4) |
|
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qed |
62 |
||
57245 | 63 |
sledgehammer_params [isar_proofs, compress = 2] |
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parents:
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|
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lemma |
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"(\<exists>c\<Colon>'a\<Colon>linordered_idom. |
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\<forall>x. abs (h x) \<le> c * abs (f x)) |
|
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\<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. abs(h x) \<le> c * abs (f x)))" |
|
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apply auto |
70 |
apply (case_tac "c = 0", simp) |
|
71 |
apply (rule_tac x = "1" in exI, simp) |
|
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apply (rule_tac x = "abs c" in exI, auto) |
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proof - |
74 |
fix c :: 'a and x :: 'b |
|
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assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
76 |
have F1: "\<forall>x\<^sub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
77 |
have F2: "\<forall>x\<^sub>2 x\<^sub>3\<Colon>'a\<Colon>linordered_idom. \<bar>x\<^sub>3\<bar> * \<bar>x\<^sub>2\<bar> = \<bar>x\<^sub>3 * x\<^sub>2\<bar>" |
36561 | 78 |
by (metis abs_mult) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
79 |
have "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^sub>1" by (metis F1 abs_mult_pos abs_one) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
80 |
hence "\<forall>x\<^sub>3. \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>" by (metis A1 abs_ge_zero order_trans) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
81 |
hence "\<forall>x\<^sub>3. 0 \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c * f x\<^sub>3\<bar>" by (metis F2 abs_mult_pos) |
36561 | 82 |
hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero) |
83 |
thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis F2) |
|
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qed |
85 |
||
57245 | 86 |
sledgehammer_params [isar_proofs, compress = 3] |
25710
4cdf7de81e1b
Replaced refs by config params; finer critical section in mets method
paulson
parents:
25592
diff
changeset
|
87 |
|
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lemma |
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"(\<exists>c\<Colon>'a\<Colon>linordered_idom. |
90 |
\<forall>x. abs (h x) \<le> c * abs (f x)) |
|
91 |
\<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. abs(h x) \<le> c * abs (f x)))" |
|
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apply auto |
93 |
apply (case_tac "c = 0", simp) |
|
94 |
apply (rule_tac x = "1" in exI, simp) |
|
36561 | 95 |
apply (rule_tac x = "abs c" in exI, auto) |
96 |
proof - |
|
97 |
fix c :: 'a and x :: 'b |
|
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assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
99 |
have F1: "\<forall>x\<^sub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
100 |
have F2: "\<forall>x\<^sub>3 x\<^sub>1\<Colon>'a\<Colon>linordered_idom. 0 \<le> x\<^sub>1 \<longrightarrow> \<bar>x\<^sub>3 * x\<^sub>1\<bar> = \<bar>x\<^sub>3\<bar> * x\<^sub>1" by (metis abs_mult_pos) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
101 |
hence "\<forall>x\<^sub>1\<ge>0. \<bar>x\<^sub>1\<Colon>'a\<Colon>linordered_idom\<bar> = x\<^sub>1" by (metis F1 abs_one) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
102 |
hence "\<forall>x\<^sub>3. 0 \<le> \<bar>f x\<^sub>3\<bar> \<longrightarrow> c * \<bar>f x\<^sub>3\<bar> = \<bar>c\<bar> * \<bar>f x\<^sub>3\<bar>" by (metis F2 A1 abs_ge_zero order_trans) |
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thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis A1 abs_ge_zero) |
23449 | 104 |
qed |
105 |
||
57245 | 106 |
sledgehammer_params [isar_proofs, compress = 4] |
24545 | 107 |
|
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lemma |
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"(\<exists>c\<Colon>'a\<Colon>linordered_idom. |
110 |
\<forall>x. abs (h x) \<le> c * abs (f x)) |
|
111 |
\<longleftrightarrow> (\<exists>c. 0 < c \<and> (\<forall>x. abs(h x) \<le> c * abs (f x)))" |
|
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apply auto |
113 |
apply (case_tac "c = 0", simp) |
|
114 |
apply (rule_tac x = "1" in exI, simp) |
|
36561 | 115 |
apply (rule_tac x = "abs c" in exI, auto) |
116 |
proof - |
|
117 |
fix c :: 'a and x :: 'b |
|
118 |
assume A1: "\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>" |
|
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
119 |
have "\<forall>x\<^sub>1\<Colon>'a\<Colon>linordered_idom. 1 * x\<^sub>1 = x\<^sub>1" by (metis mult_1) |
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
120 |
