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1 \chapter{Zermelo-Fraenkel Set Theory} |
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2 \index{set theory|(} |
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3 |
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4 The theory~\thydx{ZF} implements Zermelo-Fraenkel set |
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5 theory~\cite{halmos60,suppes72} as an extension of~\texttt{FOL}, classical |
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6 first-order logic. The theory includes a collection of derived natural |
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7 deduction rules, for use with Isabelle's classical reasoner. Some |
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8 of it is based on the work of No\"el~\cite{noel}. |
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9 |
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10 A tremendous amount of set theory has been formally developed, including the |
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11 basic properties of relations, functions, ordinals and cardinals. Significant |
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12 results have been proved, such as the Schr\"oder-Bernstein Theorem, the |
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13 Wellordering Theorem and a version of Ramsey's Theorem. \texttt{ZF} provides |
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14 both the integers and the natural numbers. General methods have been |
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15 developed for solving recursion equations over monotonic functors; these have |
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16 been applied to yield constructions of lists, trees, infinite lists, etc. |
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17 |
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18 \texttt{ZF} has a flexible package for handling inductive definitions, |
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19 such as inference systems, and datatype definitions, such as lists and |
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20 trees. Moreover it handles coinductive definitions, such as |
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21 bisimulation relations, and codatatype definitions, such as streams. It |
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22 provides a streamlined syntax for defining primitive recursive functions over |
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23 datatypes. |
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24 |
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25 Published articles~\cite{paulson-set-I,paulson-set-II} describe \texttt{ZF} |
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26 less formally than this chapter. Isabelle employs a novel treatment of |
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27 non-well-founded data structures within the standard {\sc zf} axioms including |
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28 the Axiom of Foundation~\cite{paulson-mscs}. |
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29 |
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30 |
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31 \section{Which version of axiomatic set theory?} |
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32 The two main axiom systems for set theory are Bernays-G\"odel~({\sc bg}) |
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33 and Zermelo-Fraenkel~({\sc zf}). Resolution theorem provers can use {\sc |
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34 bg} because it is finite~\cite{boyer86,quaife92}. {\sc zf} does not |
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35 have a finite axiom system because of its Axiom Scheme of Replacement. |
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36 This makes it awkward to use with many theorem provers, since instances |
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37 of the axiom scheme have to be invoked explicitly. Since Isabelle has no |
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38 difficulty with axiom schemes, we may adopt either axiom system. |
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39 |
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40 These two theories differ in their treatment of {\bf classes}, which are |
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41 collections that are `too big' to be sets. The class of all sets,~$V$, |
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42 cannot be a set without admitting Russell's Paradox. In {\sc bg}, both |
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43 classes and sets are individuals; $x\in V$ expresses that $x$ is a set. In |
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44 {\sc zf}, all variables denote sets; classes are identified with unary |
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45 predicates. The two systems define essentially the same sets and classes, |
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46 with similar properties. In particular, a class cannot belong to another |
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47 class (let alone a set). |
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48 |
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49 Modern set theorists tend to prefer {\sc zf} because they are mainly concerned |
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50 with sets, rather than classes. {\sc bg} requires tiresome proofs that various |
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51 collections are sets; for instance, showing $x\in\{x\}$ requires showing that |
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52 $x$ is a set. |
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53 |
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54 |
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55 \begin{figure} \small |
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56 \begin{center} |
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57 \begin{tabular}{rrr} |
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58 \it name &\it meta-type & \it description \\ |
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59 \cdx{Let} & $[\alpha,\alpha\To\beta]\To\beta$ & let binder\\ |
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60 \cdx{0} & $i$ & empty set\\ |
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61 \cdx{cons} & $[i,i]\To i$ & finite set constructor\\ |
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62 \cdx{Upair} & $[i,i]\To i$ & unordered pairing\\ |
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63 \cdx{Pair} & $[i,i]\To i$ & ordered pairing\\ |
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64 \cdx{Inf} & $i$ & infinite set\\ |
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65 \cdx{Pow} & $i\To i$ & powerset\\ |
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66 \cdx{Union} \cdx{Inter} & $i\To i$ & set union/intersection \\ |
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67 \cdx{split} & $[[i,i]\To i, i] \To i$ & generalized projection\\ |
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68 \cdx{fst} \cdx{snd} & $i\To i$ & projections\\ |
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69 \cdx{converse}& $i\To i$ & converse of a relation\\ |
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70 \cdx{succ} & $i\To i$ & successor\\ |
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71 \cdx{Collect} & $[i,i\To o]\To i$ & separation\\ |
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72 \cdx{Replace} & $[i, [i,i]\To o] \To i$ & replacement\\ |
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73 \cdx{PrimReplace} & $[i, [i,i]\To o] \To i$ & primitive replacement\\ |
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74 \cdx{RepFun} & $[i, i\To i] \To i$ & functional replacement\\ |
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75 \cdx{Pi} \cdx{Sigma} & $[i,i\To i]\To i$ & general product/sum\\ |
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76 \cdx{domain} & $i\To i$ & domain of a relation\\ |
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77 \cdx{range} & $i\To i$ & range of a relation\\ |
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78 \cdx{field} & $i\To i$ & field of a relation\\ |
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79 \cdx{Lambda} & $[i, i\To i]\To i$ & $\lambda$-abstraction\\ |
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80 \cdx{restrict}& $[i, i] \To i$ & restriction of a function\\ |
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81 \cdx{The} & $[i\To o]\To i$ & definite description\\ |
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82 \cdx{if} & $[o,i,i]\To i$ & conditional\\ |
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83 \cdx{Ball} \cdx{Bex} & $[i, i\To o]\To o$ & bounded quantifiers |
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84 \end{tabular} |
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85 \end{center} |
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86 \subcaption{Constants} |
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87 |
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88 \begin{center} |
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89 \index{*"`"` symbol} |
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90 \index{*"-"`"` symbol} |
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91 \index{*"` symbol}\index{function applications} |
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92 \index{*"- symbol} |
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93 \index{*": symbol} |
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94 \index{*"<"= symbol} |
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95 \begin{tabular}{rrrr} |
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96 \it symbol & \it meta-type & \it priority & \it description \\ |
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97 \tt `` & $[i,i]\To i$ & Left 90 & image \\ |
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98 \tt -`` & $[i,i]\To i$ & Left 90 & inverse image \\ |
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99 \tt ` & $[i,i]\To i$ & Left 90 & application \\ |
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100 \sdx{Int} & $[i,i]\To i$ & Left 70 & intersection ($\int$) \\ |
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101 \sdx{Un} & $[i,i]\To i$ & Left 65 & union ($\un$) \\ |
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102 \tt - & $[i,i]\To i$ & Left 65 & set difference ($-$) \\[1ex] |
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103 \tt: & $[i,i]\To o$ & Left 50 & membership ($\in$) \\ |
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104 \tt <= & $[i,i]\To o$ & Left 50 & subset ($\subseteq$) |
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105 \end{tabular} |
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106 \end{center} |
