1 (*<*) |
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2 theory Main_Doc |
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3 imports Main |
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4 begin |
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5 |
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6 ML {* |
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7 fun pretty_term_type_only ctxt (t, T) = |
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8 (if fastype_of t = Sign.certify_typ (ProofContext.theory_of ctxt) T then () |
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9 else error "term_type_only: type mismatch"; |
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10 Syntax.pretty_typ ctxt T) |
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11 |
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12 val _ = ThyOutput.antiquotation "term_type_only" (Args.term -- Args.typ_abbrev) |
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13 (fn {source, context, ...} => fn arg => |
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14 ThyOutput.output |
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15 (ThyOutput.maybe_pretty_source (pretty_term_type_only context) source [arg])); |
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16 *} |
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17 (*>*) |
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18 text{* |
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19 |
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20 \begin{abstract} |
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21 This document lists the main types, functions and syntax provided by theory @{theory Main}. It is meant as a quick overview of what is available. The sophisticated class structure is only hinted at. For details see \url{http://isabelle.in.tum.de/dist/library/HOL/}. |
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22 \end{abstract} |
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23 |
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24 \section{HOL} |
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25 |
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26 The basic logic: @{prop "x = y"}, @{const True}, @{const False}, @{prop"Not P"}, @{prop"P & Q"}, @{prop "P | Q"}, @{prop "P --> Q"}, @{prop"ALL x. P"}, @{prop"EX x. P"}, @{prop"EX! x. P"}, @{term"THE x. P"}. |
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27 \smallskip |
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28 |
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29 \begin{tabular}{@ {} l @ {~::~} l @ {}} |
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30 @{const HOL.undefined} & @{typeof HOL.undefined}\\ |
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31 @{const HOL.default} & @{typeof HOL.default}\\ |
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32 \end{tabular} |
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33 |
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34 \subsubsection*{Syntax} |
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35 |
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36 \begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
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37 @{term"~(x = y)"} & @{term[source]"\<not> (x = y)"} & (\verb$~=$)\\ |
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38 @{term[source]"P \<longleftrightarrow> Q"} & @{term"P \<longleftrightarrow> Q"} \\ |
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39 @{term"If x y z"} & @{term[source]"If x y z"}\\ |
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40 @{term"Let e\<^isub>1 (%x. e\<^isub>2)"} & @{term[source]"Let e\<^isub>1 (\<lambda>x. e\<^isub>2)"}\\ |
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41 \end{supertabular} |
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42 |
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43 |
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44 \section{Orderings} |
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45 |
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46 A collection of classes defining basic orderings: |
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47 preorder, partial order, linear order, dense linear order and wellorder. |
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48 \smallskip |
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49 |
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50 \begin{supertabular}{@ {} l @ {~::~} l l @ {}} |
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51 @{const HOL.less_eq} & @{typeof HOL.less_eq} & (\verb$<=$)\\ |
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52 @{const HOL.less} & @{typeof HOL.less}\\ |
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53 @{const Orderings.Least} & @{typeof Orderings.Least}\\ |
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54 @{const Orderings.min} & @{typeof Orderings.min}\\ |
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55 @{const Orderings.max} & @{typeof Orderings.max}\\ |
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56 @{const[source] top} & @{typeof Orderings.top}\\ |
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57 @{const[source] bot} & @{typeof Orderings.bot}\\ |
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58 @{const Orderings.mono} & @{typeof Orderings.mono}\\ |
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59 @{const Orderings.strict_mono} & @{typeof Orderings.strict_mono}\\ |
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60 \end{supertabular} |
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61 |
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62 \subsubsection*{Syntax} |
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63 |
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64 \begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
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65 @{term[source]"x \<ge> y"} & @{term"x \<ge> y"} & (\verb$>=$)\\ |
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66 @{term[source]"x > y"} & @{term"x > y"}\\ |
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67 @{term"ALL x<=y. P"} & @{term[source]"\<forall>x. x \<le> y \<longrightarrow> P"}\\ |
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68 @{term"EX x<=y. P"} & @{term[source]"\<exists>x. x \<le> y \<and> P"}\\ |
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69 \multicolumn{2}{@ {}l@ {}}{Similarly for $<$, $\ge$ and $>$}\\ |
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70 @{term"LEAST x. P"} & @{term[source]"Least (\<lambda>x. P)"}\\ |
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71 \end{supertabular} |
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72 |
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73 |
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74 \section{Lattices} |
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75 |
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76 Classes semilattice, lattice, distributive lattice and complete lattice (the |
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77 latter in theory @{theory Set}). |