hence "\<forall>x\<^sub>3. \<bar>c * \<bar>f x\<^sub>3\<bar>\<bar> = c * \<bar>f x\<^sub>3\<bar>" |
36561 | 121 |
by (metis A1 abs_ge_zero order_trans abs_mult_pos abs_one) |
122 |
hence "\<bar>h x\<bar> \<le> \<bar>c * f x\<bar>" by (metis A1 abs_ge_zero abs_mult_pos abs_mult) |
|
123 |
thus "\<bar>h x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>" by (metis abs_mult) |
|
24545 | 124 |
qed |
125 |
||
57245 | 126 |
sledgehammer_params [isar_proofs, compress = 1] |
24545 | 127 |
|
46364 | 128 |
lemma bigo_alt_def: "O(f) = {h. \<exists>c. (0 < c \<and> (\<forall>x. abs (h x) <= c * abs (f x)))}" |
23449 | 129 |
by (auto simp add: bigo_def bigo_pos_const) |
130 |
||
46364 | 131 |
lemma bigo_elt_subset [intro]: "f : O(g) \<Longrightarrow> O(f) \<le> O(g)" |
45575 | 132 |
apply (auto simp add: bigo_alt_def) |
133 |
apply (rule_tac x = "ca * c" in exI) |
|
46364 | 134 |
by (metis comm_semiring_1_class.normalizing_semiring_rules(7,19) |
135 |
mult_le_cancel_left_pos order_trans mult_pos_pos) |
|
23449 | 136 |
|
137 |
lemma bigo_refl [intro]: "f : O(f)" |
|
46364 | 138 |
unfolding bigo_def mem_Collect_eq |
36844 | 139 |
by (metis mult_1 order_refl) |
23449 | 140 |
|
141 |
lemma bigo_zero: "0 : O(g)" |
|
36561 | 142 |
apply (auto simp add: bigo_def func_zero) |
36844 | 143 |
by (metis mult_zero_left order_refl) |
23449 | 144 |
|
45575 | 145 |
lemma bigo_zero2: "O(\<lambda>x. 0) = {\<lambda>x. 0}" |
146 |
by (auto simp add: bigo_def) |
|
23449 | 147 |
|
43197 | 148 |
lemma bigo_plus_self_subset [intro]: |
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
149 |
"O(f) + O(f) <= O(f)" |
45575 | 150 |
apply (auto simp add: bigo_alt_def set_plus_def) |
151 |
apply (rule_tac x = "c + ca" in exI) |
|
152 |
apply auto |
|
153 |
apply (simp add: ring_distribs func_plus) |
|
154 |
by (metis order_trans abs_triangle_ineq add_mono) |
|
23449 | 155 |
|
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
156 |
lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)" |
45575 | 157 |
by (metis bigo_plus_self_subset bigo_zero set_eq_subset set_zero_plus2) |
23449 | 158 |
|
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
159 |
lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)" |
45575 | 160 |
apply (rule subsetI) |
161 |
apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus_def) |
|
162 |
apply (subst bigo_pos_const [symmetric])+ |
|
163 |
apply (rule_tac x = "\<lambda>n. if abs (g n) <= (abs (f n)) then x n else 0" in exI) |
|
164 |
apply (rule conjI) |
|
165 |
apply (rule_tac x = "c + c" in exI) |
|
166 |
apply clarsimp |
|
56536 | 167 |
apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)") |
168 |
apply (metis mult_2 order_trans) |
|
169 |
apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))") |
|
170 |
apply (erule order_trans) |
|
171 |
apply (simp add: ring_distribs) |
|
172 |
apply (rule mult_left_mono) |
|
173 |
apply (simp add: abs_triangle_ineq) |
|
174 |
apply (simp add: order_less_le) |
|
45575 | 175 |
apply (rule_tac x = "\<lambda>n. if (abs (f n)) < abs (g n) then x n else 0" in exI) |
176 |
apply (rule conjI) |
|
177 |
apply (rule_tac x = "c + c" in exI) |
|
178 |
apply auto |
|
56536 | 179 |
apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)") |
180 |
apply (metis order_trans mult_2) |
|
181 |
apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))") |
|
182 |
apply (erule order_trans) |
|
183 |
apply (simp add: ring_distribs) |
|
184 |
by (metis abs_triangle_ineq mult_le_cancel_left_pos) |
|
23449 | 185 |
|
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
186 |
lemma bigo_plus_subset2 [intro]: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> A + B <= O(f)" |
45575 | 187 |
by (metis bigo_plus_idemp set_plus_mono2) |
23449 | 188 |
|
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
189 |
lemma bigo_plus_eq: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) + O(g)" |
45575 | 190 |
apply (rule equalityI) |
191 |
apply (rule bigo_plus_subset) |
|
192 |
apply (simp add: bigo_alt_def set_plus_def func_plus) |
|
193 |
apply clarify |
|
194 |
(* sledgehammer *) |
|
195 |
apply (rule_tac x = "max c ca" in exI) |
|
46369 | 196 |
|
45575 | 197 |
apply (rule conjI) |
198 |
apply (metis less_max_iff_disj) |
|
199 |
apply clarify |
|
200 |
apply (drule_tac x = "xa" in spec)+ |
|
201 |
apply (subgoal_tac "0 <= f xa + g xa") |
|
202 |
apply (simp add: ring_distribs) |
|
203 |
apply (subgoal_tac "abs (a xa + b xa) <= abs (a xa) + abs (b xa)") |
|
204 |
apply (subgoal_tac "abs (a xa) + abs (b xa) <= |
|
205 |
max c ca * f xa + max c ca * g xa") |
|
206 |
apply (metis order_trans) |
|
23449 | 207 |
defer 1 |
45575 | 208 |
apply (metis abs_triangle_ineq) |
209 |
apply (metis add_nonneg_nonneg) |
|
46644 | 210 |
apply (rule add_mono) |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54230
diff
changeset
|
211 |
apply (metis max.cobounded2 linorder_linear max.absorb1 mult_right_mono xt1(6)) |
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54230