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107 \subcaption{Infixes} |
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108 \caption{Constants of ZF} \label{zf-constants} |
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109 \end{figure} |
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110 |
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111 |
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112 \section{The syntax of set theory} |
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113 The language of set theory, as studied by logicians, has no constants. The |
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114 traditional axioms merely assert the existence of empty sets, unions, |
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115 powersets, etc.; this would be intolerable for practical reasoning. The |
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116 Isabelle theory declares constants for primitive sets. It also extends |
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117 \texttt{FOL} with additional syntax for finite sets, ordered pairs, |
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118 comprehension, general union/intersection, general sums/products, and |
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119 bounded quantifiers. In most other respects, Isabelle implements precisely |
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120 Zermelo-Fraenkel set theory. |
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121 |
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122 Figure~\ref{zf-constants} lists the constants and infixes of~ZF, while |
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123 Figure~\ref{zf-trans} presents the syntax translations. Finally, |
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124 Figure~\ref{zf-syntax} presents the full grammar for set theory, including the |
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125 constructs of FOL. |
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126 |
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127 Local abbreviations can be introduced by a \isa{let} construct whose |
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128 syntax appears in Fig.\ts\ref{zf-syntax}. Internally it is translated into |
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129 the constant~\cdx{Let}. It can be expanded by rewriting with its |
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130 definition, \tdx{Let_def}. |
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131 |
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132 Apart from \isa{let}, set theory does not use polymorphism. All terms in |
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133 ZF have type~\tydx{i}, which is the type of individuals and has |
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134 class~\cldx{term}. The type of first-order formulae, remember, |
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135 is~\tydx{o}. |
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136 |
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137 Infix operators include binary union and intersection ($A\un B$ and |
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138 $A\int B$), set difference ($A-B$), and the subset and membership |
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139 relations. Note that $a$\verb|~:|$b$ is translated to $\lnot(a\in b)$, |
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140 which is equivalent to $a\notin b$. The |
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141 union and intersection operators ($\bigcup A$ and $\bigcap A$) form the |
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142 union or intersection of a set of sets; $\bigcup A$ means the same as |
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143 $\bigcup@{x\in A}x$. Of these operators, only $\bigcup A$ is primitive. |
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144 |
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145 The constant \cdx{Upair} constructs unordered pairs; thus \isa{Upair($A$,$B$)} denotes the set~$\{A,B\}$ and |
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146 \isa{Upair($A$,$A$)} denotes the singleton~$\{A\}$. General union is |
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147 used to define binary union. The Isabelle version goes on to define |
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148 the constant |
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149 \cdx{cons}: |
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150 \begin{eqnarray*} |
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151 A\cup B & \equiv & \bigcup(\isa{Upair}(A,B)) \\ |
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152 \isa{cons}(a,B) & \equiv & \isa{Upair}(a,a) \un B |
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153 \end{eqnarray*} |
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154 The $\{a@1, \ldots\}$ notation abbreviates finite sets constructed in the |
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155 obvious manner using~\isa{cons} and~$\emptyset$ (the empty set) \isasymin \begin{eqnarray*} |
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156 \{a,b,c\} & \equiv & \isa{cons}(a,\isa{cons}(b,\isa{cons}(c,\emptyset))) |
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157 \end{eqnarray*} |
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158 |
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159 The constant \cdx{Pair} constructs ordered pairs, as in \isa{Pair($a$,$b$)}. Ordered pairs may also be written within angle brackets, |
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160 as {\tt<$a$,$b$>}. The $n$-tuple {\tt<$a@1$,\ldots,$a@{n-1}$,$a@n$>} |
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161 abbreviates the nest of pairs\par\nobreak |
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162 \centerline{\isa{Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots).}} |
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163 |
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164 In ZF, a function is a set of pairs. A ZF function~$f$ is simply an |
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165 individual as far as Isabelle is concerned: its Isabelle type is~$i$, not say |
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166 $i\To i$. The infix operator~{\tt`} denotes the application of a function set |
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167 to its argument; we must write~$f{\tt`}x$, not~$f(x)$. The syntax for image |
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168 is~$f{\tt``}A$ and that for inverse image is~$f{\tt-``}A$. |
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169 |
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170 |
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171 \begin{figure} |
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172 \index{lambda abs@$\lambda$-abstractions} |
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173 \index{*"-"> symbol} |
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174 \index{*"* symbol} |
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175 \begin{center} \footnotesize\tt\frenchspacing |
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176 \begin{tabular}{rrr} |
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177 \it external & \it internal & \it description \\ |
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178 $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm negated membership\\ |
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179 \ttlbrace$a@1$, $\ldots$, $a@n$\ttrbrace & cons($a@1$,$\ldots$,cons($a@n$,0)) & |
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180 \rm finite set \\ |
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181 <$a@1$, $\ldots$, $a@{n-1}$, $a@n$> & |
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182 Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots) & |
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183 \rm ordered $n$-tuple \\ |
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184 \ttlbrace$x$:$A . P[x]$\ttrbrace & Collect($A$,$\lambda x. P[x]$) & |
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185 \rm separation \\ |
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186 \ttlbrace$y . x$:$A$, $Q[x,y]$\ttrbrace & Replace($A$,$\lambda x\,y. Q[x,y]$) & |
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187 \rm replacement \\ |
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188 \ttlbrace$b[x] . x$:$A$\ttrbrace & RepFun($A$,$\lambda x. b[x]$) & |
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189 \rm functional replacement \\ |
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190 \sdx{INT} $x$:$A . B[x]$ & Inter(\ttlbrace$B[x] . x$:$A$\ttrbrace) & |
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191 \rm general intersection \\ |
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192 \sdx{UN} $x$:$A . B[x]$ & Union(\ttlbrace$B[x] . x$:$A$\ttrbrace) & |
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193 \rm general union \\ |
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194 \sdx{PROD} $x$:$A . B[x]$ & Pi($A$,$\lambda x. B[x]$) & |
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195 \rm general product \\ |
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196 \sdx{SUM} $x$:$A . B[x]$ & Sigma($A$,$\lambda x. B[x]$) & |
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197 \rm general sum \\ |
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198 $A$ -> $B$ & Pi($A$,$\lambda x. B$) & |
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199 \rm function space \\ |
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200 $A$ * $B$ & Sigma($A$,$\lambda x. B$) & |
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201 \rm binary product \\ |
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202 \sdx{THE} $x . P[x]$ & The($\lambda x. P[x]$) & |
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203 \rm definite description \\ |
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204 \sdx{lam} $x$:$A . b[x]$ & Lambda($A$,$\lambda x. b[x]$) & |
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205 \rm $\lambda$-abstraction\\[1ex] |
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206 \sdx{ALL} $x$:$A . P[x]$ & Ball($A$,$\lambda x. P[x]$) & |
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207 \rm bounded $\forall$ \\ |
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208 \sdx{EX} $x$:$A . P[x]$ & Bex($A$,$\lambda x. P[x]$) & |
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209 \rm bounded $\exists$ |
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210 \end{tabular} |
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211 \end{center} |
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212 \caption{Translations for ZF} \label{zf-trans} |
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213 \end{figure} |
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214 |
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215 |
