author | wenzelm |
Thu, 28 Jul 2005 15:19:47 +0200 | |
changeset 16932 | 0bca871f5a21 |
parent 16908 | d374530bfaaa |
child 17161 | 57c69627d71a |
permissions | -rwxr-xr-x |
16932 | 1 |
(* Title: HOL/Library/SetsAndFunctions.thy |
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ID: $Id$ |
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Author: Jeremy Avigad and Kevin Donnelly |
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*) |
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|
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header {* Operations on sets and functions *} |
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|
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theory SetsAndFunctions |
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imports Main |
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begin |
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|
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text {* |
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This library lifts operations like addition and muliplication to sets and |
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functions of appropriate types. It was designed to support asymptotic |
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calculations. See the comments at the top of BigO.thy |
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*} |
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|
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subsection {* Basic definitions *} |
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|
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instance set :: (plus)plus |
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by intro_classes |
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|
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instance fun :: (type,plus)plus |
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by intro_classes |
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|
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defs (overloaded) |
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func_plus: "f + g == (%x. f x + g x)" |
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set_plus: "A + B == {c. EX a:A. EX b:B. c = a + b}" |
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|
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instance set :: (times)times |
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by intro_classes |
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instance fun :: (type,times)times |
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by intro_classes |
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|
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defs (overloaded) |
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func_times: "f * g == (%x. f x * g x)" |
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set_times:"A * B == {c. EX a:A. EX b:B. c = a * b}" |
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|
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instance fun :: (type,minus)minus |
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by intro_classes |
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|
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defs (overloaded) |
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func_minus: "- f == (%x. - f x)" |
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func_diff: "f - g == %x. f x - g x" |
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|
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instance fun :: (type,zero)zero |
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by intro_classes |
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|
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instance set :: (zero)zero |
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by(intro_classes) |
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|
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defs (overloaded) |
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func_zero: "0::(('a::type) => ('b::zero)) == %x. 0" |
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set_zero: "0::('a::zero)set == {0}" |
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|
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instance fun :: (type,one)one |
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by intro_classes |
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|
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instance set :: (one)one |
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by intro_classes |
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|
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defs (overloaded) |
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func_one: "1::(('a::type) => ('b::one)) == %x. 1" |
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set_one: "1::('a::one)set == {1}" |
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|
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constdefs |
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68 |
elt_set_plus :: "'a::plus => 'a set => 'a set" (infixl "+o" 70) |
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"a +o B == {c. EX b:B. c = a + b}" |
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70 |
|
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71 |
elt_set_times :: "'a::times => 'a set => 'a set" (infixl "*o" 80) |
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"a *o B == {c. EX b:B. c = a * b}" |
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|
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syntax |
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"elt_set_eq" :: "'a => 'a set => bool" (infix "=o" 50) |
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|
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translations |
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"x =o A" => "x : A" |
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|
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80 |
instance fun :: (type,semigroup_add)semigroup_add |
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81 |
apply intro_classes |
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82 |
apply (auto simp add: func_plus add_assoc) |
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done |
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|
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85 |
instance fun :: (type,comm_monoid_add)comm_monoid_add |
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86 |
apply intro_classes |
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87 |
apply (auto simp add: func_zero func_plus add_ac) |
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88 |
done |
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|
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90 |
instance fun :: (type,ab_group_add)ab_group_add |
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91 |
apply intro_classes |
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92 |
apply (simp add: func_minus func_plus func_zero) |
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93 |
apply (simp add: func_minus func_plus func_diff diff_minus) |
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94 |
done |
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95 |
|
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96 |
instance fun :: (type,semigroup_mult)semigroup_mult |
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97 |
apply intro_classes |
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98 |
apply (auto simp add: func_times mult_assoc) |
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99 |
done |
