author | haftmann |
Sat, 14 Mar 2009 12:50:29 +0100 | |
changeset 30531 | ab3d61baf66a |
parent 30352 | 047f183c43b0 |
child 30596 | 140b22f22071 |
permissions | -rw-r--r-- |
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(* Title: HOL/Set.thy |
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Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel |
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*) |
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header {* Set theory for higher-order logic *} |
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theory Set |
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imports Lattices |
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begin |
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text {* A set in HOL is simply a predicate. *} |
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subsection {* Basic syntax *} |
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global |
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types 'a set = "'a => bool" |
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consts |
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Collect :: "('a => bool) => 'a set" -- "comprehension" |
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"op :" :: "'a => 'a set => bool" -- "membership" |
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insert :: "'a => 'a set => 'a set" |
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Ball :: "'a set => ('a => bool) => bool" -- "bounded universal quantifiers" |
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Bex :: "'a set => ('a => bool) => bool" -- "bounded existential quantifiers" |
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Bex1 :: "'a set => ('a => bool) => bool" -- "bounded unique existential quantifiers" |
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Pow :: "'a set => 'a set set" -- "powerset" |
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image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90) |
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|
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local |
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notation |
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"op :" ("op :") and |
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"op :" ("(_/ : _)" [50, 51] 50) |
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abbreviation |
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"not_mem x A == ~ (x : A)" -- "non-membership" |
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notation |
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not_mem ("op ~:") and |
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not_mem ("(_/ ~: _)" [50, 51] 50) |
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notation (xsymbols) |
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"op :" ("op \<in>") and |
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"op :" ("(_/ \<in> _)" [50, 51] 50) and |
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not_mem ("op \<notin>") and |
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not_mem ("(_/ \<notin> _)" [50, 51] 50) |
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notation (HTML output) |
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"op :" ("op \<in>") and |
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"op :" ("(_/ \<in> _)" [50, 51] 50) and |
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not_mem ("op \<notin>") and |
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not_mem ("(_/ \<notin> _)" [50, 51] 50) |
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54 |
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syntax |
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"@Coll" :: "pttrn => bool => 'a set" ("(1{_./ _})") |
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57 |
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translations |
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"{x. P}" == "Collect (%x. P)" |
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60 |
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definition empty :: "'a set" ("{}") where |
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"empty \<equiv> {x. False}" |
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63 |
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definition UNIV :: "'a set" where |
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"UNIV \<equiv> {x. True}" |
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66 |
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syntax |
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"@Finset" :: "args => 'a set" ("{(_)}") |
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69 |
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70 |
translations |
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71 |
"{x, xs}" == "insert x {xs}" |
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"{x}" == "insert x {}" |
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73 |
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definition Int :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Int" 70) where |
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"A Int B \<equiv> {x. x \<in> A \<and> x \<in> B}" |
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76 |
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definition Un :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "Un" 65) where |
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"A Un B \<equiv> {x. x \<in> A \<or> x \<in> B}" |
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79 |
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80 |
notation (xsymbols) |
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81 |
"Int" (infixl "\<inter>" 70) and |
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82 |
"Un" (infixl "\<union>" 65) |
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83 |
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notation (HTML output) |
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85 |
"Int" (infixl "\<inter>" 70) and |
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86 |
"Un" (infixl "\<union>" 65) |
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87 |
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88 |
syntax |
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89 |
"_Ball" :: "pttrn => 'a set => bool => bool" ("(3ALL _:_./ _)" [0, 0, 10] 10) |
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90 |
"_Bex" :: "pttrn => 'a set => bool => bool" ("(3EX _:_./ _)" [0, 0, 10] 10) |
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91 |
"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3EX! _:_./ _)" [0, 0, 10] 10) |
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"_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST _:_./ _)" [0, 0, 10] 10) |
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93 |
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94 |
syntax (HOL) |
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95 |
"_Ball" :: "pttrn => 'a set => bool => bool" ("(3! _:_./ _)" [0, 0, 10] 10) |
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96 |
"_Bex" :: "pttrn => 'a set => bool => bool" ("(3? _:_./ _)" [0, 0, 10] 10) |
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97 |
"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3?! _:_./ _)" [0, 0, 10] 10) |
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98 |
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99 |
syntax (xsymbols) |
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100 |
"_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10) |
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101 |
"_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10) |
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102 |
"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10) |
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103 |
"_Bleast" :: "id => 'a set => bool => 'a" ("(3LEAST_\<in>_./ _)" [0, 0, 10] 10) |
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104 |
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105 |
syntax (HTML output) |
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106 |
"_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10) |
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107 |
"_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10) |
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108 |
"_Bex1" :: "pttrn => 'a set => bool => bool" ("(3\<exists>!_\<in>_./ _)" [0, 0, 10] 10) |
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109 |
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translations |
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111 |
"ALL x:A. P" == "Ball A (%x. P)" |
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"EX x:A. P" == "Bex A (%x. P)" |
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113 |
"EX! x:A. P" == "Bex1 A (%x. P)" |
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"LEAST x:A. P" => "LEAST x. x:A & P" |
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115 |
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116 |
definition INTER :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where |
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117 |
"INTER A B \<equiv> {y. \<forall>x\<in>A. y \<in> B x}" |
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118 |
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119 |
definition UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where |
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120 |
"UNION A B \<equiv> {y. \<exists>x\<in>A. y \<in> B x}" |
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121 |
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122 |
definition Inter :: "'a set set \<Rightarrow> 'a set" where |
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123 |
"Inter S \<equiv> INTER S (\<lambda>x. x)" |
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124 |
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125 |
definition Union :: "'a set set \<Rightarrow> 'a set" where |
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126 |
"Union S \<equiv> UNION S (\<lambda>x. x)" |
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127 |
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128 |
notation (xsymbols) |
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129 |
Inter ("\<Inter>_" [90] 90) and |
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130 |
Union ("\<Union>_" [90] 90) |
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131 |
|
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132 |
|
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133 |
subsection {* Additional concrete syntax *} |
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134 |
|
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135 |
syntax |
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136 |
"@SetCompr" :: "'a => idts => bool => 'a set" ("(1{_ |/_./ _})") |
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137 |
"@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ :/ _./ _})") |
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138 |
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" [0, 10] 10) |
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139 |
"@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" [0, 10] 10) |
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140 |
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" [0, 10] 10) |
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141 |
"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" [0, 10] 10) |
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142 |
|
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143 |
syntax (xsymbols) |
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144 |
"@Collect" :: "idt => 'a set => bool => 'a set" ("(1{_ \<in>/ _./ _})") |