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216 \begin{figure} |
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217 \index{*let symbol} |
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218 \index{*in symbol} |
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219 \dquotes |
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220 \[\begin{array}{rcl} |
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221 term & = & \hbox{expression of type~$i$} \\ |
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222 & | & "let"~id~"="~term";"\dots";"~id~"="~term~"in"~term \\ |
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223 & | & "if"~term~"then"~term~"else"~term \\ |
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224 & | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\ |
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225 & | & "< " term\; ("," term)^* " >" \\ |
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226 & | & "{\ttlbrace} " id ":" term " . " formula " {\ttrbrace}" \\ |
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227 & | & "{\ttlbrace} " id " . " id ":" term ", " formula " {\ttrbrace}" \\ |
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228 & | & "{\ttlbrace} " term " . " id ":" term " {\ttrbrace}" \\ |
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229 & | & term " `` " term \\ |
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230 & | & term " -`` " term \\ |
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231 & | & term " ` " term \\ |
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232 & | & term " * " term \\ |
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233 & | & term " \isasyminter " term \\ |
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234 & | & term " \isasymunion " term \\ |
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235 & | & term " - " term \\ |
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236 & | & term " -> " term \\ |
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237 & | & "THE~~" id " . " formula\\ |
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238 & | & "lam~~" id ":" term " . " term \\ |
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239 & | & "INT~~" id ":" term " . " term \\ |
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240 & | & "UN~~~" id ":" term " . " term \\ |
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241 & | & "PROD~" id ":" term " . " term \\ |
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242 & | & "SUM~~" id ":" term " . " term \\[2ex] |
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243 formula & = & \hbox{expression of type~$o$} \\ |
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244 & | & term " : " term \\ |
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245 & | & term " \ttilde: " term \\ |
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246 & | & term " <= " term \\ |
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247 & | & term " = " term \\ |
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248 & | & term " \ttilde= " term \\ |
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249 & | & "\ttilde\ " formula \\ |
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250 & | & formula " \& " formula \\ |
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251 & | & formula " | " formula \\ |
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252 & | & formula " --> " formula \\ |
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253 & | & formula " <-> " formula \\ |
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254 & | & "ALL " id ":" term " . " formula \\ |
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255 & | & "EX~~" id ":" term " . " formula \\ |
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256 & | & "ALL~" id~id^* " . " formula \\ |
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257 & | & "EX~~" id~id^* " . " formula \\ |
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258 & | & "EX!~" id~id^* " . " formula |
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259 \end{array} |
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260 \] |
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261 \caption{Full grammar for ZF} \label{zf-syntax} |
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262 \end{figure} |
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263 |
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264 |
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265 \section{Binding operators} |
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266 The constant \cdx{Collect} constructs sets by the principle of {\bf |
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267 separation}. The syntax for separation is |
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268 \hbox{\tt\ttlbrace$x$:$A$.\ $P[x]$\ttrbrace}, where $P[x]$ is a formula |
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269 that may contain free occurrences of~$x$. It abbreviates the set \isa{Collect($A$,$\lambda x. P[x]$)}, which consists of all $x\in A$ that |
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270 satisfy~$P[x]$. Note that \isa{Collect} is an unfortunate choice of |
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271 name: some set theories adopt a set-formation principle, related to |
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272 replacement, called collection. |
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273 |
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274 The constant \cdx{Replace} constructs sets by the principle of {\bf |
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275 replacement}. The syntax |
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276 \hbox{\tt\ttlbrace$y$.\ $x$:$A$,$Q[x,y]$\ttrbrace} denotes the set |
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277 \isa{Replace($A$,$\lambda x\,y. Q[x,y]$)}, which consists of all~$y$ such |
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278 that there exists $x\in A$ satisfying~$Q[x,y]$. The Replacement Axiom |
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279 has the condition that $Q$ must be single-valued over~$A$: for |
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280 all~$x\in A$ there exists at most one $y$ satisfying~$Q[x,y]$. A |
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281 single-valued binary predicate is also called a {\bf class function}. |
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282 |
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283 The constant \cdx{RepFun} expresses a special case of replacement, |
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284 where $Q[x,y]$ has the form $y=b[x]$. Such a $Q$ is trivially |
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285 single-valued, since it is just the graph of the meta-level |
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286 function~$\lambda x. b[x]$. The resulting set consists of all $b[x]$ |
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287 for~$x\in A$. This is analogous to the \ML{} functional \isa{map}, |
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288 since it applies a function to every element of a set. The syntax is |
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289 \isa{\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to |
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290 \isa{RepFun($A$,$\lambda x. b[x]$)}. |
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291 |
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292 \index{*INT symbol}\index{*UN symbol} |
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293 General unions and intersections of indexed |
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294 families of sets, namely $\bigcup@{x\in A}B[x]$ and $\bigcap@{x\in A}B[x]$, |
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295 are written \isa{UN $x$:$A$.\ $B[x]$} and \isa{INT $x$:$A$.\ $B[x]$}. |
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296 Their meaning is expressed using \isa{RepFun} as |
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297 \[ |
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298 \bigcup(\{B[x]. x\in A\}) \qquad\hbox{and}\qquad |
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299 \bigcap(\{B[x]. x\in A\}). |
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300 \] |
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301 General sums $\sum@{x\in A}B[x]$ and products $\prod@{x\in A}B[x]$ can be |
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302 constructed in set theory, where $B[x]$ is a family of sets over~$A$. They |
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303 have as special cases $A\times B$ and $A\to B$, where $B$ is simply a set. |
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304 This is similar to the situation in Constructive Type Theory (set theory |
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305 has `dependent sets') and calls for similar syntactic conventions. The |
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306 constants~\cdx{Sigma} and~\cdx{Pi} construct general sums and |
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307 products. Instead of \isa{Sigma($A$,$B$)} and \isa{Pi($A$,$B$)} we may |
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308 write |
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309 \isa{SUM $x$:$A$.\ $B[x]$} and \isa{PROD $x$:$A$.\ $B[x]$}. |
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310 \index{*SUM symbol}\index{*PROD symbol}% |
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311 The special cases as \hbox{\tt$A$*$B$} and \hbox{\tt$A$->$B$} abbreviate |
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312 general sums and products over a constant family.\footnote{Unlike normal |
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313 infix operators, {\tt*} and {\tt->} merely define abbreviations; there are |
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314 no constants~\isa{op~*} and~\isa{op~->}.} Isabelle accepts these |
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315 abbreviations in parsing and uses them whenever possible for printing. |
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316 |
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317 \index{*THE symbol} As mentioned above, whenever the axioms assert the |
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318 existence and uniqueness of a set, Isabelle's set theory declares a constant |
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319 for that set. These constants can express the {\bf definite description} |
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320 operator~$\iota x. P[x]$, which stands for the unique~$a$ satisfying~$P[a]$, |
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321 if such exists. Since all terms in ZF denote something, a description is |
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322 always meaningful, but we do not know its value unless $P[x]$ defines it |