diff
changeset
|
212 |
by (metis max.cobounded2 linorder_not_le mult_le_cancel_right order_trans) |
23449 | 213 |
|
45575 | 214 |
lemma bigo_bounded_alt: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)" |
215 |
apply (auto simp add: bigo_def) |
|
36561 | 216 |
(* Version 1: one-line proof *) |
45575 | 217 |
by (metis abs_le_D1 linorder_class.not_less order_less_le Orderings.xt1(12) abs_mult) |
23449 | 218 |
|
45575 | 219 |
lemma "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)" |
36561 | 220 |
apply (auto simp add: bigo_def) |
221 |
(* Version 2: structured proof *) |
|
222 |
proof - |
|
223 |
assume "\<forall>x. f x \<le> c * g x" |
|
224 |
thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" by (metis abs_mult abs_ge_self order_trans) |
|
23449 | 225 |
qed |
226 |
||
45575 | 227 |
lemma bigo_bounded: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= g x \<Longrightarrow> f : O(g)" |
228 |
apply (erule bigo_bounded_alt [of f 1 g]) |
|
229 |
by (metis mult_1) |
|
23449 | 230 |
|
45575 | 231 |
lemma bigo_bounded2: "\<forall>x. lb x <= f x \<Longrightarrow> \<forall>x. f x <= lb x + g x \<Longrightarrow> f : lb +o O(g)" |
36561 | 232 |
apply (rule set_minus_imp_plus) |
233 |
apply (rule bigo_bounded) |
|
46369 | 234 |
apply (metis add_le_cancel_left diff_add_cancel diff_self minus_apply |
235 |
comm_semiring_1_class.normalizing_semiring_rules(24)) |
|
236 |
by (metis add_le_cancel_left diff_add_cancel func_plus le_fun_def |
|
237 |
comm_semiring_1_class.normalizing_semiring_rules(24)) |
|
23449 | 238 |
|
45575 | 239 |
lemma bigo_abs: "(\<lambda>x. abs(f x)) =o O(f)" |
36561 | 240 |
apply (unfold bigo_def) |
241 |
apply auto |
|
36844 | 242 |
by (metis mult_1 order_refl) |
23449 | 243 |
|
45575 | 244 |
lemma bigo_abs2: "f =o O(\<lambda>x. abs(f x))" |
36561 | 245 |
apply (unfold bigo_def) |
246 |
apply auto |
|
36844 | 247 |
by (metis mult_1 order_refl) |
43197 | 248 |
|
45575 | 249 |
lemma bigo_abs3: "O(f) = O(\<lambda>x. abs(f x))" |
36561 | 250 |
proof - |
251 |
have F1: "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)" by (metis bigo_elt_subset) |
|
252 |
have F2: "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)" by (metis bigo_abs) |
|
253 |
have "\<forall>u. u \<in> O(\<lambda>R. \<bar>u R\<bar>)" by (metis bigo_abs2) |
|
254 |
thus "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" using F1 F2 by auto |
|
43197 | 255 |
qed |
23449 | 256 |
|
45575 | 257 |
lemma bigo_abs4: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. abs (f x)) =o (\<lambda>x. abs (g x)) +o O(h)" |
23449 | 258 |
apply (drule set_plus_imp_minus) |
259 |
apply (rule set_minus_imp_plus) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
260 |
apply (subst fun_diff_def) |
23449 | 261 |
proof - |
262 |
assume a: "f - g : O(h)" |
|
45575 | 263 |
have "(\<lambda>x. abs (f x) - abs (g x)) =o O(\<lambda>x. abs(abs (f x) - abs (g x)))" |
23449 | 264 |
by (rule bigo_abs2) |
45575 | 265 |
also have "... <= O(\<lambda>x. abs (f x - g x))" |
23449 | 266 |
apply (rule bigo_elt_subset) |
267 |
apply (rule bigo_bounded) |
|
46369 | 268 |
apply (metis abs_ge_zero) |
269 |
by (metis abs_triangle_ineq3) |
|
23449 | 270 |
also have "... <= O(f - g)" |
271 |
apply (rule bigo_elt_subset) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
272 |
apply (subst fun_diff_def) |
23449 | 273 |
apply (rule bigo_abs) |
274 |
done |
|
275 |
also have "... <= O(h)" |
|
23464 | 276 |
using a by (rule bigo_elt_subset) |
45575 | 277 |
finally show "(\<lambda>x. abs (f x) - abs (g x)) : O(h)". |
23449 | 278 |
qed |
279 |
||
45575 | 280 |
lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. abs(f x)) =o O(g)" |
23449 | 281 |
by (unfold bigo_def, auto) |
282 |
||
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
283 |
lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) + O(h)" |
23449 | 284 |
proof - |
285 |
assume "f : g +o O(h)" |
|
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
286 |
also have "... <= O(g) + O(h)" |
23449 | 287 |
by (auto del: subsetI) |
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
288 |
also have "... = O(\<lambda>x. abs(g x)) + O(\<lambda>x. abs(h x))" |
46369 | 289 |
by (metis bigo_abs3) |
45575 | 290 |
also have "... = O((\<lambda>x. abs(g x)) + (\<lambda>x. abs(h x)))" |
23449 | 291 |
by (rule bigo_plus_eq [symmetric], auto) |
292 |
finally have "f : ...". |
|
293 |
then have "O(f) <= ..." |
|
294 |
by (elim bigo_elt_subset) |
|
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
295 |
also have "... = O(\<lambda>x. abs(g x)) + O(\<lambda>x. abs(h x))" |
23449 | 296 |
by (rule bigo_plus_eq, auto) |
297 |
finally show ?thesis |
|
298 |
by (simp add: bigo_abs3 [symmetric]) |
|
299 |
qed |
|
300 |
||
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
301 |
lemma bigo_mult [intro]: "O(f) * O(g) <= O(f * g)" |
46369 | 302 |
apply (rule subsetI) |
303 |
apply (subst bigo_def) |
|
304 |