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145 |
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" [0, 10] 10) |
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146 |
"@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" [0, 10] 10) |
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147 |
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" [0, 10] 10) |
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148 |
"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" [0, 10] 10) |
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149 |
|
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150 |
syntax (latex output) |
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151 |
"@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
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152 |
"@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<^esub>)/ _)" [0, 10] 10) |
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153 |
"@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) |
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154 |
"@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>(00\<^bsub>_\<in>_\<^esub>)/ _)" [0, 10] 10) |
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155 |
|
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156 |
translations |
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157 |
"{x:A. P}" => "{x. x:A & P}" |
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158 |
"INT x y. B" == "INT x. INT y. B" |
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159 |
"INT x. B" == "CONST INTER CONST UNIV (%x. B)" |
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160 |
"INT x. B" == "INT x:CONST UNIV. B" |
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161 |
"INT x:A. B" == "CONST INTER A (%x. B)" |
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162 |
"UN x y. B" == "UN x. UN y. B" |
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163 |
"UN x. B" == "CONST UNION CONST UNIV (%x. B)" |
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164 |
"UN x. B" == "UN x:CONST UNIV. B" |
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165 |
"UN x:A. B" == "CONST UNION A (%x. B)" |
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166 |
|
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167 |
text {* |
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168 |
Note the difference between ordinary xsymbol syntax of indexed |
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169 |
unions and intersections (e.g.\ @{text"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}) |
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170 |
and their \LaTeX\ rendition: @{term"\<Union>a\<^isub>1\<in>A\<^isub>1. B"}. The |
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171 |
former does not make the index expression a subscript of the |
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172 |
union/intersection symbol because this leads to problems with nested |
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173 |
subscripts in Proof General. |
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174 |
*} |
2261 | 175 |
|
21333 | 176 |
abbreviation |
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177 |
subset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where |
21819 | 178 |
"subset \<equiv> less" |
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179 |
|
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180 |
abbreviation |
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181 |
subset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where |
21819 | 182 |
"subset_eq \<equiv> less_eq" |
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|
184 |
notation (output) |
|
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185 |
subset ("op <") and |
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186 |
subset ("(_/ < _)" [50, 51] 50) and |
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187 |
subset_eq ("op <=") and |
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subset_eq ("(_/ <= _)" [50, 51] 50) |
189 |
||
190 |
notation (xsymbols) |
|
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191 |
subset ("op \<subset>") and |
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192 |
subset ("(_/ \<subset> _)" [50, 51] 50) and |
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193 |
subset_eq ("op \<subseteq>") and |
21333 | 194 |
subset_eq ("(_/ \<subseteq> _)" [50, 51] 50) |
195 |
||
196 |
notation (HTML output) |
|
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197 |
subset ("op \<subset>") and |
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198 |
subset ("(_/ \<subset> _)" [50, 51] 50) and |
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199 |
subset_eq ("op \<subseteq>") and |
21333 | 200 |
subset_eq ("(_/ \<subseteq> _)" [50, 51] 50) |
201 |
||
202 |
abbreviation (input) |
|
21819 | 203 |
supset :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where |
204 |
"supset \<equiv> greater" |
|
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205 |
|
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206 |
abbreviation (input) |
21819 | 207 |
supset_eq :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool" where |
208 |
"supset_eq \<equiv> greater_eq" |
|
209 |
||
210 |
notation (xsymbols) |
|
211 |
supset ("op \<supset>") and |
|
212 |
supset ("(_/ \<supset> _)" [50, 51] 50) and |
|
213 |
supset_eq ("op \<supseteq>") and |
|
214 |
supset_eq ("(_/ \<supseteq> _)" [50, 51] 50) |
|
21333 | 215 |
|
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216 |
abbreviation |
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217 |
range :: "('a => 'b) => 'b set" where -- "of function" |
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218 |
"range f == f ` UNIV" |
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219 |
|
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220 |
|
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221 |
subsubsection "Bounded quantifiers" |
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222 |
|
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223 |
syntax (output) |
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224 |
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10) |
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225 |
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10) |
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226 |
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10) |
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227 |
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10) |
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228 |
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3EX! _<=_./ _)" [0, 0, 10] 10) |
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229 |
|
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230 |
syntax (xsymbols) |
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231 |
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) |
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232 |
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) |
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|
233 |
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) |
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234 |
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) |
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235 |
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) |
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236 |
|
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237 |
syntax (HOL output) |
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238 |
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10) |
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|
239 |
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10) |
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|
240 |
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10) |
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|
241 |
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10) |
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242 |
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3?! _<=_./ _)" [0, 0, 10] 10) |
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|
243 |
|
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244 |
syntax (HTML output) |
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245 |
"_setlessAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subset>_./ _)" [0, 0, 10] 10) |
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|
246 |
"_setlessEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subset>_./ _)" [0, 0, 10] 10) |
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247 |
"_setleAll" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<subseteq>_./ _)" [0, 0, 10] 10) |
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248 |
"_setleEx" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<subseteq>_./ _)" [0, 0, 10] 10) |
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249 |
"_setleEx1" :: "[idt, 'a, bool] => bool" ("(3\<exists>!_\<subseteq>_./ _)" [0, 0, 10] 10) |
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250 |
|
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251 |
translations |
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252 |
"\<forall>A\<subset>B. P" => "ALL A. A \<subset> B --> P" |
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253 |
"\<exists>A\<subset>B. P" => "EX A. A \<subset> B & P" |
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changeset
|
254 |
"\<forall>A\<subseteq>B. P" => "ALL A. A \<subseteq> B --> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
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parents:
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|
255 |
"\<exists>A\<subseteq>B. P" => "EX A. A \<subseteq> B & P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
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parents:
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changeset
|
256 |
"\<exists>!A\<subseteq>B. P" => "EX! A. A \<subseteq> B & P" |
14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset
|
257 |
|
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
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diff
changeset
|
258 |
print_translation {* |
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
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parents:
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diff
changeset
|
259 |
let |
22377 | 260 |
val Type (set_type, _) = @{typ "'a set"}; |
261 |
val All_binder = Syntax.binder_name @{const_syntax "All"}; |
|
262 |
val Ex_binder = Syntax.binder_name @{const_syntax "Ex"}; |
|
263 |
val impl = @{const_syntax "op -->"}; |
|
264 |
val conj = @{const_syntax "op &"}; |
|
265 |
val sbset = @{const_syntax "subset"}; |
|
266 |
val sbset_eq = @{const_syntax "subset_eq"}; |
|
21819 | 267 |
|
268 |
val trans = |
|
269 |
[((All_binder, impl, sbset), "_setlessAll"), |
|
270 |
((All_binder, impl, sbset_eq), "_setleAll"), |
|
271 |
((Ex_binder, conj, sbset), "_setlessEx"), |
|
272 |
((Ex_binder, conj, sbset_eq), "_setleEx")]; |
|
273 |
||
274 |
fun mk v v' c n P = |
|
275 |
if v = v' andalso not (Term.exists_subterm (fn Free (x, _) => x = v | _ => false) n) |
|
276 |
then Syntax.const c $ Syntax.mark_bound v' $ n $ P else raise Match; |
|
277 |
||
278 |
fun tr' q = (q, |
|
279 |
fn [Const ("_bound", _) $ Free (v, Type (T, _)), Const (c, _) $ (Const (d, _) $ (Const ("_bound", _) $ Free (v', _)) $ n) $ P] => |
|
280 |
if T = (set_type) then case AList.lookup (op =) trans (q, c, d) |
|
281 |
of NONE => raise Match |
|
282 |
| SOME l => mk v v' l n P |
|
283 |
else raise Match |
|
284 |
| _ => raise Match); |
|
14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset
|
285 |
in |
21819 | 286 |
[tr' All_binder, tr' Ex_binder] |
14804
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset
|
287 |
end |
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset
|
288 |