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323 uniquely. Using the constant~\cdx{The}, we may write descriptions as |
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324 \isa{The($\lambda x. P[x]$)} or use the syntax \isa{THE $x$.\ $P[x]$}. |
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325 |
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326 \index{*lam symbol} |
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327 Function sets may be written in $\lambda$-notation; $\lambda x\in A. b[x]$ |
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328 stands for the set of all pairs $\pair{x,b[x]}$ for $x\in A$. In order for |
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329 this to be a set, the function's domain~$A$ must be given. Using the |
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330 constant~\cdx{Lambda}, we may express function sets as \isa{Lambda($A$,$\lambda x. b[x]$)} or use the syntax \isa{lam $x$:$A$.\ $b[x]$}. |
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331 |
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332 Isabelle's set theory defines two {\bf bounded quantifiers}: |
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333 \begin{eqnarray*} |
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334 \forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\ |
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335 \exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x] |
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336 \end{eqnarray*} |
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337 The constants~\cdx{Ball} and~\cdx{Bex} are defined |
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338 accordingly. Instead of \isa{Ball($A$,$P$)} and \isa{Bex($A$,$P$)} we may |
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339 write |
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340 \isa{ALL $x$:$A$.\ $P[x]$} and \isa{EX $x$:$A$.\ $P[x]$}. |
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341 |
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342 |
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343 %%%% ZF.thy |
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344 |
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345 \begin{figure} |
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346 \begin{alltt*}\isastyleminor |
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347 \tdx{Let_def}: Let(s, f) == f(s) |
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348 |
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349 \tdx{Ball_def}: Ball(A,P) == {\isasymforall}x. x \isasymin A --> P(x) |
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350 \tdx{Bex_def}: Bex(A,P) == {\isasymexists}x. x \isasymin A & P(x) |
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351 |
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352 \tdx{subset_def}: A \isasymsubseteq B == {\isasymforall}x \isasymin A. x \isasymin B |
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353 \tdx{extension}: A = B <-> A \isasymsubseteq B & B \isasymsubseteq A |
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354 |
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355 \tdx{Union_iff}: A \isasymin Union(C) <-> ({\isasymexists}B \isasymin C. A \isasymin B) |
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356 \tdx{Pow_iff}: A \isasymin Pow(B) <-> A \isasymsubseteq B |
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357 \tdx{foundation}: A=0 | ({\isasymexists}x \isasymin A. {\isasymforall}y \isasymin x. y \isasymnotin A) |
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358 |
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359 \tdx{replacement}: ({\isasymforall}x \isasymin A. {\isasymforall}y z. P(x,y) & P(x,z) --> y=z) ==> |
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360 b \isasymin PrimReplace(A,P) <-> ({\isasymexists}x{\isasymin}A. P(x,b)) |
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361 \subcaption{The Zermelo-Fraenkel Axioms} |
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362 |
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363 \tdx{Replace_def}: Replace(A,P) == |
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364 PrimReplace(A, \%x y. (\isasymexists!z. P(x,z)) & P(x,y)) |
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365 \tdx{RepFun_def}: RepFun(A,f) == {\ttlbrace}y . x \isasymin A, y=f(x)\ttrbrace |
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366 \tdx{the_def}: The(P) == Union({\ttlbrace}y . x \isasymin {\ttlbrace}0{\ttrbrace}, P(y){\ttrbrace}) |
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367 \tdx{if_def}: if(P,a,b) == THE z. P & z=a | ~P & z=b |
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368 \tdx{Collect_def}: Collect(A,P) == {\ttlbrace}y . x \isasymin A, x=y & P(x){\ttrbrace} |
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369 \tdx{Upair_def}: Upair(a,b) == |
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370 {\ttlbrace}y. x\isasymin{}Pow(Pow(0)), x=0 & y=a | x=Pow(0) & y=b{\ttrbrace} |
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371 \subcaption{Consequences of replacement} |
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372 |
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373 \tdx{Inter_def}: Inter(A) == {\ttlbrace}x \isasymin Union(A) . {\isasymforall}y \isasymin A. x \isasymin y{\ttrbrace} |
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374 \tdx{Un_def}: A \isasymunion B == Union(Upair(A,B)) |
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375 \tdx{Int_def}: A \isasyminter B == Inter(Upair(A,B)) |
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376 \tdx{Diff_def}: A - B == {\ttlbrace}x \isasymin A . x \isasymnotin B{\ttrbrace} |
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377 \subcaption{Union, intersection, difference} |
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378 \end{alltt*} |
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379 \caption{Rules and axioms of ZF} \label{zf-rules} |
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380 \end{figure} |
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381 |
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382 |
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383 \begin{figure} |
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384 \begin{alltt*}\isastyleminor |
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385 \tdx{cons_def}: cons(a,A) == Upair(a,a) \isasymunion A |
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386 \tdx{succ_def}: succ(i) == cons(i,i) |
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387 \tdx{infinity}: 0 \isasymin Inf & ({\isasymforall}y \isasymin Inf. succ(y) \isasymin Inf) |
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388 \subcaption{Finite and infinite sets} |
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389 |
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390 \tdx{Pair_def}: <a,b> == {\ttlbrace}{\ttlbrace}a,a{\ttrbrace}, {\ttlbrace}a,b{\ttrbrace}{\ttrbrace} |
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391 \tdx{split_def}: split(c,p) == THE y. {\isasymexists}a b. p=<a,b> & y=c(a,b) |
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392 \tdx{fst_def}: fst(A) == split(\%x y. x, p) |
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393 \tdx{snd_def}: snd(A) == split(\%x y. y, p) |
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394 \tdx{Sigma_def}: Sigma(A,B) == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x,y>{\ttrbrace} |
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395 \subcaption{Ordered pairs and Cartesian products} |
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396 |
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397 \tdx{converse_def}: converse(r) == {\ttlbrace}z. w\isasymin{}r, {\isasymexists}x y. w=<x,y> & z=<y,x>{\ttrbrace} |
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398 \tdx{domain_def}: domain(r) == {\ttlbrace}x. w \isasymin r, {\isasymexists}y. w=<x,y>{\ttrbrace} |
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399 \tdx{range_def}: range(r) == domain(converse(r)) |
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400 \tdx{field_def}: field(r) == domain(r) \isasymunion range(r) |
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401 \tdx{image_def}: r `` A == {\ttlbrace}y\isasymin{}range(r) . {\isasymexists}x \isasymin A. <x,y> \isasymin r{\ttrbrace} |
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402 \tdx{vimage_def}: r -`` A == converse(r)``A |
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403 \subcaption{Operations on relations} |
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404 |
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405 \tdx{lam_def}: Lambda(A,b) == {\ttlbrace}<x,b(x)> . x \isasymin A{\ttrbrace} |
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406 \tdx{apply_def}: f`a == THE y. <a,y> \isasymin f |
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407 \tdx{Pi_def}: Pi(A,B) == {\ttlbrace}f\isasymin{}Pow(Sigma(A,B)). {\isasymforall}x\isasymin{}A. \isasymexists!y. <x,y>\isasymin{}f{\ttrbrace} |
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408 \tdx{restrict_def}: restrict(f,A) == lam x \isasymin A. f`x |
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409 \subcaption{Functions and general product} |
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410 \end{alltt*} |
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411 \caption{Further definitions of ZF} \label{zf-defs} |
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412 \end{figure} |
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413 |
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414 |
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415 |
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416 \section{The Zermelo-Fraenkel axioms} |
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417 The axioms appear in Fig.\ts \ref{zf-rules}. They resemble those |
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418 presented by Suppes~\cite{suppes72}. Most of the theory consists of |
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419 definitions. In particular, bounded quantifiers and the subset relation |
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420 appear in other axioms. Object-level quantifiers and implications have |
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421 been replaced by meta-level ones wherever possible, to simplify use of the |
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422 axioms. |
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423 |
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424 The traditional replacement axiom asserts |
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425 \[ y \in \isa{PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y)) \] |
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426 subject to the condition that $P(x,y)$ is single-valued for all~$x\in A$. |