apply (auto simp del: abs_mult mult_ac |
|
305 |
simp add: bigo_alt_def set_times_def func_times) |
|
45575 | 306 |
(* sledgehammer *) |
46369 | 307 |
apply (rule_tac x = "c * ca" in exI) |
308 |
apply (rule allI) |
|
309 |
apply (erule_tac x = x in allE)+ |
|
310 |
apply (subgoal_tac "c * ca * abs (f x * g x) = (c * abs(f x)) * (ca * abs (g x))") |
|
311 |
apply (metis (no_types) abs_ge_zero abs_mult mult_mono') |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
312 |
by (metis mult.assoc mult.left_commute abs_of_pos mult.left_commute abs_mult) |
23449 | 313 |
|
314 |
lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)" |
|
46369 | 315 |
by (metis bigo_mult bigo_refl set_times_mono3 subset_trans) |
23449 | 316 |
|
45575 | 317 |
lemma bigo_mult3: "f : O(h) \<Longrightarrow> g : O(j) \<Longrightarrow> f * g : O(h * j)" |
36561 | 318 |
by (metis bigo_mult set_rev_mp set_times_intro) |
23449 | 319 |
|
45575 | 320 |
lemma bigo_mult4 [intro]:"f : k +o O(h) \<Longrightarrow> g * f : (g * k) +o O(g * h)" |
23449 | 321 |
by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib) |
322 |
||
45575 | 323 |
lemma bigo_mult5: "\<forall>x. f x ~= 0 \<Longrightarrow> |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
324 |
O(f * g) <= (f\<Colon>'a => ('b\<Colon>linordered_field)) *o O(g)" |
23449 | 325 |
proof - |
45575 | 326 |
assume a: "\<forall>x. f x ~= 0" |
23449 | 327 |
show "O(f * g) <= f *o O(g)" |
328 |
proof |
|
329 |
fix h |
|
41541 | 330 |
assume h: "h : O(f * g)" |
45575 | 331 |
then have "(\<lambda>x. 1 / (f x)) * h : (\<lambda>x. 1 / f x) *o O(f * g)" |
23449 | 332 |
by auto |
45575 | 333 |
also have "... <= O((\<lambda>x. 1 / f x) * (f * g))" |
23449 | 334 |
by (rule bigo_mult2) |
45575 | 335 |
also have "(\<lambda>x. 1 / f x) * (f * g) = g" |
43197 | 336 |
apply (simp add: func_times) |
46369 | 337 |
by (metis (lifting, no_types) a ext mult_ac(2) nonzero_divide_eq_eq) |
45575 | 338 |
finally have "(\<lambda>x. (1\<Colon>'b) / f x) * h : O(g)". |
339 |
then have "f * ((\<lambda>x. (1\<Colon>'b) / f x) * h) : f *o O(g)" |
|
23449 | 340 |
by auto |
45575 | 341 |
also have "f * ((\<lambda>x. (1\<Colon>'b) / f x) * h) = h" |
43197 | 342 |
apply (simp add: func_times) |
46369 | 343 |
by (metis (lifting, no_types) a eq_divide_imp ext |
344 |
comm_semiring_1_class.normalizing_semiring_rules(7)) |
|
23449 | 345 |
finally show "h : f *o O(g)". |
346 |
qed |
|
347 |
qed |
|
348 |
||
46369 | 349 |
lemma bigo_mult6: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
350 |
"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = (f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) *o O(g)" |
23449 | 351 |
by (metis bigo_mult2 bigo_mult5 order_antisym) |
352 |
||
353 |
(*proof requires relaxing relevance: 2007-01-25*) |
|
45705 | 354 |
declare bigo_mult6 [simp] |
355 |
||
46369 | 356 |
lemma bigo_mult7: |
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
357 |
"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<le> O(f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) * O(g)" |
46369 | 358 |
by (metis bigo_refl bigo_mult6 set_times_mono3) |
23449 | 359 |
|
45575 | 360 |
declare bigo_mult6 [simp del] |
361 |
declare bigo_mult7 [intro!] |
|
362 |
||
46369 | 363 |
lemma bigo_mult8: |
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
364 |
"\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f\<Colon>'a \<Rightarrow> ('b\<Colon>linordered_field)) * O(g)" |
23449 | 365 |
by (metis bigo_mult bigo_mult7 order_antisym_conv) |
366 |
||
45575 | 367 |
lemma bigo_minus [intro]: "f : O(g) \<Longrightarrow> - f : O(g)" |
46369 | 368 |
by (auto simp add: bigo_def fun_Compl_def) |
23449 | 369 |
|
45575 | 370 |
lemma bigo_minus2: "f : g +o O(h) \<Longrightarrow> -f : -g +o O(h)" |
46369 | 371 |
by (metis (no_types) bigo_elt_subset bigo_minus bigo_mult4 bigo_refl |
372 |
comm_semiring_1_class.normalizing_semiring_rules(11) minus_mult_left |
|
373 |
set_plus_mono_b) |
|
23449 | 374 |
|
375 |
lemma bigo_minus3: "O(-f) = O(f)" |
|
46369 | 376 |
by (metis bigo_elt_subset bigo_minus bigo_refl equalityI minus_minus) |
23449 | 377 |
|
46369 | 378 |
lemma bigo_plus_absorb_lemma1: "f : O(g) \<Longrightarrow> f +o O(g) \<le> O(g)" |
379 |
by (metis bigo_plus_idemp set_plus_mono3) |
|
23449 | 380 |
|
46369 | 381 |
lemma bigo_plus_absorb_lemma2: "f : O(g) \<Longrightarrow> O(g) \<le> f +o O(g)" |
382 |
by (metis (no_types) bigo_minus bigo_plus_absorb_lemma1 right_minus |
|
46644 | 383 |
set_plus_mono set_plus_rearrange2 set_zero_plus subsetD subset_refl |
384 |
subset_trans) |
|
23449 | 385 |
|
45575 | 386 |
lemma bigo_plus_absorb [simp]: "f : O(g) \<Longrightarrow> f +o O(g) = O(g)" |
41865
4e8483cc2cc5
declare ext [intro]: Extensionality now available by default
paulson
parents:
41541
diff
changeset
|
387 |
by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff) |
23449 | 388 |
|
46369 | 389 |
lemma bigo_plus_absorb2 [intro]: "f : O(g) \<Longrightarrow> A <= O(g) \<Longrightarrow> f +o A \<le> O(g)" |