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78 |
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79 \begin{tabular}{@ {} l @ {~::~} l @ {}} |
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80 @{const Lattices.inf} & @{typeof Lattices.inf}\\ |
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81 @{const Lattices.sup} & @{typeof Lattices.sup}\\ |
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82 @{const Set.Inf} & @{term_type_only Set.Inf "'a set \<Rightarrow> 'a::complete_lattice"}\\ |
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83 @{const Set.Sup} & @{term_type_only Set.Sup "'a set \<Rightarrow> 'a::complete_lattice"}\\ |
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84 \end{tabular} |
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85 |
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86 \subsubsection*{Syntax} |
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87 |
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88 Available by loading theory @{text Lattice_Syntax} in directory @{text |
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89 Library}. |
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90 |
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91 \begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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92 @{text[source]"x \<sqsubseteq> y"} & @{term"x \<le> y"}\\ |
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93 @{text[source]"x \<sqsubset> y"} & @{term"x < y"}\\ |
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94 @{text[source]"x \<sqinter> y"} & @{term"inf x y"}\\ |
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95 @{text[source]"x \<squnion> y"} & @{term"sup x y"}\\ |
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96 @{text[source]"\<Sqinter> A"} & @{term"Sup A"}\\ |
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97 @{text[source]"\<Squnion> A"} & @{term"Inf A"}\\ |
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98 @{text[source]"\<top>"} & @{term[source] top}\\ |
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99 @{text[source]"\<bottom>"} & @{term[source] bot}\\ |
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100 \end{supertabular} |
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101 |
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102 |
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103 \section{Set} |
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104 |
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105 Sets are predicates: @{text[source]"'a set = 'a \<Rightarrow> bool"} |
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106 \bigskip |
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107 |
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108 \begin{supertabular}{@ {} l @ {~::~} l l @ {}} |
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109 @{const Set.empty} & @{term_type_only "Set.empty" "'a set"}\\ |
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110 @{const insert} & @{term_type_only insert "'a\<Rightarrow>'a set\<Rightarrow>'a set"}\\ |
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111 @{const Collect} & @{term_type_only Collect "('a\<Rightarrow>bool)\<Rightarrow>'a set"}\\ |
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112 @{const "op :"} & @{term_type_only "op :" "'a\<Rightarrow>'a set\<Rightarrow>bool"} & (\texttt{:})\\ |
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113 @{const Set.Un} & @{term_type_only Set.Un "'a set\<Rightarrow>'a set \<Rightarrow> 'a set"} & (\texttt{Un})\\ |
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114 @{const Set.Int} & @{term_type_only Set.Int "'a set\<Rightarrow>'a set \<Rightarrow> 'a set"} & (\texttt{Int})\\ |
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115 @{const UNION} & @{term_type_only UNION "'a set\<Rightarrow>('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"}\\ |
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116 @{const INTER} & @{term_type_only INTER "'a set\<Rightarrow>('a \<Rightarrow> 'b set) \<Rightarrow> 'b set"}\\ |
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117 @{const Union} & @{term_type_only Union "'a set set\<Rightarrow>'a set"}\\ |
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118 @{const Inter} & @{term_type_only Inter "'a set set\<Rightarrow>'a set"}\\ |
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119 @{const Pow} & @{term_type_only Pow "'a set \<Rightarrow>'a set set"}\\ |
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120 @{const UNIV} & @{term_type_only UNIV "'a set"}\\ |
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121 @{const image} & @{term_type_only image "('a\<Rightarrow>'b)\<Rightarrow>'a set\<Rightarrow>'b set"}\\ |
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122 @{const Ball} & @{term_type_only Ball "'a set\<Rightarrow>('a\<Rightarrow>bool)\<Rightarrow>bool"}\\ |
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123 @{const Bex} & @{term_type_only Bex "'a set\<Rightarrow>('a\<Rightarrow>bool)\<Rightarrow>bool"}\\ |
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124 \end{supertabular} |
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125 |
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126 \subsubsection*{Syntax} |
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127 |
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128 \begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
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129 @{text"{x\<^isub>1,\<dots>,x\<^isub>n}"} & @{text"insert x\<^isub>1 (\<dots> (insert x\<^isub>n {})\<dots>)"}\\ |
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130 @{term"x ~: A"} & @{term[source]"\<not>(x \<in> A)"}\\ |
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131 @{term"A \<subseteq> B"} & @{term[source]"A \<le> B"}\\ |
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132 @{term"A \<subset> B"} & @{term[source]"A < B"}\\ |
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133 @{term[source]"A \<supseteq> B"} & @{term[source]"B \<le> A"}\\ |
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134 @{term[source]"A \<supset> B"} & @{term[source]"B < A"}\\ |
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135 @{term"{x. P}"} & @{term[source]"Collect (\<lambda>x. P)"}\\ |
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136 @{term[mode=xsymbols]"UN x:I. A"} & @{term[source]"UNION I (\<lambda>x. A)"} & (\texttt{UN})\\ |
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137 @{term[mode=xsymbols]"UN x. A"} & @{term[source]"UNION UNIV (\<lambda>x. A)"}\\ |
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138 @{term[mode=xsymbols]"INT x:I. A"} & @{term[source]"INTER I (\<lambda>x. A)"} & (\texttt{INT})\\ |
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139 @{term[mode=xsymbols]"INT x. A"} & @{term[source]"INTER UNIV (\<lambda>x. A)"}\\ |
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140 @{term"ALL x:A. P"} & @{term[source]"Ball A (\<lambda>x. P)"}\\ |
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141 @{term"EX x:A. P"} & @{term[source]"Bex A (\<lambda>x. P)"}\\ |