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100 |
|
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101 |
instance fun :: (type,comm_monoid_mult)comm_monoid_mult |
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102 |
apply intro_classes |
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103 |
apply (auto simp add: func_one func_times mult_ac) |
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104 |
done |
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105 |
|
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106 |
instance fun :: (type,comm_ring_1)comm_ring_1 |
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107 |
apply intro_classes |
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108 |
apply (auto simp add: func_plus func_times func_minus func_diff ext |
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109 |
func_one func_zero ring_eq_simps) |
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110 |
apply (drule fun_cong) |
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111 |
apply simp |
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112 |
done |
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113 |
|
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114 |
instance set :: (semigroup_add)semigroup_add |
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115 |
apply intro_classes |
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116 |
apply (unfold set_plus) |
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117 |
apply (force simp add: add_assoc) |
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118 |
done |
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119 |
|
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120 |
instance set :: (semigroup_mult)semigroup_mult |
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121 |
apply intro_classes |
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122 |
apply (unfold set_times) |
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123 |
apply (force simp add: mult_assoc) |
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124 |
done |
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125 |
|
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126 |
instance set :: (comm_monoid_add)comm_monoid_add |
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127 |
apply intro_classes |
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128 |
apply (unfold set_plus) |
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|
129 |
apply (force simp add: add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
130 |
apply (unfold set_zero) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
131 |
apply force |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
132 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
133 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
134 |
instance set :: (comm_monoid_mult)comm_monoid_mult |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
135 |
apply intro_classes |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
136 |
apply (unfold set_times) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
137 |
apply (force simp add: mult_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
138 |
apply (unfold set_one) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
139 |
apply force |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
140 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
141 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
142 |
subsection {* Basic properties *} |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
143 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
144 |
lemma set_plus_intro [intro]: "a : C ==> b : D ==> a + b : C + D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
145 |
by (auto simp add: set_plus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
146 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
147 |
lemma set_plus_intro2 [intro]: "b : C ==> a + b : a +o C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
148 |
by (auto simp add: elt_set_plus_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
149 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
150 |
lemma set_plus_rearrange: "((a::'a::comm_monoid_add) +o C) + |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
151 |
(b +o D) = (a + b) +o (C + D)" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
152 |
apply (auto simp add: elt_set_plus_def set_plus add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
153 |
apply (rule_tac x = "ba + bb" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
154 |
apply (auto simp add: add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
155 |
apply (rule_tac x = "aa + a" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
156 |
apply (auto simp add: add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
157 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
158 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
159 |
lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) = |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
160 |
(a + b) +o C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
161 |
by (auto simp add: elt_set_plus_def add_assoc) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
162 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
163 |
lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C = |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
164 |
a +o (B + C)" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
165 |
apply (auto simp add: elt_set_plus_def set_plus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
166 |
apply (blast intro: add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
167 |
apply (rule_tac x = "a + aa" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
168 |
apply (rule conjI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
169 |
apply (rule_tac x = "aa" in bexI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
170 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
171 |
apply (rule_tac x = "ba" in bexI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
172 |
apply (auto simp add: add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
173 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
174 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
175 |
theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) = |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
176 |
a +o (C + D)" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
177 |
apply (auto intro!: subsetI simp add: elt_set_plus_def set_plus add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
178 |
apply (rule_tac x = "aa + ba" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
179 |
apply (auto simp add: add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
180 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
181 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
182 |
theorems set_plus_rearranges = set_plus_rearrange set_plus_rearrange2 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
183 |
set_plus_rearrange3 set_plus_rearrange4 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
184 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
185 |
lemma set_plus_mono [intro!]: "C <= D ==> a +o C <= a +o D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
186 |
by (auto simp add: elt_set_plus_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
187 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
188 |
lemma set_plus_mono2 [intro]: "(C::('a::plus) set) <= D ==> E <= F ==> |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
189 |
C + E <= D + F" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
190 |
by (auto simp add: set_plus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
191 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
192 |