390 |
by (metis bigo_plus_absorb set_plus_mono) |
|
23449 | 391 |
|
45575 | 392 |
lemma bigo_add_commute_imp: "f : g +o O(h) \<Longrightarrow> g : f +o O(h)" |
46369 | 393 |
by (metis bigo_minus minus_diff_eq set_plus_imp_minus set_minus_plus) |
23449 | 394 |
|
395 |
lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))" |
|
46369 | 396 |
by (metis bigo_add_commute_imp) |
23449 | 397 |
|
45575 | 398 |
lemma bigo_const1: "(\<lambda>x. c) : O(\<lambda>x. 1)" |
23449 | 399 |
by (auto simp add: bigo_def mult_ac) |
400 |
||
46369 | 401 |
lemma bigo_const2 [intro]: "O(\<lambda>x. c) \<le> O(\<lambda>x. 1)" |
41865
4e8483cc2cc5
declare ext [intro]: Extensionality now available by default
paulson
parents:
41541
diff
changeset
|
402 |
by (metis bigo_const1 bigo_elt_subset) |
23449 | 403 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
404 |
lemma bigo_const3: "(c\<Colon>'a\<Colon>linordered_field) ~= 0 \<Longrightarrow> (\<lambda>x. 1) : O(\<lambda>x. c)" |
23449 | 405 |
apply (simp add: bigo_def) |
36561 | 406 |
by (metis abs_eq_0 left_inverse order_refl) |
23449 | 407 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
408 |
lemma bigo_const4: "(c\<Colon>'a\<Colon>linordered_field) ~= 0 \<Longrightarrow> O(\<lambda>x. 1) <= O(\<lambda>x. c)" |
46369 | 409 |
by (metis bigo_elt_subset bigo_const3) |
23449 | 410 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
411 |
lemma bigo_const [simp]: "(c\<Colon>'a\<Colon>linordered_field) ~= 0 \<Longrightarrow> |
45575 | 412 |
O(\<lambda>x. c) = O(\<lambda>x. 1)" |
46369 | 413 |
by (metis bigo_const2 bigo_const4 equalityI) |
23449 | 414 |
|
45575 | 415 |
lemma bigo_const_mult1: "(\<lambda>x. c * f x) : O(f)" |
46369 | 416 |
apply (simp add: bigo_def abs_mult) |
36561 | 417 |
by (metis le_less) |
23449 | 418 |
|
46369 | 419 |
lemma bigo_const_mult2: "O(\<lambda>x. c * f x) \<le> O(f)" |
23449 | 420 |
by (rule bigo_elt_subset, rule bigo_const_mult1) |
421 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
422 |
lemma bigo_const_mult3: "(c\<Colon>'a\<Colon>linordered_field) ~= 0 \<Longrightarrow> f : O(\<lambda>x. c * f x)" |
45575 | 423 |
apply (simp add: bigo_def) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
424 |
by (metis (no_types) abs_mult mult.assoc mult_1 order_refl left_inverse) |
23449 | 425 |
|
46369 | 426 |
lemma bigo_const_mult4: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
427 |
"(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> O(f) \<le> O(\<lambda>x. c * f x)" |
46369 | 428 |
by (metis bigo_elt_subset bigo_const_mult3) |
23449 | 429 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
430 |
lemma bigo_const_mult [simp]: "(c\<Colon>'a\<Colon>linordered_field) ~= 0 \<Longrightarrow> |
45575 | 431 |
O(\<lambda>x. c * f x) = O(f)" |
46369 | 432 |
by (metis equalityI bigo_const_mult2 bigo_const_mult4) |
23449 | 433 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
434 |
lemma bigo_const_mult5 [simp]: "(c\<Colon>'a\<Colon>linordered_field) ~= 0 \<Longrightarrow> |
45575 | 435 |
(\<lambda>x. c) *o O(f) = O(f)" |
23449 | 436 |
apply (auto del: subsetI) |
437 |
apply (rule order_trans) |
|
438 |
apply (rule bigo_mult2) |
|
439 |
apply (simp add: func_times) |
|
440 |
apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times) |
|
45575 | 441 |
apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI) |
43197 | 442 |
apply (rename_tac g d) |
24942 | 443 |
apply safe |
43197 | 444 |
apply (rule_tac [2] ext) |
445 |
prefer 2 |
|
26041
c2e15e65165f
locales ACf, ACIf, ACIfSL and ACIfSLlin have been abandoned in favour of the existing algebraic classes ab_semigroup_mult, ab_semigroup_idem_mult, lower_semilattice (resp. uper_semilattice) and linorder
haftmann
parents:
25710
diff
changeset
|
446 |
apply simp |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
447 |
apply (simp add: mult.assoc [symmetric] abs_mult) |
39259
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
448 |
(* couldn't get this proof without the step above *) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
449 |
proof - |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
450 |
fix g :: "'b \<Rightarrow> 'a" and d :: 'a |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
451 |
assume A1: "c \<noteq> (0\<Colon>'a)" |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
452 |
assume A2: "\<forall>x\<Colon>'b. \<bar>g x\<bar> \<le> d * \<bar>f x\<bar>" |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
453 |
have F1: "inverse \<bar>c\<bar> = \<bar>inverse c\<bar>" using A1 by (metis nonzero_abs_inverse) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
454 |
have F2: "(0\<Colon>'a) < \<bar>c\<bar>" using A1 by (metis zero_less_abs_iff) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
455 |
have "(0\<Colon>'a) < \<bar>c\<bar> \<longrightarrow> (0\<Colon>'a) < \<bar>inverse c\<bar>" using F1 by (metis positive_imp_inverse_positive) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