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142 @{term"range f"} & @{term[source]"f ` UNIV"}\\ |
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143 \end{supertabular} |
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144 |
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145 |
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146 \section{Fun} |
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147 |
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148 \begin{supertabular}{@ {} l @ {~::~} l @ {}} |
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149 @{const "Fun.id"} & @{typeof Fun.id}\\ |
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150 @{const "Fun.comp"} & @{typeof Fun.comp}\\ |
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151 @{const "Fun.inj_on"} & @{term_type_only Fun.inj_on "('a\<Rightarrow>'b)\<Rightarrow>'a set\<Rightarrow>bool"}\\ |
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152 @{const "Fun.inj"} & @{typeof Fun.inj}\\ |
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153 @{const "Fun.surj"} & @{typeof Fun.surj}\\ |
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154 @{const "Fun.bij"} & @{typeof Fun.bij}\\ |
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155 @{const "Fun.bij_betw"} & @{term_type_only Fun.bij_betw "('a\<Rightarrow>'b)\<Rightarrow>'a set\<Rightarrow>'b set\<Rightarrow>bool"}\\ |
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156 @{const "Fun.fun_upd"} & @{typeof Fun.fun_upd}\\ |
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157 \end{supertabular} |
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158 |
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159 \subsubsection*{Syntax} |
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160 |
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161 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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162 @{term"fun_upd f x y"} & @{term[source]"fun_upd f x y"}\\ |
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163 @{text"f(x\<^isub>1:=y\<^isub>1,\<dots>,x\<^isub>n:=y\<^isub>n)"} & @{text"f(x\<^isub>1:=y\<^isub>1)\<dots>(x\<^isub>n:=y\<^isub>n)"}\\ |
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164 \end{tabular} |
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165 |
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166 |
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167 \section{Fixed Points} |
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168 |
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169 Theory: @{theory Inductive}. |
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170 |
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171 Least and greatest fixed points in a complete lattice @{typ 'a}: |
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172 |
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173 \begin{tabular}{@ {} l @ {~::~} l @ {}} |
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174 @{const Inductive.lfp} & @{typeof Inductive.lfp}\\ |
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175 @{const Inductive.gfp} & @{typeof Inductive.gfp}\\ |
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176 \end{tabular} |
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177 |
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178 Note that in particular sets (@{typ"'a \<Rightarrow> bool"}) are complete lattices. |
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179 |
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180 \section{Sum\_Type} |
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181 |
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182 Type constructor @{text"+"}. |
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183 |
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184 \begin{tabular}{@ {} l @ {~::~} l @ {}} |
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185 @{const Sum_Type.Inl} & @{typeof Sum_Type.Inl}\\ |
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186 @{const Sum_Type.Inr} & @{typeof Sum_Type.Inr}\\ |
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187 @{const Sum_Type.Plus} & @{term_type_only Sum_Type.Plus "'a set\<Rightarrow>'b set\<Rightarrow>('a+'b)set"} |
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188 \end{tabular} |
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189 |
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190 |
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191 \section{Product\_Type} |
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192 |
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193 Types @{typ unit} and @{text"\<times>"}. |
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194 |
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195 \begin{supertabular}{@ {} l @ {~::~} l @ {}} |
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196 @{const Product_Type.Unity} & @{typeof Product_Type.Unity}\\ |
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197 @{const Pair} & @{typeof Pair}\\ |
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198 @{const fst} & @{typeof fst}\\ |
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199 @{const snd} & @{typeof snd}\\ |
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200 @{const split} & @{typeof split}\\ |
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201 @{const curry} & @{typeof curry}\\ |
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202 @{const Product_Type.Sigma} & @{term_type_only Product_Type.Sigma "'a set\<Rightarrow>('a\<Rightarrow>'b set)\<Rightarrow>('a*'b)set"}\\ |
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203 \end{supertabular} |
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204 |
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205 \subsubsection*{Syntax} |
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206 |
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207 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} ll @ {}} |
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208 @{term"Pair a b"} & @{term[source]"Pair a b"}\\ |
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209 @{term"split (\<lambda>x y. t)"} & @{term[source]"split (\<lambda>x y. t)"}\\ |
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210 @{term"A <*> B"} & @{text"Sigma A (\<lambda>\<^raw:\_>. B)"} & (\verb$<*>$) |
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211 \end{tabular} |
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212 |
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213 Pairs may be nested. Nesting to the right is printed as a tuple, |
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214 e.g.\ \mbox{@{term"(a,b,c)"}} is really \mbox{@{text"(a, (b, c))"}.} |
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215 Pattern matching with pairs and tuples extends to all binders, |
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216 e.g.\ \mbox{@{prop"ALL (x,y):A. P"},} @{term"{(x,y). P}"}, etc. |
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217 |
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218 |