*} |
8de39d3e8eb6
Corrected printer bug for bounded quantifiers Q x<=y. P
nipkow
parents:
14752
diff
changeset
|
289 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
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diff
changeset
|
290 |
|
11979 | 291 |
text {* |
292 |
\medskip Translate between @{text "{e | x1...xn. P}"} and @{text |
|
293 |
"{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is |
|
294 |
only translated if @{text "[0..n] subset bvs(e)"}. |
|
295 |
*} |
|
296 |
||
297 |
parse_translation {* |
|
298 |
let |
|
299 |
val ex_tr = snd (mk_binder_tr ("EX ", "Ex")); |
|
3947 | 300 |
|
11979 | 301 |
fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1 |
302 |
| nvars _ = 1; |
|
303 |
||
304 |
fun setcompr_tr [e, idts, b] = |
|
305 |
let |
|
306 |
val eq = Syntax.const "op =" $ Bound (nvars idts) $ e; |
|
307 |
val P = Syntax.const "op &" $ eq $ b; |
|
308 |
val exP = ex_tr [idts, P]; |
|
17784 | 309 |
in Syntax.const "Collect" $ Term.absdummy (dummyT, exP) end; |
11979 | 310 |
|
311 |
in [("@SetCompr", setcompr_tr)] end; |
|
312 |
*} |
|
923 | 313 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
314 |
(* To avoid eta-contraction of body: *) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
315 |
print_translation {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
316 |
let |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
317 |
fun btr' syn [A, Abs abs] = |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
318 |
let val (x, t) = atomic_abs_tr' abs |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
319 |
in Syntax.const syn $ x $ A $ t end |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
320 |
in |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
321 |
[(@{const_syntax Ball}, btr' "_Ball"), (@{const_syntax Bex}, btr' "_Bex"), |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
322 |
(@{const_syntax UNION}, btr' "@UNION"),(@{const_syntax INTER}, btr' "@INTER")] |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
323 |
end |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
324 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
325 |
|
13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
326 |
print_translation {* |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
327 |
let |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
328 |
val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY")); |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
329 |
|
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
330 |
fun setcompr_tr' [Abs (abs as (_, _, P))] = |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
331 |
let |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
332 |
fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1) |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
333 |
| check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) = |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
334 |
n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso |
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
335 |
((0 upto (n - 1)) subset add_loose_bnos (e, 0, [])) |
13764 | 336 |
| check _ = false |
923 | 337 |
|
11979 | 338 |
fun tr' (_ $ abs) = |
339 |
let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs] |
|
340 |
in Syntax.const "@SetCompr" $ e $ idts $ Q end; |
|
13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
341 |
in if check (P, 0) then tr' P |
15535 | 342 |
else let val (x as _ $ Free(xN,_), t) = atomic_abs_tr' abs |
343 |
val M = Syntax.const "@Coll" $ x $ t |
|
344 |
in case t of |
|
345 |
Const("op &",_) |
|
346 |
$ (Const("op :",_) $ (Const("_bound",_) $ Free(yN,_)) $ A) |
|
347 |
$ P => |
|
348 |
if xN=yN then Syntax.const "@Collect" $ x $ A $ P else M |
|
349 |
| _ => M |
|
350 |
end |
|
13763
f94b569cd610
added print translations tha avoid eta contraction for important binders.
nipkow
parents:
13653
diff
changeset
|
351 |
end; |
11979 | 352 |
in [("Collect", setcompr_tr')] end; |
353 |
*} |
|
354 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
355 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
356 |
subsection {* Rules and definitions *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
357 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
358 |
text {* Isomorphisms between predicates and sets. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
359 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
360 |
defs |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
361 |
mem_def [code]: "x : S == S x" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
362 |
Collect_def [code]: "Collect P == P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
363 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
364 |
defs |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
365 |
Ball_def: "Ball A P == ALL x. x:A --> P(x)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
366 |
Bex_def: "Bex A P == EX x. x:A & P(x)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
367 |
Bex1_def: "Bex1 A P == EX! x. x:A & P(x)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
368 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
369 |
instantiation "fun" :: (type, minus) minus |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
370 |
begin |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
371 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
372 |
definition |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
373 |
fun_diff_def: "A - B = (%x. A x - B x)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
374 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
375 |
instance .. |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
376 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
377 |
end |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
378 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
379 |
instantiation bool :: minus |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
380 |
begin |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
381 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
382 |
definition |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
383 |
bool_diff_def: "A - B = (A & ~ B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
384 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
385 |
instance .. |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
386 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
387 |
end |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
388 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
389 |
instantiation "fun" :: (type, uminus) uminus |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
390 |
begin |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
391 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
392 |
definition |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
393 |
fun_Compl_def: "- A = (%x. - A x)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
394 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
395 |
instance .. |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
396 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
397 |
end |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
398 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
399 |
instantiation bool :: uminus |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
400 |
begin |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
401 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
402 |
definition |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
403 |
bool_Compl_def: "- A = (~ A)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
404 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
405 |
instance .. |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
406 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
407 |
end |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
408 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
409 |
defs |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
410 |
Pow_def: "Pow A == {B. B <= A}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
411 |
insert_def: "insert a B == {x. x=a} Un B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
412 |
image_def: "f`A == {y. EX x:A. y = f(x)}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
413 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
414 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
415 |
subsection {* Lemmas and proof tool setup *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
416 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
417 |
subsubsection {* Relating predicates and sets *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
418 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
419 |
lemma mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
420 |
by (simp add: Collect_def mem_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
421 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
422 |
lemma Collect_mem_eq [simp]: "{x. x:A} = A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
423 |
by (simp add: Collect_def mem_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
424 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
425 |
lemma CollectI: "P(a) ==> a : {x. P(x)}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
426 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
427 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
428 |
lemma CollectD: "a : {x. P(x)} ==> P(a)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
429 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
430 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
431 |
lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
432 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
433 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
434 |
lemmas CollectE = CollectD [elim_format] |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
435 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
436 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
437 |
subsubsection {* Bounded quantifiers *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
438 |
|
11979 | 439 |
lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x" |
440 |
by (simp add: Ball_def) |
|
441 |
||
442 |
lemmas strip = impI allI ballI |
|
443 |
||
444 |
lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x" |
|
445 |
by (simp add: Ball_def) |
|
446 |
||
447 |
lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q" |
|
448 |
by (unfold Ball_def) blast |
|
22139 | 449 |
|
450 |
ML {* bind_thm ("rev_ballE", permute_prems 1 1 @{thm ballE}) *} |
|
11979 | 451 |
|
452 |
text {* |
|
453 |
\medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and |
|
454 |
@{prop "a:A"}; creates assumption @{prop "P a"}. |
|
455 |
*} |
|
456 |
||
457 |
ML {* |
|
22139 | 458 |
fun ball_tac i = etac @{thm ballE} i THEN contr_tac (i + 1) |
11979 | 459 |
*} |
460 |
||
461 |
text {* |
|
462 |
Gives better instantiation for bound: |
|
463 |
*} |
|
464 |
||
26339 | 465 |
declaration {* fn _ => |
466 |
Classical.map_cs (fn cs => cs addbefore ("bspec", datac @{thm bspec} 1)) |
|
11979 | 467 |
*} |
468 |
||
469 |
lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x" |
|
470 |
-- {* Normally the best argument order: @{prop "P x"} constrains the |
|
471 |
choice of @{prop "x:A"}. *} |
|
472 |
by (unfold Bex_def) blast |
|
473 |
||
13113 | 474 |
lemma rev_bexI [intro?]: "x:A ==> P x ==> EX x:A. P x" |
11979 | 475 |
-- {* The best argument order when there is only one @{prop "x:A"}. *} |
476 |
by (unfold Bex_def) blast |
|
477 |
||
478 |
lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x" |
|
479 |
by (unfold Bex_def) blast |
|
480 |
||
481 |
lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q" |
|
482 |
by (unfold Bex_def) blast |
|
483 |
||
484 |
lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)" |
|
485 |
-- {* Trival rewrite rule. *} |
|
486 |
by (simp add: Ball_def) |
|
487 |
||
488 |
lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)" |
|
489 |
-- {* Dual form for existentials. *} |
|
490 |
by (simp add: Bex_def) |