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427 The Isabelle theory defines \cdx{Replace} to apply |
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428 \cdx{PrimReplace} to the single-valued part of~$P$, namely |
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429 \[ (\exists!z. P(x,z)) \conj P(x,y). \] |
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430 Thus $y\in \isa{Replace}(A,P)$ if and only if there is some~$x$ such that |
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431 $P(x,-)$ holds uniquely for~$y$. Because the equivalence is unconditional, |
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432 \isa{Replace} is much easier to use than \isa{PrimReplace}; it defines the |
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433 same set, if $P(x,y)$ is single-valued. The nice syntax for replacement |
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434 expands to \isa{Replace}. |
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435 |
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436 Other consequences of replacement include replacement for |
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437 meta-level functions |
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438 (\cdx{RepFun}) and definite descriptions (\cdx{The}). |
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439 Axioms for separation (\cdx{Collect}) and unordered pairs |
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440 (\cdx{Upair}) are traditionally assumed, but they actually follow |
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441 from replacement~\cite[pages 237--8]{suppes72}. |
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442 |
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443 The definitions of general intersection, etc., are straightforward. Note |
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444 the definition of \isa{cons}, which underlies the finite set notation. |
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445 The axiom of infinity gives us a set that contains~0 and is closed under |
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446 successor (\cdx{succ}). Although this set is not uniquely defined, |
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447 the theory names it (\cdx{Inf}) in order to simplify the |
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448 construction of the natural numbers. |
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449 |
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450 Further definitions appear in Fig.\ts\ref{zf-defs}. Ordered pairs are |
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451 defined in the standard way, $\pair{a,b}\equiv\{\{a\},\{a,b\}\}$. Recall |
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452 that \cdx{Sigma}$(A,B)$ generalizes the Cartesian product of two |
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453 sets. It is defined to be the union of all singleton sets |
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454 $\{\pair{x,y}\}$, for $x\in A$ and $y\in B(x)$. This is a typical usage of |
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455 general union. |
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456 |
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457 The projections \cdx{fst} and~\cdx{snd} are defined in terms of the |
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458 generalized projection \cdx{split}. The latter has been borrowed from |
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459 Martin-L\"of's Type Theory, and is often easier to use than \cdx{fst} |
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460 and~\cdx{snd}. |
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461 |
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462 Operations on relations include converse, domain, range, and image. The |
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463 set $\isa{Pi}(A,B)$ generalizes the space of functions between two sets. |
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464 Note the simple definitions of $\lambda$-abstraction (using |
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465 \cdx{RepFun}) and application (using a definite description). The |
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466 function \cdx{restrict}$(f,A)$ has the same values as~$f$, but only |
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467 over the domain~$A$. |
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468 |
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469 |
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470 %%%% zf.thy |
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471 |
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472 \begin{figure} |
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473 \begin{alltt*}\isastyleminor |
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474 \tdx{ballI}: [| !!x. x\isasymin{}A ==> P(x) |] ==> {\isasymforall}x\isasymin{}A. P(x) |
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475 \tdx{bspec}: [| {\isasymforall}x\isasymin{}A. P(x); x\isasymin{}A |] ==> P(x) |
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476 \tdx{ballE}: [| {\isasymforall}x\isasymin{}A. P(x); P(x) ==> Q; x \isasymnotin A ==> Q |] ==> Q |
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477 |
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478 \tdx{ball_cong}: [| A=A'; !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==> |
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479 ({\isasymforall}x\isasymin{}A. P(x)) <-> ({\isasymforall}x\isasymin{}A'. P'(x)) |
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480 |
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481 \tdx{bexI}: [| P(x); x\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x) |
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482 \tdx{bexCI}: [| {\isasymforall}x\isasymin{}A. ~P(x) ==> P(a); a\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x) |
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483 \tdx{bexE}: [| {\isasymexists}x\isasymin{}A. P(x); !!x. [| x\isasymin{}A; P(x) |] ==> Q |] ==> Q |
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484 |
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485 \tdx{bex_cong}: [| A=A'; !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==> |
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486 ({\isasymexists}x\isasymin{}A. P(x)) <-> ({\isasymexists}x\isasymin{}A'. P'(x)) |
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487 \subcaption{Bounded quantifiers} |
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488 |
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489 \tdx{subsetI}: (!!x. x \isasymin A ==> x \isasymin B) ==> A \isasymsubseteq B |
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490 \tdx{subsetD}: [| A \isasymsubseteq B; c \isasymin A |] ==> c \isasymin B |
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491 \tdx{subsetCE}: [| A \isasymsubseteq B; c \isasymnotin A ==> P; c \isasymin B ==> P |] ==> P |
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492 \tdx{subset_refl}: A \isasymsubseteq A |
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493 \tdx{subset_trans}: [| A \isasymsubseteq B; B \isasymsubseteq C |] ==> A \isasymsubseteq C |
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494 |
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495 \tdx{equalityI}: [| A \isasymsubseteq B; B \isasymsubseteq A |] ==> A = B |
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496 \tdx{equalityD1}: A = B ==> A \isasymsubseteq B |
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497 \tdx{equalityD2}: A = B ==> B \isasymsubseteq A |
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498 \tdx{equalityE}: [| A = B; [| A \isasymsubseteq B; B \isasymsubseteq A |] ==> P |] ==> P |
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499 \subcaption{Subsets and extensionality} |
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500 |
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501 \tdx{emptyE}: a \isasymin 0 ==> P |
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502 \tdx{empty_subsetI}: 0 \isasymsubseteq A |
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503 \tdx{equals0I}: [| !!y. y \isasymin A ==> False |] ==> A=0 |
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504 \tdx{equals0D}: [| A=0; a \isasymin A |] ==> P |
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505 |
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506 \tdx{PowI}: A \isasymsubseteq B ==> A \isasymin Pow(B) |
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507 \tdx{PowD}: A \isasymin Pow(B) ==> A \isasymsubseteq B |
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508 \subcaption{The empty set; power sets} |
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509 \end{alltt*} |
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510 \caption{Basic derived rules for ZF} \label{zf-lemmas1} |
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511 \end{figure} |
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512 |
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513 |
|
514 \section{From basic lemmas to function spaces} |
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515 Faced with so many definitions, it is essential to prove lemmas. Even |
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516 trivial theorems like $A \int B = B \int A$ would be difficult to |
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517 prove from the definitions alone. Isabelle's set theory derives many |
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518 rules using a natural deduction style. Ideally, a natural deduction |
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519 rule should introduce or eliminate just one operator, but this is not |
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520 always practical. For most operators, we may forget its definition |
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521 and use its derived rules instead. |
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522 |
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523 \subsection{Fundamental lemmas} |
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524 Figure~\ref{zf-lemmas1} presents the derived rules for the most basic |
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525 operators. The rules for the bounded quantifiers resemble those for the |
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526 ordinary quantifiers, but note that \tdx{ballE} uses a negated assumption |
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527 in the style of Isabelle's classical reasoner. The \rmindex{congruence |
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528 rules} \tdx{ball_cong} and \tdx{bex_cong} are required by Isabelle's |
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529 simplifier, but have few other uses. Congruence rules must be specially |
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530 derived for all binding operators, and henceforth will not be shown. |