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219 \section{Relation} |
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220 |
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221 \begin{supertabular}{@ {} l @ {~::~} l @ {}} |
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222 @{const Relation.converse} & @{term_type_only Relation.converse "('a * 'b)set \<Rightarrow> ('b*'a)set"}\\ |
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223 @{const Relation.rel_comp} & @{term_type_only Relation.rel_comp "('a*'b)set\<Rightarrow>('c*'a)set\<Rightarrow>('c*'b)set"}\\ |
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224 @{const Relation.Image} & @{term_type_only Relation.Image "('a*'b)set\<Rightarrow>'a set\<Rightarrow>'b set"}\\ |
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225 @{const Relation.inv_image} & @{term_type_only Relation.inv_image "('a*'a)set\<Rightarrow>('b\<Rightarrow>'a)\<Rightarrow>('b*'b)set"}\\ |
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226 @{const Relation.Id_on} & @{term_type_only Relation.Id_on "'a set\<Rightarrow>('a*'a)set"}\\ |
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227 @{const Relation.Id} & @{term_type_only Relation.Id "('a*'a)set"}\\ |
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228 @{const Relation.Domain} & @{term_type_only Relation.Domain "('a*'b)set\<Rightarrow>'a set"}\\ |
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229 @{const Relation.Range} & @{term_type_only Relation.Range "('a*'b)set\<Rightarrow>'b set"}\\ |
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230 @{const Relation.Field} & @{term_type_only Relation.Field "('a*'a)set\<Rightarrow>'a set"}\\ |
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231 @{const Relation.refl_on} & @{term_type_only Relation.refl_on "'a set\<Rightarrow>('a*'a)set\<Rightarrow>bool"}\\ |
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232 @{const Relation.refl} & @{term_type_only Relation.refl "('a*'a)set\<Rightarrow>bool"}\\ |
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233 @{const Relation.sym} & @{term_type_only Relation.sym "('a*'a)set\<Rightarrow>bool"}\\ |
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234 @{const Relation.antisym} & @{term_type_only Relation.antisym "('a*'a)set\<Rightarrow>bool"}\\ |
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235 @{const Relation.trans} & @{term_type_only Relation.trans "('a*'a)set\<Rightarrow>bool"}\\ |
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236 @{const Relation.irrefl} & @{term_type_only Relation.irrefl "('a*'a)set\<Rightarrow>bool"}\\ |
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237 @{const Relation.total_on} & @{term_type_only Relation.total_on "'a set\<Rightarrow>('a*'a)set\<Rightarrow>bool"}\\ |
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238 @{const Relation.total} & @{term_type_only Relation.total "('a*'a)set\<Rightarrow>bool"}\\ |
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239 \end{supertabular} |
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240 |
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241 \subsubsection*{Syntax} |
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242 |
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243 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
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244 @{term"converse r"} & @{term[source]"converse r"} & (\verb$^-1$) |
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245 \end{tabular} |
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246 |
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247 \section{Equiv\_Relations} |
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248 |
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249 \begin{supertabular}{@ {} l @ {~::~} l @ {}} |
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250 @{const Equiv_Relations.equiv} & @{term_type_only Equiv_Relations.equiv "'a set \<Rightarrow> ('a*'a)set\<Rightarrow>bool"}\\ |
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251 @{const Equiv_Relations.quotient} & @{term_type_only Equiv_Relations.quotient "'a set \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> 'a set set"}\\ |
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252 @{const Equiv_Relations.congruent} & @{term_type_only Equiv_Relations.congruent "('a*'a)set\<Rightarrow>('a\<Rightarrow>'b)\<Rightarrow>bool"}\\ |
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253 @{const Equiv_Relations.congruent2} & @{term_type_only Equiv_Relations.congruent2 "('a*'a)set\<Rightarrow>('b*'b)set\<Rightarrow>('a\<Rightarrow>'b\<Rightarrow>'c)\<Rightarrow>bool"}\\ |
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254 %@ {const Equiv_Relations.} & @ {term_type_only Equiv_Relations. ""}\\ |
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255 \end{supertabular} |
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256 |
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257 \subsubsection*{Syntax} |
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258 |
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259 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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260 @{term"congruent r f"} & @{term[source]"congruent r f"}\\ |
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261 @{term"congruent2 r r f"} & @{term[source]"congruent2 r r f"}\\ |
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262 \end{tabular} |
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263 |
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264 |
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265 \section{Transitive\_Closure} |
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266 |
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267 \begin{tabular}{@ {} l @ {~::~} l @ {}} |
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268 @{const Transitive_Closure.rtrancl} & @{term_type_only Transitive_Closure.rtrancl "('a*'a)set\<Rightarrow>('a*'a)set"}\\ |
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269 @{const Transitive_Closure.trancl} & @{term_type_only Transitive_Closure.trancl "('a*'a)set\<Rightarrow>('a*'a)set"}\\ |
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270 @{const Transitive_Closure.reflcl} & @{term_type_only Transitive_Closure.reflcl "('a*'a)set\<Rightarrow>('a*'a)set"}\\ |
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271 \end{tabular} |
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272 |
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273 \subsubsection*{Syntax} |
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274 |
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275 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
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276 @{term"rtrancl r"} & @{term[source]"rtrancl r"} & (\verb$^*$)\\ |
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277 @{term"trancl r"} & @{term[source]"trancl r"} & (\verb$^+$)\\ |
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278 @{term"reflcl r"} & @{term[source]"reflcl r"} & (\verb$^=$) |
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279 \end{tabular} |