|
491 |
||
492 |
lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)" |
|
493 |
by blast |
|
494 |
||
495 |
lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)" |
|
496 |
by blast |
|
497 |
||
498 |
lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)" |
|
499 |
by blast |
|
500 |
||
501 |
lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)" |
|
502 |
by blast |
|
503 |
||
504 |
lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)" |
|
505 |
by blast |
|
506 |
||
507 |
lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)" |
|
508 |
by blast |
|
509 |
||
26480 | 510 |
ML {* |
13462 | 511 |
local |
22139 | 512 |
val unfold_bex_tac = unfold_tac @{thms "Bex_def"}; |
18328 | 513 |
fun prove_bex_tac ss = unfold_bex_tac ss THEN Quantifier1.prove_one_point_ex_tac; |
11979 | 514 |
val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac; |
515 |
||
22139 | 516 |
val unfold_ball_tac = unfold_tac @{thms "Ball_def"}; |
18328 | 517 |
fun prove_ball_tac ss = unfold_ball_tac ss THEN Quantifier1.prove_one_point_all_tac; |
11979 | 518 |
val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac; |
519 |
in |
|
18328 | 520 |
val defBEX_regroup = Simplifier.simproc (the_context ()) |
13462 | 521 |
"defined BEX" ["EX x:A. P x & Q x"] rearrange_bex; |
18328 | 522 |
val defBALL_regroup = Simplifier.simproc (the_context ()) |
13462 | 523 |
"defined BALL" ["ALL x:A. P x --> Q x"] rearrange_ball; |
11979 | 524 |
end; |
13462 | 525 |
|
526 |
Addsimprocs [defBALL_regroup, defBEX_regroup]; |
|
11979 | 527 |
*} |
528 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
529 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
530 |
subsubsection {* Congruence rules *} |
11979 | 531 |
|
16636
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
532 |
lemma ball_cong: |
11979 | 533 |
"A = B ==> (!!x. x:B ==> P x = Q x) ==> |
534 |
(ALL x:A. P x) = (ALL x:B. Q x)" |
|
535 |
by (simp add: Ball_def) |
|
536 |
||
16636
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
537 |
lemma strong_ball_cong [cong]: |
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
538 |
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> |
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
539 |
(ALL x:A. P x) = (ALL x:B. Q x)" |
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
540 |
by (simp add: simp_implies_def Ball_def) |
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
541 |
|
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
542 |
lemma bex_cong: |
11979 | 543 |
"A = B ==> (!!x. x:B ==> P x = Q x) ==> |
544 |
(EX x:A. P x) = (EX x:B. Q x)" |
|
545 |
by (simp add: Bex_def cong: conj_cong) |
|
1273 | 546 |
|
16636
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
547 |
lemma strong_bex_cong [cong]: |
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
548 |
"A = B ==> (!!x. x:B =simp=> P x = Q x) ==> |
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
549 |
(EX x:A. P x) = (EX x:B. Q x)" |
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
550 |
by (simp add: simp_implies_def Bex_def cong: conj_cong) |
1ed737a98198
Added strong_ball_cong and strong_bex_cong (these are now the standard
berghofe
parents:
15950
diff
changeset
|
551 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
552 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
553 |
subsubsection {* Subsets *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
554 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
555 |
lemma subsetI [atp,intro!]: "(!!x. x:A ==> x:B) ==> A \<subseteq> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
556 |
by (auto simp add: mem_def intro: predicate1I) |
30352 | 557 |
|
11979 | 558 |
text {* |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
559 |
\medskip Map the type @{text "'a set => anything"} to just @{typ |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
560 |
'a}; for overloading constants whose first argument has type @{typ |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
561 |
"'a set"}. |
11979 | 562 |
*} |
563 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
564 |
lemma subsetD [elim]: "A \<subseteq> B ==> c \<in> A ==> c \<in> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
565 |
-- {* Rule in Modus Ponens style. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
566 |
by (unfold mem_def) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
567 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
568 |
declare subsetD [intro?] -- FIXME |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
569 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
570 |
lemma rev_subsetD: "c \<in> A ==> A \<subseteq> B ==> c \<in> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
571 |
-- {* The same, with reversed premises for use with @{text erule} -- |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
572 |
cf @{text rev_mp}. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
573 |
by (rule subsetD) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
574 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
575 |
declare rev_subsetD [intro?] -- FIXME |
11979 | 576 |
|
577 |
text {* |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
578 |
\medskip Converts @{prop "A \<subseteq> B"} to @{prop "x \<in> A ==> x \<in> B"}. |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
579 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
580 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
581 |
ML {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
582 |
fun impOfSubs th = th RSN (2, @{thm rev_subsetD}) |
11979 | 583 |
*} |
584 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
585 |
lemma subsetCE [elim]: "A \<subseteq> B ==> (c \<notin> A ==> P) ==> (c \<in> B ==> P) ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
586 |
-- {* Classical elimination rule. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
587 |
by (unfold mem_def) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
588 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
589 |
lemma subset_eq: "A \<le> B = (\<forall>x\<in>A. x \<in> B)" by blast |
2388 | 590 |
|
11979 | 591 |
text {* |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
592 |
\medskip Takes assumptions @{prop "A \<subseteq> B"}; @{prop "c \<in> A"} and |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
593 |
creates the assumption @{prop "c \<in> B"}. |
30352 | 594 |
*} |
595 |
||
596 |
ML {* |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
597 |
fun set_mp_tac i = etac @{thm subsetCE} i THEN mp_tac i |
11979 | 598 |
*} |
599 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
600 |
lemma contra_subsetD: "A \<subseteq> B ==> c \<notin> B ==> c \<notin> A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
601 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
602 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
603 |
lemma subset_refl [simp,atp]: "A \<subseteq> A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
604 |
by fast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
605 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
606 |
lemma subset_trans: "A \<subseteq> B ==> B \<subseteq> C ==> A \<subseteq> C" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
607 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
608 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
609 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
610 |
subsubsection {* Equality *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
611 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
612 |
lemma set_ext: assumes prem: "(!!x. (x:A) = (x:B))" shows "A = B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
613 |
apply (rule prem [THEN ext, THEN arg_cong, THEN box_equals]) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
614 |
apply (rule Collect_mem_eq) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
615 |
apply (rule Collect_mem_eq) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
616 |
done |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
617 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
618 |
(* Due to Brian Huffman *) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
619 |
lemma expand_set_eq: "(A = B) = (ALL x. (x:A) = (x:B))" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
620 |
by(auto intro:set_ext) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
621 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
622 |
lemma subset_antisym [intro!]: "A \<subseteq> B ==> B \<subseteq> A ==> A = B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
623 |
-- {* Anti-symmetry of the subset relation. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
624 |
by (iprover intro: set_ext subsetD) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
625 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
626 |
lemmas equalityI [intro!] = subset_antisym |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
627 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
628 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
629 |
\medskip Equality rules from ZF set theory -- are they appropriate |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
630 |
here? |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
631 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
632 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
633 |
lemma equalityD1: "A = B ==> A \<subseteq> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
634 |
by (simp add: subset_refl) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
635 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
636 |
lemma equalityD2: "A = B ==> B \<subseteq> A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
637 |
by (simp add: subset_refl) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
638 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
639 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
640 |
\medskip Be careful when adding this to the claset as @{text |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
641 |
subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
642 |
\<subseteq> A"} and @{prop "A \<subseteq> {}"} and then back to @{prop "A = {}"}! |
30352 | 643 |
*} |
644 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
645 |
lemma equalityE: "A = B ==> (A \<subseteq> B ==> B \<subseteq> A ==> P) ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
646 |
by (simp add: subset_refl) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
647 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
648 |
lemma equalityCE [elim]: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
649 |
"A = B ==> (c \<in> A ==> c \<in> B ==> P) ==> (c \<notin> A ==> c \<notin> B ==> P) ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
650 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
651 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
652 |
lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
653 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
654 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
655 |
lemma eqelem_imp_iff: "x = y ==> (x : A) = (y : A)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
656 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
657 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
658 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
659 |
subsubsection {* The universal set -- UNIV *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
660 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
661 |
lemma UNIV_I [simp]: "x : UNIV" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
662 |
by (simp add: UNIV_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
663 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
664 |
declare UNIV_I [intro] -- {* unsafe makes it less likely to cause problems *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
665 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
666 |
lemma UNIV_witness [intro?]: "EX x. x : UNIV" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
667 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