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531 |
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532 Figure~\ref{zf-lemmas1} also shows rules for the subset and equality |
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533 relations (proof by extensionality), and rules about the empty set and the |
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534 power set operator. |
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535 |
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536 Figure~\ref{zf-lemmas2} presents rules for replacement and separation. |
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537 The rules for \cdx{Replace} and \cdx{RepFun} are much simpler than |
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538 comparable rules for \isa{PrimReplace} would be. The principle of |
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539 separation is proved explicitly, although most proofs should use the |
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540 natural deduction rules for \isa{Collect}. The elimination rule |
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541 \tdx{CollectE} is equivalent to the two destruction rules |
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542 \tdx{CollectD1} and \tdx{CollectD2}, but each rule is suited to |
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543 particular circumstances. Although too many rules can be confusing, there |
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544 is no reason to aim for a minimal set of rules. |
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545 |
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546 Figure~\ref{zf-lemmas3} presents rules for general union and intersection. |
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547 The empty intersection should be undefined. We cannot have |
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548 $\bigcap(\emptyset)=V$ because $V$, the universal class, is not a set. All |
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549 expressions denote something in ZF set theory; the definition of |
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550 intersection implies $\bigcap(\emptyset)=\emptyset$, but this value is |
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551 arbitrary. The rule \tdx{InterI} must have a premise to exclude |
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552 the empty intersection. Some of the laws governing intersections require |
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553 similar premises. |
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554 |
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555 |
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556 %the [p] gives better page breaking for the book |
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557 \begin{figure}[p] |
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558 \begin{alltt*}\isastyleminor |
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559 \tdx{ReplaceI}: [| x\isasymin{}A; P(x,b); !!y. P(x,y) ==> y=b |] ==> |
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560 b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace} |
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561 |
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562 \tdx{ReplaceE}: [| b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace}; |
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563 !!x. [| x\isasymin{}A; P(x,b); {\isasymforall}y. P(x,y)-->y=b |] ==> R |
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564 |] ==> R |
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565 |
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566 \tdx{RepFunI}: [| a\isasymin{}A |] ==> f(a)\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace} |
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567 \tdx{RepFunE}: [| b\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace}; |
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568 !!x.[| x\isasymin{}A; b=f(x) |] ==> P |] ==> P |
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569 |
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570 \tdx{separation}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} <-> a\isasymin{}A & P(a) |
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571 \tdx{CollectI}: [| a\isasymin{}A; P(a) |] ==> a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} |
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572 \tdx{CollectE}: [| a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace}; [| a\isasymin{}A; P(a) |] ==> R |] ==> R |
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573 \tdx{CollectD1}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> a\isasymin{}A |
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574 \tdx{CollectD2}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> P(a) |
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575 \end{alltt*} |
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576 \caption{Replacement and separation} \label{zf-lemmas2} |
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577 \end{figure} |
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578 |
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579 |
|
580 \begin{figure} |
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581 \begin{alltt*}\isastyleminor |
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582 \tdx{UnionI}: [| B\isasymin{}C; A\isasymin{}B |] ==> A\isasymin{}Union(C) |
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583 \tdx{UnionE}: [| A\isasymin{}Union(C); !!B.[| A\isasymin{}B; B\isasymin{}C |] ==> R |] ==> R |
|
584 |
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585 \tdx{InterI}: [| !!x. x\isasymin{}C ==> A\isasymin{}x; c\isasymin{}C |] ==> A\isasymin{}Inter(C) |
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586 \tdx{InterD}: [| A\isasymin{}Inter(C); B\isasymin{}C |] ==> A\isasymin{}B |
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587 \tdx{InterE}: [| A\isasymin{}Inter(C); A\isasymin{}B ==> R; B \isasymnotin C ==> R |] ==> R |
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588 |
|
589 \tdx{UN_I}: [| a\isasymin{}A; b\isasymin{}B(a) |] ==> b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x)) |
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590 \tdx{UN_E}: [| b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x)); !!x.[| x\isasymin{}A; b\isasymin{}B(x) |] ==> R |
|
591 |] ==> R |
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592 |
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593 \tdx{INT_I}: [| !!x. x\isasymin{}A ==> b\isasymin{}B(x); a\isasymin{}A |] ==> b\isasymin{}({\isasymInter}x\isasymin{}A. B(x)) |
|
594 \tdx{INT_E}: [| b\isasymin{}({\isasymInter}x\isasymin{}A. B(x)); a\isasymin{}A |] ==> b\isasymin{}B(a) |
|
595 \end{alltt*} |
|
596 \caption{General union and intersection} \label{zf-lemmas3} |
|
597 \end{figure} |
|
598 |
|
599 |
|
600 %%% upair.thy |
|
601 |
|
602 \begin{figure} |
|
603 \begin{alltt*}\isastyleminor |
|
604 \tdx{pairing}: a\isasymin{}Upair(b,c) <-> (a=b | a=c) |
|
605 \tdx{UpairI1}: a\isasymin{}Upair(a,b) |
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606 \tdx{UpairI2}: b\isasymin{}Upair(a,b) |
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607 \tdx{UpairE}: [| a\isasymin{}Upair(b,c); a=b ==> P; a=c ==> P |] ==> P |
|
608 \end{alltt*} |
|
609 \caption{Unordered pairs} \label{zf-upair1} |
|
610 \end{figure} |
|
611 |
|
612 |
|
613 \begin{figure} |
|
614 \begin{alltt*}\isastyleminor |
|
615 \tdx{UnI1}: c\isasymin{}A ==> c\isasymin{}A \isasymunion B |
|
616 \tdx{UnI2}: c\isasymin{}B ==> c\isasymin{}A \isasymunion B |
|
617 \tdx{UnCI}: (c \isasymnotin B ==> c\isasymin{}A) ==> c\isasymin{}A \isasymunion B |
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618 \tdx{UnE}: [| c\isasymin{}A \isasymunion B; c\isasymin{}A ==> P; c\isasymin{}B ==> P |] ==> P |
|
619 |
|
620 \tdx{IntI}: [| c\isasymin{}A; c\isasymin{}B |] ==> c\isasymin{}A \isasyminter B |
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621 \tdx{IntD1}: c\isasymin{}A \isasyminter B ==> c\isasymin{}A |
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622 \tdx{IntD2}: c\isasymin{}A \isasyminter B ==> c\isasymin{}B |
|
623 \tdx{IntE}: [| c\isasymin{}A \isasyminter B; [| c\isasymin{}A; c\isasymin{}B |] ==> P |] ==> P |
|
624 |
|
625 \tdx{DiffI}: [| c\isasymin{}A; c \isasymnotin B |] ==> c\isasymin{}A - B |
|
626 \tdx{DiffD1}: c\isasymin{}A - B ==> c\isasymin{}A |
|
627 \tdx{DiffD2}: c\isasymin{}A - B ==> c \isasymnotin B |
|
628 \tdx{DiffE}: [| c\isasymin{}A - B; [| c\isasymin{}A; c \isasymnotin B |] ==> P |] ==> P |
|
629 \end{alltt*} |
|
630 \caption{Union, intersection, difference} \label{zf-Un} |
|
631 \end{figure} |
|
632 |
|
633 |
|
634 \begin{figure} |
|
635 \begin{alltt*}\isastyleminor |
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636 \tdx{consI1}: a\isasymin{}cons(a,B) |
|
637 \tdx{consI2}: a\isasymin{}B ==> a\isasymin{}cons(b,B) |
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638 \tdx{consCI}: (a \isasymnotin B ==> a=b) ==> a\isasymin{}cons(b,B) |
|
639 \tdx{consE}: [| a\isasymin{}cons(b,A); a=b ==> P; a\isasymin{}A ==> P |] ==> P |
|
640 |
|
641 \tdx{singletonI}: a\isasymin{}{\ttlbrace}a{\ttrbrace} |
|
642 \tdx{singletonE}: [| a\isasymin{}{\ttlbrace}b{\ttrbrace}; a=b ==> P |] ==> P |
|
643 \end{alltt*} |
|
644 \caption{Finite and singleton sets} \label{zf-upair2} |
|
645 \end{figure} |
|
646 |
|
647 |
|
648 \begin{figure} |
|
649 \begin{alltt*}\isastyleminor |
|
650 \tdx{succI1}: i\isasymin{}succ(i) |
|
651 \tdx{succI2}: i\isasymin{}j ==> i\isasymin{}succ(j) |
|
652 \tdx{succCI}: (i \isasymnotin j ==> i=j) ==> i\isasymin{}succ(j) |
|
653 \tdx{succE}: [| i\isasymin{}succ(j); i=j ==> P; i\isasymin{}j ==> P |] ==> P |
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654 \tdx{succ_neq_0}: [| succ(n)=0 |] ==> P |
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655 \tdx{succ_inject}: succ(m) = succ(n) ==> m=n |
|
656 \end{alltt*} |
|
657 \caption{The successor function} \label{zf-succ} |
|
658 \end{figure} |
|
659 |
|
660 |
|
661 \begin{figure} |
|
662 \begin{alltt*}\isastyleminor |
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663 \tdx{the_equality}: [| P(a); !!x. P(x) ==> x=a |] ==> (THE x. P(x))=a |
|
664 \tdx{theI}: \isasymexists! x. P(x) ==> P(THE x. P(x)) |
|
665 |
|
666 \tdx{if_P}: P ==> (if P then a else b) = a |
|
667 \tdx{if_not_P}: ~P ==> (if P then a else b) = b |
|
668 |
|
669 \tdx{mem_asym}: [| a\isasymin{}b; b\isasymin{}a |] ==> P |
|
670 \tdx{mem_irrefl}: a\isasymin{}a ==> P |
|
671 \end{alltt*} |
|
672 \caption{Descriptions; non-circularity} \label{zf-the} |
|
673 \end{figure} |
|
674 |
|
675 |
|
676 \subsection{Unordered pairs and finite sets} |
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677 Figure~\ref{zf-upair1} presents the principle of unordered pairing, along |
|
678 with its derived rules. Binary union and intersection are defined in terms |
|
679 of ordered pairs (Fig.\ts\ref{zf-Un}). Set difference is also included. The |
|