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280 |
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281 |
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282 \section{Algebra} |
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283 |
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284 Theories @{theory OrderedGroup}, @{theory Ring_and_Field} and @{theory |
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285 Divides} define a large collection of classes describing common algebraic |
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286 structures from semigroups up to fields. Everything is done in terms of |
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287 overloaded operators: |
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288 |
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289 \begin{supertabular}{@ {} l @ {~::~} l l @ {}} |
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290 @{text "0"} & @{typeof zero}\\ |
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291 @{text "1"} & @{typeof one}\\ |
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292 @{const plus} & @{typeof plus}\\ |
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293 @{const minus} & @{typeof minus}\\ |
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294 @{const uminus} & @{typeof uminus} & (\verb$-$)\\ |
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295 @{const times} & @{typeof times}\\ |
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296 @{const inverse} & @{typeof inverse}\\ |
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297 @{const divide} & @{typeof divide}\\ |
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298 @{const abs} & @{typeof abs}\\ |
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299 @{const sgn} & @{typeof sgn}\\ |
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300 @{const dvd_class.dvd} & @{typeof "dvd_class.dvd"}\\ |
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301 @{const div_class.div} & @{typeof "div_class.div"}\\ |
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302 @{const div_class.mod} & @{typeof "div_class.mod"}\\ |
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303 \end{supertabular} |
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304 |
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305 \subsubsection*{Syntax} |
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306 |
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307 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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308 @{term"abs x"} & @{term[source]"abs x"} |
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309 \end{tabular} |
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310 |
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311 |
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312 \section{Nat} |
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313 |
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314 @{datatype nat} |
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315 \bigskip |
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316 |
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317 \begin{tabular}{@ {} lllllll @ {}} |
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318 @{term "op + :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
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319 @{term "op - :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
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320 @{term "op * :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
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321 @{term "op ^ :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
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322 @{term "op div :: nat \<Rightarrow> nat \<Rightarrow> nat"}& |
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323 @{term "op mod :: nat \<Rightarrow> nat \<Rightarrow> nat"}& |
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324 @{term "op dvd :: nat \<Rightarrow> nat \<Rightarrow> bool"}\\ |
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325 @{term "op \<le> :: nat \<Rightarrow> nat \<Rightarrow> bool"} & |
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326 @{term "op < :: nat \<Rightarrow> nat \<Rightarrow> bool"} & |
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327 @{term "min :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
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328 @{term "max :: nat \<Rightarrow> nat \<Rightarrow> nat"} & |
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329 @{term "Min :: nat set \<Rightarrow> nat"} & |
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330 @{term "Max :: nat set \<Rightarrow> nat"}\\ |
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331 \end{tabular} |
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332 |
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333 \begin{tabular}{@ {} l @ {~::~} l @ {}} |
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334 @{const Nat.of_nat} & @{typeof Nat.of_nat} |
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335 \end{tabular} |
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336 |
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337 \section{Int} |
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338 |
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339 Type @{typ int} |
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340 \bigskip |
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341 |
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342 \begin{tabular}{@ {} llllllll @ {}} |
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343 @{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} & |
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344 @{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} & |
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345 @{term "uminus :: int \<Rightarrow> int"} & |
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346 @{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} & |
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347 @{term "op ^ :: int \<Rightarrow> nat \<Rightarrow> int"} & |
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348 @{term "op div :: int \<Rightarrow> int \<Rightarrow> int"}& |
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349 @{term "op mod :: int \<Rightarrow> int \<Rightarrow> int"}& |
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350 @{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"}\\ |
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351 @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} & |
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352 @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} & |
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353 @{term "min :: int \<Rightarrow> int \<Rightarrow> int"} & |
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354 @{term "max :: int \<Rightarrow> int \<Rightarrow> int"} & |
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355 @{term "Min :: int set \<Rightarrow> int"} & |