456 |
hence "(0\<Colon>'a) < \<bar>inverse c\<bar>" using F2 by metis |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
457 |
hence F3: "(0\<Colon>'a) \<le> \<bar>inverse c\<bar>" by (metis order_le_less) |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
458 |
have "\<exists>(u\<Colon>'a) SKF\<^sub>7\<Colon>'a \<Rightarrow> 'b. \<bar>g (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar>" |
39259
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
459 |
using A2 by metis |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
460 |
hence F4: "\<exists>(u\<Colon>'a) SKF\<^sub>7\<Colon>'a \<Rightarrow> 'b. \<bar>g (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<le> u * \<bar>f (SKF\<^sub>7 (\<bar>inverse c\<bar> * u))\<bar> \<and> (0\<Colon>'a) \<le> \<bar>inverse c\<bar>" |
39259
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
461 |
using F3 by metis |
53015
a1119cf551e8
standardized symbols via "isabelle update_sub_sup", excluding src/Pure and src/Tools/WWW_Find;
wenzelm
parents:
51130
diff
changeset
|
462 |
hence "\<exists>(v\<Colon>'a) (u\<Colon>'a) SKF\<^sub>7\<Colon>'a \<Rightarrow> 'b. \<bar>inverse c\<bar> * \<bar>g (SKF\<^sub>7 (u * v))\<bar> \<le> u * (v * \<bar>f (SKF\<^sub>7 (u * v))\<bar>)" |
39259
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
463 |
by (metis comm_mult_left_mono) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
464 |
thus "\<exists>ca\<Colon>'a. \<forall>x\<Colon>'b. \<bar>inverse c\<bar> * \<bar>g x\<bar> \<le> ca * \<bar>f x\<bar>" |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
465 |
using A2 F4 by (metis ab_semigroup_mult_class.mult_ac(1) comm_mult_left_mono) |
194014eb4f9f
replace two slow "metis" proofs with faster proofs
blanchet
parents:
38991
diff
changeset
|
466 |
qed |
23449 | 467 |
|
45575 | 468 |
lemma bigo_const_mult6 [intro]: "(\<lambda>x. c) *o O(f) <= O(f)" |
23449 | 469 |
apply (auto intro!: subsetI |
470 |
simp add: bigo_def elt_set_times_def func_times |
|
471 |
simp del: abs_mult mult_ac) |
|
45575 | 472 |
(* sledgehammer *) |
23449 | 473 |
apply (rule_tac x = "ca * (abs c)" in exI) |
474 |
apply (rule allI) |
|
475 |
apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))") |
|
476 |
apply (erule ssubst) |
|
477 |
apply (subst abs_mult) |
|
478 |
apply (rule mult_left_mono) |
|
479 |
apply (erule spec) |
|
480 |
apply simp |
|
46369 | 481 |
apply (simp add: mult_ac) |
23449 | 482 |
done |
483 |
||
45575 | 484 |
lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)" |
46369 | 485 |
by (metis bigo_const_mult1 bigo_elt_subset order_less_le psubsetD) |
23449 | 486 |
|
45575 | 487 |
lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f(k x)) =o O(\<lambda>x. g(k x))" |
23449 | 488 |
by (unfold bigo_def, auto) |
489 |
||
46369 | 490 |
lemma bigo_compose2: |
491 |
"f =o g +o O(h) \<Longrightarrow> (\<lambda>x. f(k x)) =o (\<lambda>x. g(k x)) +o O(\<lambda>x. h(k x))" |
|
54230
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53015
diff
changeset
|
492 |
apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53015
diff
changeset
|
493 |
apply (drule bigo_compose1 [of "f - g" h k]) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53015
diff
changeset
|
494 |
apply (simp add: fun_diff_def) |
b1d955791529
more simplification rules on unary and binary minus
haftmann
parents:
53015
diff
changeset
|
495 |
done |
23449 | 496 |
|
497 |
subsection {* Setsum *} |
|
498 |
||
45575 | 499 |
lemma bigo_setsum_main: "\<forall>x. \<forall>y \<in> A x. 0 <= h x y \<Longrightarrow> |
500 |
\<exists>c. \<forall>x. \<forall>y \<in> A x. abs (f x y) <= c * (h x y) \<Longrightarrow> |
|
501 |
(\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)" |
|
46369 | 502 |
apply (auto simp add: bigo_def) |
503 |
apply (rule_tac x = "abs c" in exI) |
|
504 |
apply (subst abs_of_nonneg) back back |
|
505 |
apply (rule setsum_nonneg) |
|
506 |
apply force |
|
507 |
apply (subst setsum_right_distrib) |
|
508 |
apply (rule allI) |
|
509 |
apply (rule order_trans) |
|
510 |
apply (rule setsum_abs) |
|
511 |
apply (rule setsum_mono) |
|
512 |
by (metis abs_ge_self abs_mult_pos order_trans) |
|
23449 | 513 |
|
45575 | 514 |
lemma bigo_setsum1: "\<forall>x y. 0 <= h x y \<Longrightarrow> |
515 |
\<exists>c. \<forall>x y. abs (f x y) <= c * (h x y) \<Longrightarrow> |
|
516 |
(\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)" |
|
517 |
by (metis (no_types) bigo_setsum_main) |
|
23449 | 518 |
|
45575 | 519 |
lemma bigo_setsum2: "\<forall>y. 0 <= h y \<Longrightarrow> |
46369 | 520 |
\<exists>c. \<forall>y. abs (f y) <= c * (h y) \<Longrightarrow> |
45575 | 521 |
(\<lambda>x. SUM y : A x. f y) =o O(\<lambda>x. SUM y : A x. h y)" |
46369 | 522 |
apply (rule bigo_setsum1) |
523 |
by metis+ |
|
23449 | 524 |
|
45575 | 525 |
lemma bigo_setsum3: "f =o O(h) \<Longrightarrow> |
526 |