668 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
669 |
lemma subset_UNIV [simp]: "A \<subseteq> UNIV" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
670 |
by (rule subsetI) (rule UNIV_I) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
671 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
672 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
673 |
\medskip Eta-contracting these two rules (to remove @{text P}) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
674 |
causes them to be ignored because of their interaction with |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
675 |
congruence rules. |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
676 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
677 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
678 |
lemma ball_UNIV [simp]: "Ball UNIV P = All P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
679 |
by (simp add: Ball_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
680 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
681 |
lemma bex_UNIV [simp]: "Bex UNIV P = Ex P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
682 |
by (simp add: Bex_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
683 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
684 |
lemma UNIV_eq_I: "(\<And>x. x \<in> A) \<Longrightarrow> UNIV = A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
685 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
686 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
687 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
688 |
subsubsection {* The empty set *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
689 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
690 |
lemma empty_iff [simp]: "(c : {}) = False" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
691 |
by (simp add: empty_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
692 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
693 |
lemma emptyE [elim!]: "a : {} ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
694 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
695 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
696 |
lemma empty_subsetI [iff]: "{} \<subseteq> A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
697 |
-- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
698 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
699 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
700 |
lemma equals0I: "(!!y. y \<in> A ==> False) ==> A = {}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
701 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
702 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
703 |
lemma equals0D: "A = {} ==> a \<notin> A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
704 |
-- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
705 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
706 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
707 |
lemma ball_empty [simp]: "Ball {} P = True" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
708 |
by (simp add: Ball_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
709 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
710 |
lemma bex_empty [simp]: "Bex {} P = False" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
711 |
by (simp add: Bex_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
712 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
713 |
lemma UNIV_not_empty [iff]: "UNIV ~= {}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
714 |
by (blast elim: equalityE) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
715 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
716 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
717 |
subsubsection {* The Powerset operator -- Pow *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
718 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
719 |
lemma Pow_iff [iff]: "(A \<in> Pow B) = (A \<subseteq> B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
720 |
by (simp add: Pow_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
721 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
722 |
lemma PowI: "A \<subseteq> B ==> A \<in> Pow B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
723 |
by (simp add: Pow_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
724 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
725 |
lemma PowD: "A \<in> Pow B ==> A \<subseteq> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
726 |
by (simp add: Pow_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
727 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
728 |
lemma Pow_bottom: "{} \<in> Pow B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
729 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
730 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
731 |
lemma Pow_top: "A \<in> Pow A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
732 |
by (simp add: subset_refl) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
733 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
734 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
735 |
subsubsection {* Set complement *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
736 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
737 |
lemma Compl_iff [simp]: "(c \<in> -A) = (c \<notin> A)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
738 |
by (simp add: mem_def fun_Compl_def bool_Compl_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
739 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
740 |
lemma ComplI [intro!]: "(c \<in> A ==> False) ==> c \<in> -A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
741 |
by (unfold mem_def fun_Compl_def bool_Compl_def) blast |
923 | 742 |
|
11979 | 743 |
text {* |
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
744 |
\medskip This form, with negated conclusion, works well with the |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
745 |
Classical prover. Negated assumptions behave like formulae on the |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
746 |
right side of the notional turnstile ... *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
747 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
748 |
lemma ComplD [dest!]: "c : -A ==> c~:A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
749 |
by (simp add: mem_def fun_Compl_def bool_Compl_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
750 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
751 |
lemmas ComplE = ComplD [elim_format] |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
752 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
753 |
lemma Compl_eq: "- A = {x. ~ x : A}" by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
754 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
755 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
756 |
subsubsection {* Binary union -- Un *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
757 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
758 |
lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
759 |
by (unfold Un_def) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
760 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
761 |
lemma UnI1 [elim?]: "c:A ==> c : A Un B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
762 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
763 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
764 |
lemma UnI2 [elim?]: "c:B ==> c : A Un B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
765 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
766 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
767 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
768 |
\medskip Classical introduction rule: no commitment to @{prop A} vs |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
769 |
@{prop B}. |
11979 | 770 |
*} |
771 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
772 |
lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
773 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
774 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
775 |
lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
776 |
by (unfold Un_def) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
777 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
778 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
779 |
subsubsection {* Binary intersection -- Int *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
780 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
781 |
lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
782 |
by (unfold Int_def) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
783 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
784 |
lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
785 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
786 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
787 |
lemma IntD1: "c : A Int B ==> c:A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
788 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
789 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
790 |
lemma IntD2: "c : A Int B ==> c:B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
791 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
792 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
793 |
lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
794 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
795 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
796 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
797 |
subsubsection {* Set difference *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
798 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
799 |
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
800 |
by (simp add: mem_def fun_diff_def bool_diff_def) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
801 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
802 |
lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
803 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
804 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
805 |
lemma DiffD1: "c : A - B ==> c : A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
806 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
807 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
808 |
lemma DiffD2: "c : A - B ==> c : B ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
809 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
810 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
811 |
lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
812 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
813 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
814 |
lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
815 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
816 |
lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
817 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
818 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
819 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
820 |
subsubsection {* Augmenting a set -- insert *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
821 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
822 |
lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
823 |
by (unfold insert_def) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
824 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
825 |
lemma insertI1: "a : insert a B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
826 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
827 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
828 |
lemma insertI2: "a : B ==> a : insert b B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
829 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
830 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
831 |
lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
832 |
by (unfold insert_def) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
833 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
834 |
lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