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356 @{term "Max :: int set \<Rightarrow> int"}\\ |
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357 @{term "abs :: int \<Rightarrow> int"} & |
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358 @{term "sgn :: int \<Rightarrow> int"}\\ |
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359 \end{tabular} |
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360 |
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361 \begin{tabular}{@ {} l @ {~::~} l l @ {}} |
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362 @{const Int.nat} & @{typeof Int.nat}\\ |
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363 @{const Int.of_int} & @{typeof Int.of_int}\\ |
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364 @{const Int.Ints} & @{term_type_only Int.Ints "'a::ring_1 set"} & (\verb$Ints$) |
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365 \end{tabular} |
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366 |
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367 \subsubsection*{Syntax} |
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368 |
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369 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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370 @{term"of_nat::nat\<Rightarrow>int"} & @{term[source]"of_nat"}\\ |
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371 \end{tabular} |
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372 |
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373 |
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374 \section{Finite\_Set} |
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375 |
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376 |
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377 \begin{supertabular}{@ {} l @ {~::~} l @ {}} |
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378 @{const Finite_Set.finite} & @{term_type_only Finite_Set.finite "'a set\<Rightarrow>bool"}\\ |
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379 @{const Finite_Set.card} & @{term_type_only Finite_Set.card "'a set => nat"}\\ |
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380 @{const Finite_Set.fold} & @{term_type_only Finite_Set.fold "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"}\\ |
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381 @{const Finite_Set.fold_image} & @{typ "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"}\\ |
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382 @{const Finite_Set.setsum} & @{term_type_only Finite_Set.setsum "('a => 'b) => 'a set => 'b::comm_monoid_add"}\\ |
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383 @{const Finite_Set.setprod} & @{term_type_only Finite_Set.setprod "('a => 'b) => 'a set => 'b::comm_monoid_mult"}\\ |
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384 \end{supertabular} |
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385 |
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386 |
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387 \subsubsection*{Syntax} |
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388 |
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389 \begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l l @ {}} |
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390 @{term"setsum (%x. x) A"} & @{term[source]"setsum (\<lambda>x. x) A"} & (\verb$SUM$)\\ |
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391 @{term"setsum (%x. t) A"} & @{term[source]"setsum (\<lambda>x. t) A"}\\ |
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392 @{term[source]"\<Sum>x|P. t"} & @{term"\<Sum>x|P. t"}\\ |
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393 \multicolumn{2}{@ {}l@ {}}{Similarly for @{text"\<Prod>"} instead of @{text"\<Sum>"}} & (\verb$PROD$)\\ |
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394 \end{supertabular} |
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395 |
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396 |
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397 \section{Wellfounded} |
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398 |
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399 \begin{supertabular}{@ {} l @ {~::~} l @ {}} |
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400 @{const Wellfounded.wf} & @{term_type_only Wellfounded.wf "('a*'a)set\<Rightarrow>bool"}\\ |
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401 @{const Wellfounded.acyclic} & @{term_type_only Wellfounded.acyclic "('a*'a)set\<Rightarrow>bool"}\\ |
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402 @{const Wellfounded.acc} & @{term_type_only Wellfounded.acc "('a*'a)set\<Rightarrow>'a set"}\\ |
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403 @{const Wellfounded.measure} & @{term_type_only Wellfounded.measure "('a\<Rightarrow>nat)\<Rightarrow>('a*'a)set"}\\ |
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404 @{const Wellfounded.lex_prod} & @{term_type_only Wellfounded.lex_prod "('a*'a)set\<Rightarrow>('b*'b)set\<Rightarrow>(('a*'b)*('a*'b))set"}\\ |
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405 @{const Wellfounded.mlex_prod} & @{term_type_only Wellfounded.mlex_prod "('a\<Rightarrow>nat)\<Rightarrow>('a*'a)set\<Rightarrow>('a*'a)set"}\\ |
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406 @{const Wellfounded.less_than} & @{term_type_only Wellfounded.less_than "(nat*nat)set"}\\ |
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407 @{const Wellfounded.pred_nat} & @{term_type_only Wellfounded.pred_nat "(nat*nat)set"}\\ |
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408 \end{supertabular} |
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409 |
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410 |
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411 \section{SetInterval} |
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412 |
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413 \begin{supertabular}{@ {} l @ {~::~} l @ {}} |
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414 @{const lessThan} & @{term_type_only lessThan "'a::ord \<Rightarrow> 'a set"}\\ |
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415 @{const atMost} & @{term_type_only atMost "'a::ord \<Rightarrow> 'a set"}\\ |
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416 @{const greaterThan} & @{term_type_only greaterThan "'a::ord \<Rightarrow> 'a set"}\\ |
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417 @{const atLeast} & @{term_type_only atLeast "'a::ord \<Rightarrow> 'a set"}\\ |
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418 @{const greaterThanLessThan} & @{term_type_only greaterThanLessThan "'a::ord \<Rightarrow> 'a \<Rightarrow> 'a set"}\\ |
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419 @{const atLeastLessThan} & @{term_type_only atLeastLessThan "'a::ord \<Rightarrow> 'a \<Rightarrow> 'a set"}\\ |
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420 @{const greaterThanAtMost} & @{term_type_only greaterThanAtMost "'a::ord \<Rightarrow> 'a \<Rightarrow> 'a set"}\\ |