680 rule \tdx{UnCI} is useful for classical reasoning about unions, |
|
681 like \isa{disjCI}\@; it supersedes \tdx{UnI1} and |
|
682 \tdx{UnI2}, but these rules are often easier to work with. For |
|
683 intersection and difference we have both elimination and destruction rules. |
|
684 Again, there is no reason to provide a minimal rule set. |
|
685 |
|
686 Figure~\ref{zf-upair2} is concerned with finite sets: it presents rules |
|
687 for~\isa{cons}, the finite set constructor, and rules for singleton |
|
688 sets. Figure~\ref{zf-succ} presents derived rules for the successor |
|
689 function, which is defined in terms of~\isa{cons}. The proof that |
|
690 \isa{succ} is injective appears to require the Axiom of Foundation. |
|
691 |
|
692 Definite descriptions (\sdx{THE}) are defined in terms of the singleton |
|
693 set~$\{0\}$, but their derived rules fortunately hide this |
|
694 (Fig.\ts\ref{zf-the}). The rule~\tdx{theI} is difficult to apply |
|
695 because of the two occurrences of~$\Var{P}$. However, |
|
696 \tdx{the_equality} does not have this problem and the files contain |
|
697 many examples of its use. |
|
698 |
|
699 Finally, the impossibility of having both $a\in b$ and $b\in a$ |
|
700 (\tdx{mem_asym}) is proved by applying the Axiom of Foundation to |
|
701 the set $\{a,b\}$. The impossibility of $a\in a$ is a trivial consequence. |
|
702 |
|
703 |
|
704 %%% subset.thy? |
|
705 |
|
706 \begin{figure} |
|
707 \begin{alltt*}\isastyleminor |
|
708 \tdx{Union_upper}: B\isasymin{}A ==> B \isasymsubseteq Union(A) |
|
709 \tdx{Union_least}: [| !!x. x\isasymin{}A ==> x \isasymsubseteq C |] ==> Union(A) \isasymsubseteq C |
|
710 |
|
711 \tdx{Inter_lower}: B\isasymin{}A ==> Inter(A) \isasymsubseteq B |
|
712 \tdx{Inter_greatest}: [| a\isasymin{}A; !!x. x\isasymin{}A ==> C \isasymsubseteq x |] ==> C\isasymsubseteq{}Inter(A) |
|
713 |
|
714 \tdx{Un_upper1}: A \isasymsubseteq A \isasymunion B |
|
715 \tdx{Un_upper2}: B \isasymsubseteq A \isasymunion B |
|
716 \tdx{Un_least}: [| A \isasymsubseteq C; B \isasymsubseteq C |] ==> A \isasymunion B \isasymsubseteq C |
|
717 |
|
718 \tdx{Int_lower1}: A \isasyminter B \isasymsubseteq A |
|
719 \tdx{Int_lower2}: A \isasyminter B \isasymsubseteq B |
|
720 \tdx{Int_greatest}: [| C \isasymsubseteq A; C \isasymsubseteq B |] ==> C \isasymsubseteq A \isasyminter B |
|
721 |
|
722 \tdx{Diff_subset}: A-B \isasymsubseteq A |
|
723 \tdx{Diff_contains}: [| C \isasymsubseteq A; C \isasyminter B = 0 |] ==> C \isasymsubseteq A-B |
|
724 |
|
725 \tdx{Collect_subset}: Collect(A,P) \isasymsubseteq A |
|
726 \end{alltt*} |
|
727 \caption{Subset and lattice properties} \label{zf-subset} |
|
728 \end{figure} |
|
729 |
|
730 |
|
731 \subsection{Subset and lattice properties} |
|
732 The subset relation is a complete lattice. Unions form least upper bounds; |
|
733 non-empty intersections form greatest lower bounds. Figure~\ref{zf-subset} |
|
734 shows the corresponding rules. A few other laws involving subsets are |
|
735 included. |
|
736 Reasoning directly about subsets often yields clearer proofs than |
|
737 reasoning about the membership relation. Section~\ref{sec:ZF-pow-example} |
|
738 below presents an example of this, proving the equation |
|
739 ${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$. |
|
740 |
|
741 %%% pair.thy |
|
742 |
|
743 \begin{figure} |
|
744 \begin{alltt*}\isastyleminor |
|
745 \tdx{Pair_inject1}: <a,b> = <c,d> ==> a=c |
|
746 \tdx{Pair_inject2}: <a,b> = <c,d> ==> b=d |
|
747 \tdx{Pair_inject}: [| <a,b> = <c,d>; [| a=c; b=d |] ==> P |] ==> P |
|
748 \tdx{Pair_neq_0}: <a,b>=0 ==> P |
|
749 |
|
750 \tdx{fst_conv}: fst(<a,b>) = a |
|
751 \tdx{snd_conv}: snd(<a,b>) = b |
|
752 \tdx{split}: split(\%x y. c(x,y), <a,b>) = c(a,b) |
|
753 |
|
754 \tdx{SigmaI}: [| a\isasymin{}A; b\isasymin{}B(a) |] ==> <a,b>\isasymin{}Sigma(A,B) |
|
755 |
|
756 \tdx{SigmaE}: [| c\isasymin{}Sigma(A,B); |
|
757 !!x y.[| x\isasymin{}A; y\isasymin{}B(x); c=<x,y> |] ==> P |] ==> P |
|
758 |
|
759 \tdx{SigmaE2}: [| <a,b>\isasymin{}Sigma(A,B); |
|
760 [| a\isasymin{}A; b\isasymin{}B(a) |] ==> P |] ==> P |
|
761 \end{alltt*} |
|
762 \caption{Ordered pairs; projections; general sums} \label{zf-pair} |
|
763 \end{figure} |
|
764 |
|
765 |
|
766 \subsection{Ordered pairs} \label{sec:pairs} |
|
767 |
|
768 Figure~\ref{zf-pair} presents the rules governing ordered pairs, |
|
769 projections and general sums --- in particular, that |
|
770 $\{\{a\},\{a,b\}\}$ functions as an ordered pair. This property is |
|
771 expressed as two destruction rules, |
|
772 \tdx{Pair_inject1} and \tdx{Pair_inject2}, and equivalently |
|
773 as the elimination rule \tdx{Pair_inject}. |
|
774 |
|
775 The rule \tdx{Pair_neq_0} asserts $\pair{a,b}\neq\emptyset$. This |
|
776 is a property of $\{\{a\},\{a,b\}\}$, and need not hold for other |
|
777 encodings of ordered pairs. The non-standard ordered pairs mentioned below |
|
778 satisfy $\pair{\emptyset;\emptyset}=\emptyset$. |
|
779 |
|
780 The natural deduction rules \tdx{SigmaI} and \tdx{SigmaE} |
|
781 assert that \cdx{Sigma}$(A,B)$ consists of all pairs of the form |
|
782 $\pair{x,y}$, for $x\in A$ and $y\in B(x)$. The rule \tdx{SigmaE2} |
|
783 merely states that $\pair{a,b}\in \isa{Sigma}(A,B)$ implies $a\in A$ and |
|
784 $b\in B(a)$. |
|
785 |
|
786 In addition, it is possible to use tuples as patterns in abstractions: |
|
787 \begin{center} |
|
788 {\tt\%<$x$,$y$>. $t$} \quad stands for\quad \isa{split(\%$x$ $y$.\ $t$)} |
|
789 \end{center} |
|
790 Nested patterns are translated recursively: |
|
791 {\tt\%<$x$,$y$,$z$>. $t$} $\leadsto$ {\tt\%<$x$,<$y$,$z$>>. $t$} $\leadsto$ |
|
792 \isa{split(\%$x$.\%<$y$,$z$>. $t$)} $\leadsto$ \isa{split(\%$x$. split(\%$y$ |
|
793 $z$.\ $t$))}. The reverse translation is performed upon printing. |
|
794 \begin{warn} |
|
795 The translation between patterns and \isa{split} is performed automatically |
|
796 by the parser and printer. Thus the internal and external form of a term |
|
797 may differ, which affects proofs. For example the term \isa{(\%<x,y>.<y,x>)<a,b>} requires the theorem \isa{split} to rewrite to |
|
798 {\tt<b,a>}. |
|
799 \end{warn} |
|
800 In addition to explicit $\lambda$-abstractions, patterns can be used in any |
|
801 variable binding construct which is internally described by a |
|
802 $\lambda$-abstraction. Here are some important examples: |
|
803 \begin{description} |
|
804 \item[Let:] \isa{let {\it pattern} = $t$ in $u$} |
|
805 \item[Choice:] \isa{THE~{\it pattern}~.~$P$} |
|
806 \item[Set operations:] \isa{\isasymUnion~{\it pattern}:$A$.~$B$} |
|
807 \item[Comprehension:] \isa{{\ttlbrace}~{\it pattern}:$A$~.~$P$~{\ttrbrace}} |
|
808 \end{description} |
|
809 |
|
810 |
|
811 %%% domrange.thy? |
|
812 |
|
813 \begin{figure} |
|
814 \begin{alltt*}\isastyleminor |
|
815 \tdx{domainI}: <a,b>\isasymin{}r ==> a\isasymin{}domain(r) |
|
816 \tdx{domainE}: [| a\isasymin{}domain(r); !!y. <a,y>\isasymin{}r ==> P |] ==> P |
|
817 \tdx{domain_subset}: domain(Sigma(A,B)) \isasymsubseteq A |
|
818 |
|
819 \tdx{rangeI}: <a,b>\isasymin{}r ==> b\isasymin{}range(r) |
|
820 \tdx{rangeE}: [| b\isasymin{}range(r); !!x. <x,b>\isasymin{}r ==> P |] ==> P |
|
821 \tdx{range_subset}: range(A*B) \isasymsubseteq B |
|
822 |
|
823 \tdx{fieldI1}: <a,b>\isasymin{}r ==> a\isasymin{}field(r) |
|
824 \tdx{fieldI2}: <a,b>\isasymin{}r ==> b\isasymin{}field(r) |
|
825 \tdx{fieldCI}: (<c,a> \isasymnotin r ==> <a,b>\isasymin{}r) ==> a\isasymin{}field(r) |
|
826 |
|
827 \tdx{fieldE}: [| a\isasymin{}field(r); |
|
828 !!x. <a,x>\isasymin{}r ==> P; |
|
829 !!x. <x,a>\isasymin{}r ==> P |
|
830 |] ==> P |
|
831 |
|
832 \tdx{field_subset}: field(A*A) \isasymsubseteq A |
|
833 \end{alltt*} |
|
834 \caption{Domain, range and field of a relation} \label{zf-domrange} |
|
835 \end{figure} |
|
836 |
|
837 \begin{figure} |
|
838 \begin{alltt*}\isastyleminor |
|
839 \tdx{imageI}: [| <a,b>\isasymin{}r; a\isasymin{}A |] ==> b\isasymin{}r``A |
|
840 \tdx{imageE}: [| b\isasymin{}r``A; !!x.[| <x,b>\isasymin{}r; x\isasymin{}A |] ==> P |] ==> P |
|
841 |
|
842 \tdx{vimageI}: [| <a,b>\isasymin{}r; b\isasymin{}B |] ==> a\isasymin{}r-``B |
|
843 \tdx{vimageE}: [| a\isasymin{}r-``B; !!x.[| <a,x>\isasymin{}r; x\isasymin{}B |] ==> P |] ==> P |
|
844 \end{alltt*} |
|
845 \caption{Image and inverse image} \label{zf-domrange2} |
|
846 \end{figure} |
|
847 |
|
848 |
|
849 \subsection{Relations} |
|
850 Figure~\ref{zf-domrange} presents rules involving relations, which are sets |
|
851 of ordered pairs. The converse of a relation~$r$ is the set of all pairs |
|
852 $\pair{y,x}$ such that $\pair{x,y}\in r$; if $r$ is a function, then |
|
853 {\cdx{converse}$(r)$} is its inverse. The rules for the domain |
|
854 operation, namely \tdx{domainI} and~\tdx{domainE}, assert that |
|
855 \cdx{domain}$(r)$ consists of all~$x$ such that $r$ contains |
|
856 some pair of the form~$\pair{x,y}$. The range operation is similar, and |
|
857 the field of a relation is merely the union of its domain and range. |
|
858 |
|
859 Figure~\ref{zf-domrange2} presents rules for images and inverse images. |
|
860 Note that these operations are generalisations of range and domain, |
|
861 respectively. |
|
862 |
|
863 |
|
864 %%% func.thy |
|
865 |
|
866 \begin{figure} |
|
867 \begin{alltt*}\isastyleminor |
|
868 \tdx{fun_is_rel}: f\isasymin{}Pi(A,B) ==> f \isasymsubseteq Sigma(A,B) |
|
869 |
|
870 \tdx{apply_equality}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> f`a = b |
|
871 \tdx{apply_equality2}: [| <a,b>\isasymin{}f; <a,c>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b=c |
|
872 |
|
873 \tdx{apply_type}: [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> f`a\isasymin{}B(a) |
|
874 \tdx{apply_Pair}: [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> <a,f`a>\isasymin{}f |
|
875 \tdx{apply_iff}: f\isasymin{}Pi(A,B) ==> <a,b>\isasymin{}f <-> a\isasymin{}A & f`a = b |
|
876 |
|
877 \tdx{fun_extension}: [| f\isasymin{}Pi(A,B); g\isasymin{}Pi(A,D); |
|
878 !!x. x\isasymin{}A ==> f`x = g`x |] ==> f=g |
|
879 |
|
880 \tdx{domain_type}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> a\isasymin{}A |
|
881 \tdx{range_type}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b\isasymin{}B(a) |
|
882 |
|
883 \tdx{Pi_type}: [| f\isasymin{}A->C; !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==> f\isasymin{}Pi(A,B) |
|
884 \tdx{domain_of_fun}: f\isasymin{}Pi(A,B) ==> domain(f)=A |
|
885 \tdx{range_of_fun}: f\isasymin{}Pi(A,B) ==> f\isasymin{}A->range(f) |
|
886 |
|
887 \tdx{restrict}: a\isasymin{}A ==> restrict(f,A) ` a = f`a |
|
888 \tdx{restrict_type}: [| !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==> |
|
889 restrict(f,A)\isasymin{}Pi(A,B) |
|
890 \end{alltt*} |
|
891 \caption{Functions} \label{zf-func1} |
|
892 \end{figure} |
|
893 |
|
894 |
|
895 \begin{figure} |
|
896 \begin{alltt*}\isastyleminor |
|
897 \tdx{lamI}: a\isasymin{}A ==> <a,b(a)>\isasymin{}(lam x\isasymin{}A. b(x)) |
|
898 \tdx{lamE}: [| p\isasymin{}(lam x\isasymin{}A. b(x)); !!x.[| x\isasymin{}A; p=<x,b(x)> |] ==> P |
|
899 |] ==> P |
|
900 |
|
901 \tdx{lam_type}: [| !!x. x\isasymin{}A ==> b(x)\isasymin{}B(x) |] ==> (lam x\isasymin{}A. b(x))\isasymin{}Pi(A,B) |