835 |
-- {* Classical introduction rule. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
836 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
837 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
838 |
lemma subset_insert_iff: "(A \<subseteq> insert x B) = (if x:A then A - {x} \<subseteq> B else A \<subseteq> B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
839 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
840 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
841 |
lemma set_insert: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
842 |
assumes "x \<in> A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
843 |
obtains B where "A = insert x B" and "x \<notin> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
844 |
proof |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
845 |
from assms show "A = insert x (A - {x})" by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
846 |
next |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
847 |
show "x \<notin> A - {x}" by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
848 |
qed |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
849 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
850 |
lemma insert_ident: "x ~: A ==> x ~: B ==> (insert x A = insert x B) = (A = B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
851 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
852 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
853 |
subsubsection {* Singletons, using insert *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
854 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
855 |
lemma singletonI [intro!,noatp]: "a : {a}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
856 |
-- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
857 |
by (rule insertI1) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
858 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
859 |
lemma singletonD [dest!,noatp]: "b : {a} ==> b = a" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
860 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
861 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
862 |
lemmas singletonE = singletonD [elim_format] |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
863 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
864 |
lemma singleton_iff: "(b : {a}) = (b = a)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
865 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
866 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
867 |
lemma singleton_inject [dest!]: "{a} = {b} ==> a = b" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
868 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
869 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
870 |
lemma singleton_insert_inj_eq [iff,noatp]: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
871 |
"({b} = insert a A) = (a = b & A \<subseteq> {b})" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
872 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
873 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
874 |
lemma singleton_insert_inj_eq' [iff,noatp]: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
875 |
"(insert a A = {b}) = (a = b & A \<subseteq> {b})" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
876 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
877 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
878 |
lemma subset_singletonD: "A \<subseteq> {x} ==> A = {} | A = {x}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
879 |
by fast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
880 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
881 |
lemma singleton_conv [simp]: "{x. x = a} = {a}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
882 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
883 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
884 |
lemma singleton_conv2 [simp]: "{x. a = x} = {a}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
885 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
886 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
887 |
lemma diff_single_insert: "A - {x} \<subseteq> B ==> x \<in> A ==> A \<subseteq> insert x B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
888 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
889 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
890 |
lemma doubleton_eq_iff: "({a,b} = {c,d}) = (a=c & b=d | a=d & b=c)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
891 |
by (blast elim: equalityE) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
892 |
|
11979 | 893 |
|
894 |
subsubsection {* Unions of families *} |
|
895 |
||
896 |
text {* |
|
897 |
@{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}. |
|
898 |
*} |
|
899 |
||
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset
|
900 |
declare UNION_def [noatp] |
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset
|
901 |
|
11979 | 902 |
lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)" |
903 |
by (unfold UNION_def) blast |
|
904 |
||
905 |
lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)" |
|
906 |
-- {* The order of the premises presupposes that @{term A} is rigid; |
|
907 |
@{term b} may be flexible. *} |
|
908 |
by auto |
|
909 |
||
910 |
lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R" |
|
911 |
by (unfold UNION_def) blast |
|
923 | 912 |
|
11979 | 913 |
lemma UN_cong [cong]: |
914 |
"A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" |
|
915 |
by (simp add: UNION_def) |
|
916 |
||
29691 | 917 |
lemma strong_UN_cong: |
918 |
"A = B ==> (!!x. x:B =simp=> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)" |
|
919 |
by (simp add: UNION_def simp_implies_def) |
|
920 |
||
11979 | 921 |
|
922 |
subsubsection {* Intersections of families *} |
|
923 |
||
924 |
text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *} |
|
925 |
||
926 |
lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)" |
|
927 |
by (unfold INTER_def) blast |
|
923 | 928 |
|
11979 | 929 |
lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)" |
930 |
by (unfold INTER_def) blast |
|
931 |
||
932 |
lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a" |
|
933 |
by auto |
|
934 |
||
935 |
lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R" |
|
936 |
-- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *} |
|
937 |
by (unfold INTER_def) blast |
|
938 |
||
939 |
lemma INT_cong [cong]: |
|
940 |
"A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)" |
|
941 |
by (simp add: INTER_def) |
|
7238
36e58620ffc8
replaced HOL_quantifiers flag by "HOL" print mode;
wenzelm
parents:
5931
diff
changeset
|
942 |
|
923 | 943 |
|
11979 | 944 |
subsubsection {* Union *} |
945 |
||
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset
|
946 |
lemma Union_iff [simp,noatp]: "(A : Union C) = (EX X:C. A:X)" |
11979 | 947 |
by (unfold Union_def) blast |
948 |
||
949 |
lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C" |
|
950 |
-- {* The order of the premises presupposes that @{term C} is rigid; |
|
951 |
@{term A} may be flexible. *} |
|
952 |
by auto |
|
953 |
||
954 |
lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R" |
|
955 |
by (unfold Union_def) blast |
|
956 |
||
957 |
||
958 |
subsubsection {* Inter *} |
|
959 |
||
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset
|
960 |
lemma Inter_iff [simp,noatp]: "(A : Inter C) = (ALL X:C. A:X)" |
11979 | 961 |
by (unfold Inter_def) blast |
962 |
||
963 |
lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C" |
|
964 |
by (simp add: Inter_def) |
|
965 |
||
966 |
text {* |
|
967 |
\medskip A ``destruct'' rule -- every @{term X} in @{term C} |
|
968 |
contains @{term A} as an element, but @{prop "A:X"} can hold when |
|
969 |
@{prop "X:C"} does not! This rule is analogous to @{text spec}. |
|
970 |
*} |
|
971 |
||
972 |
lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X" |
|
973 |
by auto |
|
974 |
||
975 |
lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R" |
|
976 |
-- {* ``Classical'' elimination rule -- does not require proving |
|
977 |
@{prop "X:C"}. *} |
|
978 |
by (unfold Inter_def) blast |
|
979 |
||
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
980 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
981 |
\medskip Image of a set under a function. Frequently @{term b} does |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
982 |
not have the syntactic form of @{term "f x"}. |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
983 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
984 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
985 |
declare image_def [noatp] |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
986 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
987 |
lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
988 |
by (unfold image_def) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
989 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
990 |
lemma imageI: "x : A ==> f x : f ` A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
991 |
by (rule image_eqI) (rule refl) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
992 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
993 |
lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
994 |
-- {* This version's more effective when we already have the |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
995 |
required @{term x}. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
996 |
by (unfold image_def) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
997 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
998 |
lemma imageE [elim!]: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
999 |
"b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1000 |
-- {* The eta-expansion gives variable-name preservation. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1001 |
by (unfold image_def) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1002 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1003 |
lemma image_Un: "f`(A Un B) = f`A Un f`B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1004 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1005 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1006 |
lemma image_eq_UN: "f`A = (UN x:A. {f x})" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1007 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1008 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1009 |
lemma image_iff: "(z : f`A) = (EX x:A. z = f x)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1010 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1011 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1012 |
lemma image_subset_iff: "(f`A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1013 |
-- {* This rewrite rule would confuse users if made default. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1014 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1015 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1016 |
lemma subset_image_iff: "(B \<subseteq> f`A) = (EX AA. AA \<subseteq> A & B = f`AA)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1017 |
apply safe |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1018 |
prefer 2 apply fast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1019 |
apply (rule_tac x = "{a. a : A & f a : B}" in exI, fast) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1020 |
done |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1021 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1022 |
lemma image_subsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f`A \<subseteq> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1023 |
-- {* Replaces the three steps @{text subsetI}, @{text imageE}, |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1024 |
@{text hypsubst}, but breaks too many existing proofs. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1025 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1026 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1027 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1028 |