(\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o |
|
527 |
O(\<lambda>x. SUM y : A x. abs(l x y * h(k x y)))" |
|
528 |
apply (rule bigo_setsum1) |
|
529 |
apply (rule allI)+ |
|
530 |
apply (rule abs_ge_zero) |
|
531 |
apply (unfold bigo_def) |
|
532 |
apply (auto simp add: abs_mult) |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
533 |
by (metis abs_ge_zero mult.left_commute mult_left_mono) |
23449 | 534 |
|
45575 | 535 |
lemma bigo_setsum4: "f =o g +o O(h) \<Longrightarrow> |
536 |
(\<lambda>x. SUM y : A x. l x y * f(k x y)) =o |
|
537 |
(\<lambda>x. SUM y : A x. l x y * g(k x y)) +o |
|
538 |
O(\<lambda>x. SUM y : A x. abs(l x y * h(k x y)))" |
|
539 |
apply (rule set_minus_imp_plus) |
|
540 |
apply (subst fun_diff_def) |
|
541 |
apply (subst setsum_subtractf [symmetric]) |
|
542 |
apply (subst right_diff_distrib [symmetric]) |
|
543 |
apply (rule bigo_setsum3) |
|
46369 | 544 |
by (metis (lifting, no_types) fun_diff_def set_plus_imp_minus ext) |
23449 | 545 |
|
45575 | 546 |
lemma bigo_setsum5: "f =o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow> |
547 |
\<forall>x. 0 <= h x \<Longrightarrow> |
|
548 |
(\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o |
|
549 |
O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))" |
|
46369 | 550 |
apply (subgoal_tac "(\<lambda>x. SUM y : A x. (l x y) * h(k x y)) = |
45575 | 551 |
(\<lambda>x. SUM y : A x. abs((l x y) * h(k x y)))") |
46369 | 552 |
apply (erule ssubst) |
553 |
apply (erule bigo_setsum3) |
|
554 |
apply (rule ext) |
|
57418 | 555 |
apply (rule setsum.cong) |
556 |
apply (rule refl) |
|
46369 | 557 |
by (metis abs_of_nonneg zero_le_mult_iff) |
23449 | 558 |
|
45575 | 559 |
lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow> |
560 |
\<forall>x. 0 <= h x \<Longrightarrow> |
|
561 |
(\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o |
|
562 |
(\<lambda>x. SUM y : A x. (l x y) * g(k x y)) +o |
|
563 |
O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))" |
|
23449 | 564 |
apply (rule set_minus_imp_plus) |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
565 |
apply (subst fun_diff_def) |
23449 | 566 |
apply (subst setsum_subtractf [symmetric]) |
567 |
apply (subst right_diff_distrib [symmetric]) |
|
568 |
apply (rule bigo_setsum5) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
569 |
apply (subst fun_diff_def [symmetric]) |
23449 | 570 |
apply (drule set_plus_imp_minus) |
571 |
apply auto |
|
572 |
done |
|
573 |
||
574 |
subsection {* Misc useful stuff *} |
|
575 |
||
45575 | 576 |
lemma bigo_useful_intro: "A <= O(f) \<Longrightarrow> B <= O(f) \<Longrightarrow> |
47445
69e96e5500df
Set_Algebras: removed syntax \<oplus> and \<otimes>, in favour of plain + and *
krauss
parents:
47108
diff
changeset
|
577 |
A + B <= O(f)" |
23449 | 578 |
apply (subst bigo_plus_idemp [symmetric]) |
579 |
apply (rule set_plus_mono2) |
|
580 |
apply assumption+ |
|
581 |
done |
|
582 |
||
45575 | 583 |
lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)" |
23449 | 584 |
apply (subst bigo_plus_idemp [symmetric]) |
585 |
apply (rule set_plus_intro) |
|
586 |
apply assumption+ |
|
587 |
done |
|
43197 | 588 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
589 |
lemma bigo_useful_const_mult: "(c\<Colon>'a\<Colon>linordered_field) ~= 0 \<Longrightarrow> |
45575 | 590 |
(\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)" |
23449 | 591 |
apply (rule subsetD) |
45575 | 592 |
apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) <= O(h)") |
23449 | 593 |
apply assumption |
594 |
apply (rule bigo_const_mult6) |
|
45575 | 595 |
apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)") |
23449 | 596 |
apply (erule ssubst) |
597 |
apply (erule set_times_intro2) |
|
43197 | 598 |
apply (simp add: func_times) |
23449 | 599 |
done |
600 |
||
45575 | 601 |
lemma bigo_fix: "(\<lambda>x. f ((x\<Colon>nat) + 1)) =o O(\<lambda>x. h(x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow> |
23449 | 602 |
f =o O(h)" |
45575 | 603 |
apply (simp add: bigo_alt_def) |
604 |
by (metis abs_ge_zero abs_mult abs_of_pos abs_zero not0_implies_Suc) |
|
23449 | 605 |
|
43197 | 606 |
lemma bigo_fix2: |
45575 | 607 |
"(\<lambda>x. f ((x\<Colon>nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow> |
608 |
f 0 = g 0 \<Longrightarrow> f =o g +o O(h)" |
|
23449 | 609 |
apply (rule set_minus_imp_plus) |
610 |
apply (rule bigo_fix) |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
611 |
apply (subst fun_diff_def) |
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
612 |
apply (subst fun_diff_def [symmetric]) |
23449 | 613 |
apply (rule set_plus_imp_minus) |
614 |
apply simp |
|
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
615 |
apply (simp add: fun_diff_def) |
23449 | 616 |
done |
617 |
||
618 |
subsection {* Less than or equal to *} |
|
619 |
||
45575 | 620 |
definition lesso :: "('a => 'b\<Colon>linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where |
621 |