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421 @{const atLeastAtMost} & @{term_type_only atLeastAtMost "'a::ord \<Rightarrow> 'a \<Rightarrow> 'a set"}\\ |
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422 \end{supertabular} |
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423 |
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424 \subsubsection*{Syntax} |
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425 |
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426 \begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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427 @{term "lessThan y"} & @{term[source] "lessThan y"}\\ |
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428 @{term "atMost y"} & @{term[source] "atMost y"}\\ |
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429 @{term "greaterThan x"} & @{term[source] "greaterThan x"}\\ |
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430 @{term "atLeast x"} & @{term[source] "atLeast x"}\\ |
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431 @{term "greaterThanLessThan x y"} & @{term[source] "greaterThanLessThan x y"}\\ |
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432 @{term "atLeastLessThan x y"} & @{term[source] "atLeastLessThan x y"}\\ |
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433 @{term "greaterThanAtMost x y"} & @{term[source] "greaterThanAtMost x y"}\\ |
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434 @{term "atLeastAtMost x y"} & @{term[source] "atLeastAtMost x y"}\\ |
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435 @{term[mode=xsymbols] "UN i:{..n}. A"} & @{term[source] "\<Union> i \<in> {..n}. A"}\\ |
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436 @{term[mode=xsymbols] "UN i:{..<n}. A"} & @{term[source] "\<Union> i \<in> {..<n}. A"}\\ |
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437 \multicolumn{2}{@ {}l@ {}}{Similarly for @{text"\<Inter>"} instead of @{text"\<Union>"}}\\ |
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438 @{term "setsum (%x. t) {a..b}"} & @{term[source] "setsum (\<lambda>x. t) {a..b}"}\\ |
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439 @{term "setsum (%x. t) {a..<b}"} & @{term[source] "setsum (\<lambda>x. t) {a..<b}"}\\ |
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440 @{term "setsum (%x. t) {..b}"} & @{term[source] "setsum (\<lambda>x. t) {..b}"}\\ |
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441 @{term "setsum (%x. t) {..<b}"} & @{term[source] "setsum (\<lambda>x. t) {..<b}"}\\ |
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442 \multicolumn{2}{@ {}l@ {}}{Similarly for @{text"\<Prod>"} instead of @{text"\<Sum>"}}\\ |
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443 \end{supertabular} |
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444 |
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445 |
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446 \section{Power} |
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447 |
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448 \begin{tabular}{@ {} l @ {~::~} l @ {}} |
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449 @{const Power.power} & @{typeof Power.power} |
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450 \end{tabular} |
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451 |
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452 |
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453 \section{Iterated Functions and Relations} |
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454 |
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455 Theory: @{theory Relation_Power} |
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456 |
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457 Iterated functions \ @{term[source]"(f::'a\<Rightarrow>'a) ^ n"} \ |
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458 and relations \ @{term[source]"(r::('a\<times>'a)set) ^ n"}. |
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459 |
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460 |
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461 \section{Option} |
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462 |
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463 @{datatype option} |
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464 \bigskip |
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465 |
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466 \begin{tabular}{@ {} l @ {~::~} l @ {}} |
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467 @{const Option.the} & @{typeof Option.the}\\ |
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468 @{const Option.map} & @{typ[source]"('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b option"}\\ |
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469 @{const Option.set} & @{term_type_only Option.set "'a option \<Rightarrow> 'a set"} |
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470 \end{tabular} |
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471 |
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472 \section{List} |
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473 |
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474 @{datatype list} |
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475 \bigskip |
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476 |
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477 \begin{supertabular}{@ {} l @ {~::~} l @ {}} |
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478 @{const List.append} & @{typeof List.append}\\ |
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479 @{const List.butlast} & @{typeof List.butlast}\\ |
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480 @{const List.concat} & @{typeof List.concat}\\ |
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481 @{const List.distinct} & @{typeof List.distinct}\\ |
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482 @{const List.drop} & @{typeof List.drop}\\ |
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483 @{const List.dropWhile} & @{typeof List.dropWhile}\\ |
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484 @{const List.filter} & @{typeof List.filter}\\ |
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485 @{const List.foldl} & @{typeof List.foldl}\\ |
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486 @{const List.foldr} & @{typeof List.foldr}\\ |
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487 @{const List.hd} & @{typeof List.hd}\\ |
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488 @{const List.last} & @{typeof List.last}\\ |
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489 @{const List.length} & @{typeof List.length}\\ |
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490 @{const List.lenlex} & @{term_type_only List.lenlex "('a*'a)set\<Rightarrow>('a list * 'a list)set"}\\ |
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491 @{const List.lex} & @{term_type_only List.lex "('a*'a)set\<Rightarrow>('a list * 'a list)set"}\\ |
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492 @{const List.lexn} & @{term_type_only List.lexn "('a*'a)set\<Rightarrow>nat\<Rightarrow>('a list * 'a list)set"}\\ |