lemma set_plus_mono3 [intro]: "a : C ==> a +o D <= C + D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
193 |
by (auto simp add: elt_set_plus_def set_plus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
194 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
195 |
lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) : C ==> |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
196 |
a +o D <= D + C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
197 |
by (auto simp add: elt_set_plus_def set_plus add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
198 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
199 |
lemma set_plus_mono5: "a:C ==> B <= D ==> a +o B <= C + D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
200 |
apply (subgoal_tac "a +o B <= a +o D") |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
201 |
apply (erule order_trans) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
202 |
apply (erule set_plus_mono3) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
203 |
apply (erule set_plus_mono) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
204 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
205 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
206 |
lemma set_plus_mono_b: "C <= D ==> x : a +o C |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
207 |
==> x : a +o D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
208 |
apply (frule set_plus_mono) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
209 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
210 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
211 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
212 |
lemma set_plus_mono2_b: "C <= D ==> E <= F ==> x : C + E ==> |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
213 |
x : D + F" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
214 |
apply (frule set_plus_mono2) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
215 |
prefer 2 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
216 |
apply force |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
217 |
apply assumption |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
218 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
219 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
220 |
lemma set_plus_mono3_b: "a : C ==> x : a +o D ==> x : C + D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
221 |
apply (frule set_plus_mono3) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
222 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
223 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
224 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
225 |
lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C ==> |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
226 |
x : a +o D ==> x : D + C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
227 |
apply (frule set_plus_mono4) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
228 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
229 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
230 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
231 |
lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
232 |
by (auto simp add: elt_set_plus_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
233 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
234 |
lemma set_zero_plus2: "(0::'a::comm_monoid_add) : A ==> B <= A + B" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
235 |
apply (auto intro!: subsetI simp add: set_plus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
236 |
apply (rule_tac x = 0 in bexI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
237 |
apply (rule_tac x = x in bexI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
238 |
apply (auto simp add: add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
239 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
240 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
241 |
lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C ==> (a - b) : C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
242 |
by (auto simp add: elt_set_plus_def add_ac diff_minus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
243 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
244 |
lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C ==> a : b +o C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
245 |
apply (auto simp add: elt_set_plus_def add_ac diff_minus) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
246 |
apply (subgoal_tac "a = (a + - b) + b") |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
247 |
apply (rule bexI, assumption, assumption) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
248 |
apply (auto simp add: add_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
249 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
250 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
251 |
lemma set_minus_plus: "((a::'a::ab_group_add) - b : C) = (a : b +o C)" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
252 |
by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus, |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
253 |
assumption) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
254 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
255 |
lemma set_times_intro [intro]: "a : C ==> b : D ==> a * b : C * D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
256 |
by (auto simp add: set_times) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
257 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
258 |
lemma set_times_intro2 [intro!]: "b : C ==> a * b : a *o C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
259 |
by (auto simp add: elt_set_times_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
260 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
261 |
lemma set_times_rearrange: "((a::'a::comm_monoid_mult) *o C) * |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
262 |
(b *o D) = (a * b) *o (C * D)" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
263 |
apply (auto simp add: elt_set_times_def set_times) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
264 |
apply (rule_tac x = "ba * bb" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
265 |
apply (auto simp add: mult_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
266 |
apply (rule_tac x = "aa * a" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
267 |
apply (auto simp add: mult_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
268 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
269 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
270 |
lemma set_times_rearrange2: "(a::'a::semigroup_mult) *o (b *o C) = |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
271 |
(a * b) *o C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
272 |
by (auto simp add: elt_set_times_def mult_assoc) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
273 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
274 |
lemma set_times_rearrange3: "((a::'a::semigroup_mult) *o B) * C = |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
275 |
a *o (B * C)" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
276 |
apply (auto simp add: elt_set_times_def set_times) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
277 |
apply (blast intro: mult_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
278 |
apply (rule_tac x = "a * aa" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
279 |
apply (rule conjI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
280 |
apply (rule_tac x = "aa" in bexI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
281 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
282 |
apply (rule_tac x = "ba" in bexI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
283 |
apply (auto simp add: mult_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