"f <o g == (\<lambda>x. max (f x - g x) 0)" |
|
23449 | 622 |
|
45575 | 623 |
lemma bigo_lesseq1: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) <= abs (f x) \<Longrightarrow> |
23449 | 624 |
g =o O(h)" |
625 |
apply (unfold bigo_def) |
|
626 |
apply clarsimp |
|
43197 | 627 |
apply (blast intro: order_trans) |
23449 | 628 |
done |
629 |
||
45575 | 630 |
lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. abs (g x) <= f x \<Longrightarrow> |
23449 | 631 |
g =o O(h)" |
632 |
apply (erule bigo_lesseq1) |
|
43197 | 633 |
apply (blast intro: abs_ge_self order_trans) |
23449 | 634 |
done |
635 |
||
45575 | 636 |
lemma bigo_lesseq3: "f =o O(h) \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= f x \<Longrightarrow> |
23449 | 637 |
g =o O(h)" |
638 |
apply (erule bigo_lesseq2) |
|
639 |
apply (rule allI) |
|
640 |
apply (subst abs_of_nonneg) |
|
641 |
apply (erule spec)+ |
|
642 |
done |
|
643 |
||
45575 | 644 |
lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow> |
645 |
\<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= abs (f x) \<Longrightarrow> |
|
23449 | 646 |
g =o O(h)" |
647 |
apply (erule bigo_lesseq1) |
|
648 |
apply (rule allI) |
|
649 |
apply (subst abs_of_nonneg) |
|
650 |
apply (erule spec)+ |
|
651 |
done |
|
652 |
||
45575 | 653 |
lemma bigo_lesso1: "\<forall>x. f x <= g x \<Longrightarrow> f <o g =o O(h)" |
36561 | 654 |
apply (unfold lesso_def) |
45575 | 655 |
apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0") |
656 |
apply (metis bigo_zero) |
|
46364 | 657 |
by (metis (lifting, no_types) func_zero le_fun_def le_iff_diff_le_0 |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54230
diff
changeset
|
658 |
max.absorb2 order_eq_iff) |
23449 | 659 |
|
45575 | 660 |
lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow> |
661 |
\<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. k x <= f x \<Longrightarrow> |
|
23449 | 662 |
k <o g =o O(h)" |
663 |
apply (unfold lesso_def) |
|
664 |
apply (rule bigo_lesseq4) |
|
665 |
apply (erule set_plus_imp_minus) |
|
666 |
apply (rule allI) |
|
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54230
diff
changeset
|
667 |
apply (rule max.cobounded2) |
23449 | 668 |
apply (rule allI) |
26814
b3e8d5ec721d
Replaced + and * on sets by \<oplus> and \<otimes>, to avoid clash with
berghofe
parents:
26645
diff
changeset
|
669 |
apply (subst fun_diff_def) |
23449 | 670 |
apply (erule thin_rl) |
45575 | 671 |
(* sledgehammer *) |
672 |
apply (case_tac "0 <= k x - g x") |
|
46644 | 673 |
apply (metis (lifting) abs_le_D1 linorder_linear min_diff_distrib_left |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54230
diff
changeset
|
674 |
min.absorb1 min.absorb2 max.absorb1) |
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54230
diff
changeset
|
675 |
by (metis abs_ge_zero le_cases max.absorb2) |
23449 | 676 |
|
45575 | 677 |
lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow> |
678 |
\<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. g x <= k x \<Longrightarrow> |
|
23449 | 679 |
f <o k =o O(h)" |
46644 | 680 |
apply (unfold lesso_def) |
681 |
apply (rule bigo_lesseq4) |
|
23449 | 682 |
apply (erule set_plus_imp_minus) |
46644 | 683 |
apply (rule allI) |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54230
diff
changeset
|
684 |
apply (rule max.cobounded2) |
46644 | 685 |
apply (rule allI) |
686 |
apply (subst fun_diff_def) |
|
687 |
apply (erule thin_rl) |
|
688 |
(* sledgehammer *) |
|
689 |
apply (case_tac "0 <= f x - k x") |
|
690 |
apply simp |
|
691 |
apply (subst abs_of_nonneg) |
|
23449 | 692 |
apply (drule_tac x = x in spec) back |
45705 | 693 |
apply (metis diff_less_0_iff_less linorder_not_le not_leE xt1(12) xt1(6)) |
45575 | 694 |
apply (metis add_minus_cancel diff_le_eq le_diff_eq uminus_add_conv_diff) |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54230
diff
changeset
|
695 |
by (metis abs_ge_zero linorder_linear max.absorb1 max.commute) |
23449 | 696 |
|
45705 | 697 |
lemma bigo_lesso4: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46644
diff
changeset
|
698 |
"f <o g =o O(k\<Colon>'a=>'b\<Colon>{linordered_field}) \<Longrightarrow> |
45705 | 699 |
g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)" |
700 |
apply (unfold lesso_def) |
|
701 |
apply (drule set_plus_imp_minus) |
|
702 |
apply (drule bigo_abs5) back |
|
703 |
apply (simp add: fun_diff_def) |
|
704 |
apply (drule bigo_useful_add, assumption) |
|
705 |
apply (erule bigo_lesseq2) back |
|
706 |
apply (rule allI) |
|
707 |
by (auto simp add: func_plus fun_diff_def algebra_simps |
|
23449 | 708 |
split: split_max abs_split) |
709 |
||
45705 | 710 |
lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x <= g x + C * abs (h x)" |
711 |
apply (simp only: lesso_def bigo_alt_def) |
|
712 |
apply clarsimp |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57418
diff
changeset
|
713 |
by (metis add.commute diff_le_eq) |
23449 | 714 |
|
715 |
end |