\medskip Range of a function -- just a translation for image! |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1029 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1030 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1031 |
lemma range_eqI: "b = f x ==> b \<in> range f" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1032 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1033 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1034 |
lemma rangeI: "f x \<in> range f" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1035 |
by simp |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1036 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1037 |
lemma rangeE [elim?]: "b \<in> range (\<lambda>x. f x) ==> (!!x. b = f x ==> P) ==> P" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1038 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1039 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1040 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1041 |
subsubsection {* Set reasoning tools *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1042 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1043 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1044 |
Rewrite rules for boolean case-splitting: faster than @{text |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1045 |
"split_if [split]"}. |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1046 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1047 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1048 |
lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1049 |
by (rule split_if) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1050 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1051 |
lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1052 |
by (rule split_if) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1053 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1054 |
text {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1055 |
Split ifs on either side of the membership relation. Not for @{text |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1056 |
"[simp]"} -- can cause goals to blow up! |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1057 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1058 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1059 |
lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1060 |
by (rule split_if) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1061 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1062 |
lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1063 |
by (rule split_if [where P="%S. a : S"]) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1064 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1065 |
lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2 |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1066 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1067 |
lemmas mem_simps = |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1068 |
insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1069 |
mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1070 |
-- {* Each of these has ALREADY been added @{text "[simp]"} above. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1071 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1072 |
(*Would like to add these, but the existing code only searches for the |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1073 |
outer-level constant, which in this case is just "op :"; we instead need |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1074 |
to use term-nets to associate patterns with rules. Also, if a rule fails to |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1075 |
apply, then the formula should be kept. |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1076 |
[("HOL.uminus", Compl_iff RS iffD1), ("HOL.minus", [Diff_iff RS iffD1]), |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1077 |
("Int", [IntD1,IntD2]), |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1078 |
("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])] |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1079 |
*) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1080 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1081 |
ML {* |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1082 |
val mksimps_pairs = [(@{const_name Ball}, @{thms bspec})] @ mksimps_pairs; |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1083 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1084 |
declaration {* fn _ => |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1085 |
Simplifier.map_ss (fn ss => ss setmksimps (mksimps mksimps_pairs)) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1086 |
*} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1087 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1088 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1089 |
subsubsection {* The ``proper subset'' relation *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1090 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1091 |
lemma psubsetI [intro!,noatp]: "A \<subseteq> B ==> A \<noteq> B ==> A \<subset> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1092 |
by (unfold less_le) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1093 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1094 |
lemma psubsetE [elim!,noatp]: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1095 |
"[|A \<subset> B; [|A \<subseteq> B; ~ (B\<subseteq>A)|] ==> R|] ==> R" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1096 |
by (unfold less_le) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1097 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1098 |
lemma psubset_insert_iff: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1099 |
"(A \<subset> insert x B) = (if x \<in> B then A \<subset> B else if x \<in> A then A - {x} \<subset> B else A \<subseteq> B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1100 |
by (auto simp add: less_le subset_insert_iff) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1101 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1102 |
lemma psubset_eq: "(A \<subset> B) = (A \<subseteq> B & A \<noteq> B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1103 |
by (simp only: less_le) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1104 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1105 |
lemma psubset_imp_subset: "A \<subset> B ==> A \<subseteq> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1106 |
by (simp add: psubset_eq) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1107 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1108 |
lemma psubset_trans: "[| A \<subset> B; B \<subset> C |] ==> A \<subset> C" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1109 |
apply (unfold less_le) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1110 |
apply (auto dest: subset_antisym) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1111 |
done |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1112 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1113 |
lemma psubsetD: "[| A \<subset> B; c \<in> A |] ==> c \<in> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1114 |
apply (unfold less_le) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1115 |
apply (auto dest: subsetD) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1116 |
done |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1117 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1118 |
lemma psubset_subset_trans: "A \<subset> B ==> B \<subseteq> C ==> A \<subset> C" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1119 |
by (auto simp add: psubset_eq) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1120 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1121 |
lemma subset_psubset_trans: "A \<subseteq> B ==> B \<subset> C ==> A \<subset> C" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1122 |
by (auto simp add: psubset_eq) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1123 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1124 |
lemma psubset_imp_ex_mem: "A \<subset> B ==> \<exists>b. b \<in> (B - A)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1125 |
by (unfold less_le) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1126 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1127 |
lemma atomize_ball: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1128 |
"(!!x. x \<in> A ==> P x) == Trueprop (\<forall>x\<in>A. P x)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1129 |
by (simp only: Ball_def atomize_all atomize_imp) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1130 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1131 |
lemmas [symmetric, rulify] = atomize_ball |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1132 |
and [symmetric, defn] = atomize_ball |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1133 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1134 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1135 |
subsection {* Further set-theory lemmas *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1136 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1137 |
subsubsection {* Derived rules involving subsets. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1138 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1139 |
text {* @{text insert}. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1140 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1141 |
lemma subset_insertI: "B \<subseteq> insert a B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1142 |
by (rule subsetI) (erule insertI2) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1143 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1144 |
lemma subset_insertI2: "A \<subseteq> B \<Longrightarrow> A \<subseteq> insert b B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1145 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1146 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1147 |
lemma subset_insert: "x \<notin> A ==> (A \<subseteq> insert x B) = (A \<subseteq> B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1148 |
by blast |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1149 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1150 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1151 |
text {* \medskip Big Union -- least upper bound of a set. *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1152 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1153 |
lemma Union_upper: "B \<in> A ==> B \<subseteq> Union A" |
17589 | 1154 |
by (iprover intro: subsetI UnionI) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1155 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1156 |
lemma Union_least: "(!!X. X \<in> A ==> X \<subseteq> C) ==> Union A \<subseteq> C" |
17589 | 1157 |
by (iprover intro: subsetI elim: UnionE dest: subsetD) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1158 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1159 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1160 |
text {* \medskip General union. *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1161 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1162 |
lemma UN_upper: "a \<in> A ==> B a \<subseteq> (\<Union>x\<in>A. B x)" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1163 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1164 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1165 |
lemma UN_least: "(!!x. x \<in> A ==> B x \<subseteq> C) ==> (\<Union>x\<in>A. B x) \<subseteq> C" |
17589 | 1166 |
by (iprover intro: subsetI elim: UN_E dest: subsetD) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1167 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1168 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1169 |
text {* \medskip Big Intersection -- greatest lower bound of a set. *} |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1170 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1171 |