284 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
285 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
286 |
theorem set_times_rearrange4: "C * ((a::'a::comm_monoid_mult) *o D) = |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
287 |
a *o (C * D)" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
288 |
apply (auto intro!: subsetI simp add: elt_set_times_def set_times |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
289 |
mult_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
290 |
apply (rule_tac x = "aa * ba" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
291 |
apply (auto simp add: mult_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
292 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
293 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
294 |
theorems set_times_rearranges = set_times_rearrange set_times_rearrange2 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
295 |
set_times_rearrange3 set_times_rearrange4 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
296 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
297 |
lemma set_times_mono [intro]: "C <= D ==> a *o C <= a *o D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
298 |
by (auto simp add: elt_set_times_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
299 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
300 |
lemma set_times_mono2 [intro]: "(C::('a::times) set) <= D ==> E <= F ==> |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
301 |
C * E <= D * F" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
302 |
by (auto simp add: set_times) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
303 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
304 |
lemma set_times_mono3 [intro]: "a : C ==> a *o D <= C * D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
305 |
by (auto simp add: elt_set_times_def set_times) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
306 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
307 |
lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C ==> |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
308 |
a *o D <= D * C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
309 |
by (auto simp add: elt_set_times_def set_times mult_ac) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
310 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
311 |
lemma set_times_mono5: "a:C ==> B <= D ==> a *o B <= C * D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
312 |
apply (subgoal_tac "a *o B <= a *o D") |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
313 |
apply (erule order_trans) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
314 |
apply (erule set_times_mono3) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
315 |
apply (erule set_times_mono) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
316 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
317 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
318 |
lemma set_times_mono_b: "C <= D ==> x : a *o C |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
319 |
==> x : a *o D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
320 |
apply (frule set_times_mono) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
321 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
322 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
323 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
324 |
lemma set_times_mono2_b: "C <= D ==> E <= F ==> x : C * E ==> |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
325 |
x : D * F" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
326 |
apply (frule set_times_mono2) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
327 |
prefer 2 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
328 |
apply force |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
329 |
apply assumption |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
330 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
331 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
332 |
lemma set_times_mono3_b: "a : C ==> x : a *o D ==> x : C * D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
333 |
apply (frule set_times_mono3) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
334 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
335 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
336 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
337 |
lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) : C ==> |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
338 |
x : a *o D ==> x : D * C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
339 |
apply (frule set_times_mono4) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
340 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
341 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
342 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
343 |
lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
344 |
by (auto simp add: elt_set_times_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
345 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
346 |
lemma set_times_plus_distrib: "(a::'a::semiring) *o (b +o C)= |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
347 |
(a * b) +o (a *o C)" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
348 |
by (auto simp add: elt_set_plus_def elt_set_times_def ring_distrib) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
349 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
350 |
lemma set_times_plus_distrib2: "(a::'a::semiring) *o (B + C) = |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
351 |
(a *o B) + (a *o C)" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
352 |
apply (auto simp add: set_plus elt_set_times_def ring_distrib) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
353 |
apply blast |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
354 |
apply (rule_tac x = "b + bb" in exI) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
355 |
apply (auto simp add: ring_distrib) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
356 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
357 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
358 |
lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D <= |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
359 |
a *o D + C * D" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
360 |
apply (auto intro!: subsetI simp add: |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
361 |
elt_set_plus_def elt_set_times_def set_times |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
362 |
set_plus ring_distrib) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
363 |
apply auto |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
364 |
done |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
365 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
366 |
theorems set_times_plus_distribs = set_times_plus_distrib |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
367 |
set_times_plus_distrib2 |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
368 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
369 |
lemma set_neg_intro: "(a::'a::ring_1) : (- 1) *o C ==> |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
370 |
- a : C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
371 |
by (auto simp add: elt_set_times_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
372 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
373 |
lemma set_neg_intro2: "(a::'a::ring_1) : C ==> |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
374 |
- a : (- 1) *o C" |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
375 |
by (auto simp add: elt_set_times_def) |
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
376 |
|
d374530bfaaa
Added two new theories to HOL/Library: SetsAndFunctions.thy and BigO.thy
avigad
parents:
diff
changeset
|
377 |
end |