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493 @{const List.lexord} & @{term_type_only List.lexord "('a*'a)set\<Rightarrow>('a list * 'a list)set"}\\ |
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494 @{const List.listrel} & @{term_type_only List.listrel "('a*'a)set\<Rightarrow>('a list * 'a list)set"}\\ |
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495 @{const List.lists} & @{term_type_only List.lists "'a set\<Rightarrow>'a list set"}\\ |
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496 @{const List.listset} & @{term_type_only List.listset "'a set list \<Rightarrow> 'a list set"}\\ |
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497 @{const List.listsum} & @{typeof List.listsum}\\ |
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498 @{const List.list_all2} & @{typeof List.list_all2}\\ |
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499 @{const List.list_update} & @{typeof List.list_update}\\ |
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500 @{const List.map} & @{typeof List.map}\\ |
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501 @{const List.measures} & @{term_type_only List.measures "('a\<Rightarrow>nat)list\<Rightarrow>('a*'a)set"}\\ |
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502 @{const List.remdups} & @{typeof List.remdups}\\ |
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503 @{const List.removeAll} & @{typeof List.removeAll}\\ |
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504 @{const List.remove1} & @{typeof List.remove1}\\ |
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505 @{const List.replicate} & @{typeof List.replicate}\\ |
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506 @{const List.rev} & @{typeof List.rev}\\ |
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507 @{const List.rotate} & @{typeof List.rotate}\\ |
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508 @{const List.rotate1} & @{typeof List.rotate1}\\ |
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509 @{const List.set} & @{term_type_only List.set "'a list \<Rightarrow> 'a set"}\\ |
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510 @{const List.sort} & @{typeof List.sort}\\ |
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511 @{const List.sorted} & @{typeof List.sorted}\\ |
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512 @{const List.splice} & @{typeof List.splice}\\ |
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513 @{const List.sublist} & @{typeof List.sublist}\\ |
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514 @{const List.take} & @{typeof List.take}\\ |
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515 @{const List.takeWhile} & @{typeof List.takeWhile}\\ |
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516 @{const List.tl} & @{typeof List.tl}\\ |
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517 @{const List.upt} & @{typeof List.upt}\\ |
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518 @{const List.upto} & @{typeof List.upto}\\ |
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519 @{const List.zip} & @{typeof List.zip}\\ |
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520 \end{supertabular} |
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521 |
|
522 \subsubsection*{Syntax} |
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523 |
|
524 \begin{supertabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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525 @{text"[x\<^isub>1,\<dots>,x\<^isub>n]"} & @{text"x\<^isub>1 # \<dots> # x\<^isub>n # []"}\\ |
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526 @{term"[m..<n]"} & @{term[source]"upt m n"}\\ |
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527 @{term"[i..j]"} & @{term[source]"upto i j"}\\ |
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528 @{text"[e. x \<leftarrow> xs]"} & @{term"map (%x. e) xs"}\\ |
|
529 @{term"[x \<leftarrow> xs. b]"} & @{term[source]"filter (\<lambda>x. b) xs"} \\ |
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530 @{term"xs[n := x]"} & @{term[source]"list_update xs n x"}\\ |
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531 @{term"\<Sum>x\<leftarrow>xs. e"} & @{term[source]"listsum (map (\<lambda>x. e) xs)"}\\ |
|
532 \end{supertabular} |
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533 \medskip |
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534 |
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535 List comprehension: @{text"[e. q\<^isub>1, \<dots>, q\<^isub>n]"} where each |
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536 qualifier @{text q\<^isub>i} is either a generator \mbox{@{text"pat \<leftarrow> e"}} or a |
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537 guard, i.e.\ boolean expression. |
|
538 |
|
539 \section{Map} |
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540 |
|
541 Maps model partial functions and are often used as finite tables. However, |
|
542 the domain of a map may be infinite. |
|
543 |
|
544 @{text"'a \<rightharpoonup> 'b = 'a \<Rightarrow> 'b option"} |
|
545 \bigskip |
|
546 |
|
547 \begin{supertabular}{@ {} l @ {~::~} l @ {}} |
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548 @{const Map.empty} & @{typeof Map.empty}\\ |
|
549 @{const Map.map_add} & @{typeof Map.map_add}\\ |
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550 @{const Map.map_comp} & @{typeof Map.map_comp}\\ |
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551 @{const Map.restrict_map} & @{term_type_only Map.restrict_map "('a\<Rightarrow>'b option)\<Rightarrow>'a set\<Rightarrow>('a\<Rightarrow>'b option)"}\\ |
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552 @{const Map.dom} & @{term_type_only Map.dom "('a\<Rightarrow>'b option)\<Rightarrow>'a set"}\\ |
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553 @{const Map.ran} & @{term_type_only Map.ran "('a\<Rightarrow>'b option)\<Rightarrow>'b set"}\\ |
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554 @{const Map.map_le} & @{typeof Map.map_le}\\ |
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555 @{const Map.map_of} & @{typeof Map.map_of}\\ |
|
556 @{const Map.map_upds} & @{typeof Map.map_upds}\\ |
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557 \end{supertabular} |
|
558 |
|
559 \subsubsection*{Syntax} |
|
560 |
|
561 \begin{tabular}{@ {} l @ {\quad$\equiv$\quad} l @ {}} |
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562 @{term"Map.empty"} & @{term"\<lambda>x. None"}\\ |
|
563 @{term"m(x:=Some y)"} & @{term[source]"m(x:=Some y)"}\\ |
|
564 @{text"m(x\<^isub>1\<mapsto>y\<^isub>1,\<dots>,x\<^isub>n\<mapsto>y\<^isub>n)"} & @{text[source]"m(x\<^isub>1\<mapsto>y\<^isub>1)\<dots>(x\<^isub>n\<mapsto>y\<^isub>n)"}\\ |
|
565 @{text"[x\<^isub>1\<mapsto>y\<^isub>1,\<dots>,x\<^isub>n\<mapsto>y\<^isub>n]"} & @{text[source]"Map.empty(x\<^isub>1\<mapsto>y\<^isub>1,\<dots>,x\<^isub>n\<mapsto>y\<^isub>n)"}\\ |
|
566 @{term"map_upds m xs ys"} & @{term[source]"map_upds m xs ys"}\\ |
|
567 \end{tabular} |
|
568 |
|
569 *} |
|
570 (*<*) |
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571 end |
|
572 (*>*) |
|