lemma Inter_lower: "B \<in> A ==> Inter A \<subseteq> B" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1172 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1173 |
|
14551 | 1174 |
lemma Inter_subset: |
1175 |
"[| !!X. X \<in> A ==> X \<subseteq> B; A ~= {} |] ==> \<Inter>A \<subseteq> B" |
|
1176 |
by blast |
|
1177 |
||
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1178 |
lemma Inter_greatest: "(!!X. X \<in> A ==> C \<subseteq> X) ==> C \<subseteq> Inter A" |
17589 | 1179 |
by (iprover intro: InterI subsetI dest: subsetD) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1180 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1181 |
lemma INT_lower: "a \<in> A ==> (\<Inter>x\<in>A. B x) \<subseteq> B a" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1182 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1183 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1184 |
lemma INT_greatest: "(!!x. x \<in> A ==> C \<subseteq> B x) ==> C \<subseteq> (\<Inter>x\<in>A. B x)" |
17589 | 1185 |
by (iprover intro: INT_I subsetI dest: subsetD) |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1186 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1187 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1188 |
text {* \medskip Finite Union -- the least upper bound of two sets. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1189 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1190 |
lemma Un_upper1: "A \<subseteq> A \<union> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1191 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1192 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1193 |
lemma Un_upper2: "B \<subseteq> A \<union> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1194 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1195 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1196 |
lemma Un_least: "A \<subseteq> C ==> B \<subseteq> C ==> A \<union> B \<subseteq> C" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1197 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1198 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1199 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1200 |
text {* \medskip Finite Intersection -- the greatest lower bound of two sets. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1201 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1202 |
lemma Int_lower1: "A \<inter> B \<subseteq> A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1203 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1204 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1205 |
lemma Int_lower2: "A \<inter> B \<subseteq> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1206 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1207 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1208 |
lemma Int_greatest: "C \<subseteq> A ==> C \<subseteq> B ==> C \<subseteq> A \<inter> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1209 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1210 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1211 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1212 |
text {* \medskip Set difference. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1213 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1214 |
lemma Diff_subset: "A - B \<subseteq> A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1215 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1216 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1217 |
lemma Diff_subset_conv: "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1218 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1219 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1220 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1221 |
subsubsection {* Equalities involving union, intersection, inclusion, etc. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1222 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1223 |
text {* @{text "{}"}. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1224 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1225 |
lemma Collect_const [simp]: "{s. P} = (if P then UNIV else {})" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1226 |
-- {* supersedes @{text "Collect_False_empty"} *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1227 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1228 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1229 |
lemma subset_empty [simp]: "(A \<subseteq> {}) = (A = {})" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1230 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1231 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1232 |
lemma not_psubset_empty [iff]: "\<not> (A < {})" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1233 |
by (unfold less_le) blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1234 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1235 |
lemma Collect_empty_eq [simp]: "(Collect P = {}) = (\<forall>x. \<not> P x)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1236 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1237 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1238 |
lemma empty_Collect_eq [simp]: "({} = Collect P) = (\<forall>x. \<not> P x)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1239 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1240 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1241 |
lemma Collect_neg_eq: "{x. \<not> P x} = - {x. P x}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1242 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1243 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1244 |
lemma Collect_disj_eq: "{x. P x | Q x} = {x. P x} \<union> {x. Q x}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1245 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1246 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1247 |
lemma Collect_imp_eq: "{x. P x --> Q x} = -{x. P x} \<union> {x. Q x}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1248 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1249 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1250 |
lemma Collect_conj_eq: "{x. P x & Q x} = {x. P x} \<inter> {x. Q x}" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1251 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1252 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1253 |
lemma Collect_all_eq: "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1254 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1255 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1256 |
lemma Collect_ball_eq: "{x. \<forall>y\<in>A. P x y} = (\<Inter>y\<in>A. {x. P x y})" |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1257 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1258 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset
|
1259 |
lemma Collect_ex_eq [noatp]: "{x. \<exists>y. P x y} = (\<Union>y. {x. P x y})" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1260 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1261 |
|
24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24280
diff
changeset
|
1262 |
lemma Collect_bex_eq [noatp]: "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})" |
12897
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1263 |
by blast |
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1264 |
|
f4d10ad0ea7b
converted/deleted equalities.ML, mono.ML, subset.ML (see Set.thy);
wenzelm
parents:
12633
diff
changeset
|
1265 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1266 |
text {* \medskip @{text insert}. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1267 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1268 |
lemma insert_is_Un: "insert a A = {a} Un A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1269 |
-- {* NOT SUITABLE FOR REWRITING since @{text "{a} == insert a {}"} *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1270 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1271 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1272 |
lemma insert_not_empty [simp]: "insert a A \<noteq> {}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1273 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1274 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1275 |
lemmas empty_not_insert = insert_not_empty [symmetric, standard] |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1276 |
declare empty_not_insert [simp] |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1277 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1278 |
lemma insert_absorb: "a \<in> A ==> insert a A = A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1279 |
-- {* @{text "[simp]"} causes recursive calls when there are nested inserts *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1280 |
-- {* with \emph{quadratic} running time *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1281 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1282 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1283 |
lemma insert_absorb2 [simp]: "insert x (insert x A) = insert x A" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1284 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1285 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1286 |
lemma insert_commute: "insert x (insert y A) = insert y (insert x A)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1287 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1288 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1289 |
lemma insert_subset [simp]: "(insert x A \<subseteq> B) = (x \<in> B & A \<subseteq> B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1290 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1291 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1292 |
lemma mk_disjoint_insert: "a \<in> A ==> \<exists>B. A = insert a B & a \<notin> B" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1293 |
-- {* use new @{text B} rather than @{text "A - {a}"} to avoid infinite unfolding *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1294 |
apply (rule_tac x = "A - {a}" in exI, blast) |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1295 |
done |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1296 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1297 |
lemma insert_Collect: "insert a (Collect P) = {u. u \<noteq> a --> P u}" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1298 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1299 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1300 |
lemma UN_insert_distrib: "u \<in> A ==> (\<Union>x\<in>A. insert a (B x)) = insert a (\<Union>x\<in>A. B x)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1301 |
by blast |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1302 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1303 |
lemma insert_inter_insert[simp]: "insert a A \<inter> insert a B = insert a (A \<inter> B)" |
14742 | 1304 |
by blast |
14302 | 1305 |
|
30531
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1306 |
lemma insert_disjoint [simp,noatp]: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1307 |
"(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1308 |
"({} = insert a A \<inter> B) = (a \<notin> B \<and> {} = A \<inter> B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1309 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1310 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1311 |
lemma disjoint_insert [simp,noatp]: |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1312 |
"(B \<inter> insert a A = {}) = (a \<notin> B \<and> B \<inter> A = {})" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1313 |
"({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)" |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1314 |
by auto |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1315 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1316 |
text {* \medskip @{text image}. *} |
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1317 |
|
ab3d61baf66a
reverted to old version of Set.thy -- strange effects have to be traced first
haftmann
parents:
30352
diff
changeset
|
1318 |
lemma image_empty [simp]: "f`{} = {}" |