|
902 |
|
903 \tdx{beta}: a\isasymin{}A ==> (lam x\isasymin{}A. b(x)) ` a = b(a) |
|
904 \tdx{eta}: f\isasymin{}Pi(A,B) ==> (lam x\isasymin{}A. f`x) = f |
|
905 \end{alltt*} |
|
906 \caption{$\lambda$-abstraction} \label{zf-lam} |
|
907 \end{figure} |
|
908 |
|
909 |
|
910 \begin{figure} |
|
911 \begin{alltt*}\isastyleminor |
|
912 \tdx{fun_empty}: 0\isasymin{}0->0 |
|
913 \tdx{fun_single}: {\ttlbrace}<a,b>{\ttrbrace}\isasymin{}{\ttlbrace}a{\ttrbrace} -> {\ttlbrace}b{\ttrbrace} |
|
914 |
|
915 \tdx{fun_disjoint_Un}: [| f\isasymin{}A->B; g\isasymin{}C->D; A \isasyminter C = 0 |] ==> |
|
916 (f \isasymunion g)\isasymin{}(A \isasymunion C) -> (B \isasymunion D) |
|
917 |
|
918 \tdx{fun_disjoint_apply1}: [| a\isasymin{}A; f\isasymin{}A->B; g\isasymin{}C->D; A\isasyminter{}C = 0 |] ==> |
|
919 (f \isasymunion g)`a = f`a |
|
920 |
|
921 \tdx{fun_disjoint_apply2}: [| c\isasymin{}C; f\isasymin{}A->B; g\isasymin{}C->D; A\isasyminter{}C = 0 |] ==> |
|
922 (f \isasymunion g)`c = g`c |
|
923 \end{alltt*} |
|
924 \caption{Constructing functions from smaller sets} \label{zf-func2} |
|
925 \end{figure} |
|
926 |
|
927 |
|
928 \subsection{Functions} |
|
929 Functions, represented by graphs, are notoriously difficult to reason |
|
930 about. The ZF theory provides many derived rules, which overlap more |
|
931 than they ought. This section presents the more important rules. |
|
932 |
|
933 Figure~\ref{zf-func1} presents the basic properties of \cdx{Pi}$(A,B)$, |
|
934 the generalized function space. For example, if $f$ is a function and |
|
935 $\pair{a,b}\in f$, then $f`a=b$ (\tdx{apply_equality}). Two functions |
|
936 are equal provided they have equal domains and deliver equals results |
|
937 (\tdx{fun_extension}). |
|
938 |
|
939 By \tdx{Pi_type}, a function typing of the form $f\in A\to C$ can be |
|
940 refined to the dependent typing $f\in\prod@{x\in A}B(x)$, given a suitable |
|
941 family of sets $\{B(x)\}@{x\in A}$. Conversely, by \tdx{range_of_fun}, |
|
942 any dependent typing can be flattened to yield a function type of the form |
|
943 $A\to C$; here, $C=\isa{range}(f)$. |
|
944 |
|
945 Among the laws for $\lambda$-abstraction, \tdx{lamI} and \tdx{lamE} |
|
946 describe the graph of the generated function, while \tdx{beta} and |
|
947 \tdx{eta} are the standard conversions. We essentially have a |
|
948 dependently-typed $\lambda$-calculus (Fig.\ts\ref{zf-lam}). |
|
949 |
|
950 Figure~\ref{zf-func2} presents some rules that can be used to construct |
|
951 functions explicitly. We start with functions consisting of at most one |
|
952 pair, and may form the union of two functions provided their domains are |
|
953 disjoint. |
|
954 |
|
955 |
|
956 \begin{figure} |
|
957 \begin{alltt*}\isastyleminor |
|
958 \tdx{Int_absorb}: A \isasyminter A = A |
|
959 \tdx{Int_commute}: A \isasyminter B = B \isasyminter A |
|
960 \tdx{Int_assoc}: (A \isasyminter B) \isasyminter C = A \isasyminter (B \isasyminter C) |
|
961 \tdx{Int_Un_distrib}: (A \isasymunion B) \isasyminter C = (A \isasyminter C) \isasymunion (B \isasyminter C) |
|
962 |
|
963 \tdx{Un_absorb}: A \isasymunion A = A |
|
964 \tdx{Un_commute}: A \isasymunion B = B \isasymunion A |
|
965 \tdx{Un_assoc}: (A \isasymunion B) \isasymunion C = A \isasymunion (B \isasymunion C) |
|
966 \tdx{Un_Int_distrib}: (A \isasyminter B) \isasymunion C = (A \isasymunion C) \isasyminter (B \isasymunion C) |
|
967 |
|
968 \tdx{Diff_cancel}: A-A = 0 |
|
969 \tdx{Diff_disjoint}: A \isasyminter (B-A) = 0 |
|
970 \tdx{Diff_partition}: A \isasymsubseteq B ==> A \isasymunion (B-A) = B |
|
971 \tdx{double_complement}: [| A \isasymsubseteq B; B \isasymsubseteq C |] ==> (B - (C-A)) = A |
|
972 \tdx{Diff_Un}: A - (B \isasymunion C) = (A-B) \isasyminter (A-C) |
|
973 \tdx{Diff_Int}: A - (B \isasyminter C) = (A-B) \isasymunion (A-C) |
|
974 |
|
975 \tdx{Union_Un_distrib}: Union(A \isasymunion B) = Union(A) \isasymunion Union(B) |
|
976 \tdx{Inter_Un_distrib}: [| a \isasymin A; b \isasymin B |] ==> |
|
977 Inter(A \isasymunion B) = Inter(A) \isasyminter Inter(B) |
|
978 |
|
979 \tdx{Int_Union_RepFun}: A \isasyminter Union(B) = ({\isasymUnion}C \isasymin B. A \isasyminter C) |
|
980 |
|
981 \tdx{Un_Inter_RepFun}: b \isasymin B ==> |
|
982 A \isasymunion Inter(B) = ({\isasymInter}C \isasymin B. A \isasymunion C) |
|
983 |
|
984 \tdx{SUM_Un_distrib1}: (SUM x \isasymin A \isasymunion B. C(x)) = |
|
985 (SUM x \isasymin A. C(x)) \isasymunion (SUM x \isasymin B. C(x)) |
|
986 |
|
987 \tdx{SUM_Un_distrib2}: (SUM x \isasymin C. A(x) \isasymunion B(x)) = |
|
988 (SUM x \isasymin C. A(x)) \isasymunion (SUM x \isasymin C. B(x)) |
|
989 |
|
990 \tdx{SUM_Int_distrib1}: (SUM x \isasymin A \isasyminter B. C(x)) = |
|
991 (SUM x \isasymin A. C(x)) \isasyminter (SUM x \isasymin B. C(x)) |
|
992 |
|
993 \tdx{SUM_Int_distrib2}: (SUM x \isasymin C. A(x) \isasyminter B(x)) = |
|
994 (SUM x \isasymin C. A(x)) \isasyminter (SUM x \isasymin C. B(x)) |
|
995 \end{alltt*} |
|
996 \caption{Equalities} \label{zf-equalities} |
|
997 \end{figure} |
|
998 |
|
999 |
|
1000 \begin{figure} |
|
1001 %\begin{constants} |
|
1002 % \cdx{1} & $i$ & & $\{\emptyset\}$ \\ |
|
1003 % \cdx{bool} & $i$ & & the set $\{\emptyset,1\}$ \\ |
|
1004 % \cdx{cond} & $[i,i,i]\To i$ & & conditional for \isa{bool} \\ |
|
1005 % \cdx{not} & $i\To i$ & & negation for \isa{bool} \\ |
|
1006 % \sdx{and} & $[i,i]\To i$ & Left 70 & conjunction for \isa{bool} \\ |
|
1007 % \sdx{or} & $[i,i]\To i$ & Left 65 & disjunction for \isa{bool} \\ |
|
1008 % \sdx{xor} & $[i,i]\To i$ & Left 65 & exclusive-or for \isa{bool} |
|
1009 %\end{constants} |
|
1010 % |
|
1011 \begin{alltt*}\isastyleminor |
|
1012 \tdx{bool_def}: bool == {\ttlbrace}0,1{\ttrbrace} |
|
1013 \tdx{cond_def}: cond(b,c,d) == if b=1 then c else d |
|
1014 \tdx{not_def}: not(b) == cond(b,0,1) |
|
1015 \tdx{and_def}: a and b == cond(a,b,0) |
|
1016 \tdx{or_def}: a or b == cond(a,1,b) |
|
1017 \tdx{xor_def}: a xor b == cond(a,not(b),b) |
|
1018 |
|
1019 \tdx{bool_1I}: 1 \isasymin bool |
|
1020 \tdx{bool_0I}: 0 \isasymin bool |
|
1021 \tdx{boolE}: [| c \isasymin bool; c=1 ==> P; c=0 ==> P |] ==> P |
|
1022 \tdx{cond_1}: cond(1,c,d) = c |
|
1023 \tdx{cond_0}: cond(0,c,d) = d |
|
1024 \end{alltt*} |
|
1025 \caption{The booleans} \label{zf-bool} |
|
1026 \end{figure} |
|
1027 |
|
1028 |
|
1029 \section{Further developments} |
|
1030 The next group of developments is complex and extensive, and only |
|
1031 highlights can be covered here. It involves many theories and proofs. |
|
1032 |
|
1033 Figure~\ref{zf-equalities} presents commutative, associative, distributive, |
|
1034 and idempotency laws of union and intersection, along with other equations. |
|
1035 |
|
1036 Theory \thydx{Bool} defines $\{0,1\}$ as a set of booleans, with the usual |
|
1037 operators including a conditional (Fig.\ts\ref{zf-bool}). Although ZF is a |
|
1038 first-order theory, you can obtain the effect of higher-order logic using |
|
1039 \isa{bool}-valued functions, for example. The constant~\isa{1} is |
|
1040 translated to \isa{succ(0)}. |
|
1041 |
|
1042 \begin{figure} |
|
1043 \index{*"+ symbol} |
|
1044 \begin{constants} |
|
1045 \it symbol & \it meta-type & \it priority & \it description \\ |
|
1046 \tt + & $[i,i]\To i$ & Right 65 & disjoint union operator\\ |
|
1047 \cdx{Inl}~~\cdx{Inr} & $i\To i$ & & injections\\ |
|
1048 \cdx{case} & $[i\To i,i\To i, i]\To i$ & & conditional for $A+B$ |
|
1049 \end{constants} |
|
1050 \begin{alltt*}\isastyleminor |
|
1051 \tdx{sum_def}: A+B == {\ttlbrace}0{\ttrbrace}*A \isasymunion {\ttlbrace}1{\ttrbrace}*B |
|
1052 \tdx{Inl_def}: Inl(a) == <0,a> |
|
1053 \tdx{Inr_def}: Inr(b) == <1,b> |
|
1054 \tdx{case_def}: case(c,d,u) == split(\%y z. cond(y, d(z), c(z)), u) |
|
1055 |
|
1056 \tdx{InlI}: a \isasymin A ==> Inl(a) \isasymin A+B |
|
1057 \tdx{InrI}: b \isasymin B ==> Inr(b) \isasymin A+B |
|
1058 |
|
1059 \tdx{Inl_inject}: Inl(a)=Inl(b) ==> a=b |
|
1060 \tdx{Inr_inject}: Inr(a)=Inr(b) ==> a=b |
|
1061 \tdx{Inl_neq_Inr}: Inl(a)=Inr(b) ==> P |
|
1062 |
|
1063 \tdx{sum_iff}: u \isasymin A+B <-> ({\isasymexists}x\isasymin{}A. u=Inl(x)) | ({\isasymexists}y\isasymin{}B. u=Inr(y)) |
|
1064 |
|
1065 \tdx{case_Inl}: case(c,d,Inl(a)) = c(a) |
|
1066 \tdx{case_Inr}: case(c,d,Inr(b)) = d(b) |
|
1067 \end{alltt*} |
|
1068 \caption{Disjoint unions} \label{zf-sum} |
|
1069 \end{figure} |
|
1070 |
|
1071 |
|
1072 \subsection{Disjoint unions} |
|
1073 |
|
1074 Theory \thydx{Sum} defines the disjoint union of two sets, with |
|
1075 injections and a case analysis operator (Fig.\ts\ref{zf-sum}). Disjoint |
|
1076 unions play a role in datatype definitions, particularly when there is |
|
1077 mutual recursion~\cite{paulson-set-II}. |
|
1078 |
|
1079 \begin{figure} |
|
1080 \begin{alltt*}\isastyleminor |
|
1081 \tdx{QPair_def}: <a;b> == a+b |
|
1082 \tdx{qsplit_def}: qsplit(c,p) == THE y. {\isasymexists}a b. p=<a;b> & y=c(a,b) |
|
1083 \tdx{qfsplit_def}: qfsplit(R,z) == {\isasymexists}x y. z=<x;y> & R(x,y) |
|
1084 \tdx{qconverse_def}: qconverse(r) == {\ttlbrace}z. w \isasymin r, {\isasymexists}x y. w=<x;y> & z=<y;x>{\ttrbrace} |
|
1085 \tdx{QSigma_def}: QSigma(A,B) == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x;y>{\ttrbrace} |
|
1086 |
|
1087 \tdx{qsum_def}: A <+> B == ({\ttlbrace}0{\ttrbrace} <*> A) \isasymunion ({\ttlbrace}1{\ttrbrace} <*> B) |
|
1088 \tdx{QInl_def}: QInl(a) == <0;a> |
|
1089 \tdx{QInr_def}: QInr(b) == <1;b> |
|
1090 \tdx{qcase_def}: qcase(c,d) == qsplit(\%y z. cond(y, d(z), c(z))) |
|
1091 \end{alltt*} |
|
1092 \caption{Non-standard pairs, products and sums} \label{zf-qpair} |
|
1093 \end{figure} |
|
1094 |
|
1095 |
|
1096 \subsection{Non-standard ordered pairs} |
|
1097 |
|
1098 Theory \thydx{QPair} defines a notion of ordered pair that admits |
|
1099 non-well-founded tupling (Fig.\ts\ref{zf-qpair}). Such pairs are written |
|
1100 {\tt<$a$;$b$>}. It also defines the eliminator \cdx{qsplit}, the |
|
1101 converse operator \cdx{qconverse}, and the summation operator |
|
1102 \cdx{QSigma}. These are completely analogous to the corresponding |
|
1103 versions for standard ordered pairs. The theory goes on to define a |
|
1104 non-standard notion of disjoint sum using non-standard pairs. All of these |
|
1105 concepts satisfy the same properties as their standard counterparts; in |
|
1106 addition, {\tt<$a$;$b$>} is continuous. The theory supports coinductive |
|
1107 definitions, for example of infinite lists~\cite{paulson-mscs}. |
|
1108 |
|
1109 \begin{figure} |
|
1110 \begin{alltt*}\isastyleminor |
|
1111 \tdx{bnd_mono_def}: bnd_mono(D,h) == |
|
1112 h(D)\isasymsubseteq{}D & ({\isasymforall}W X. W\isasymsubseteq{}X --> X\isasymsubseteq{}D --> h(W)\isasymsubseteq{}h(X)) |
|
1113 |
|
1114 \tdx{lfp_def}: lfp(D,h) == Inter({\ttlbrace}X \isasymin Pow(D). h(X) \isasymsubseteq X{\ttrbrace}) |
|
1115 \tdx{gfp_def}: gfp(D,h) == Union({\ttlbrace}X \isasymin Pow(D). X \isasymsubseteq h(X){\ttrbrace}) |
|
1116 |
|
1117 |
|
1118 \tdx{lfp_lowerbound}: [| h(A) \isasymsubseteq A; A \isasymsubseteq D |] ==> lfp(D,h) \isasymsubseteq A |
|
1119 |
|
1120 \tdx{lfp_subset}: lfp(D,h) \isasymsubseteq D |
|
1121 |
|
1122 \tdx{lfp_greatest}: [| bnd_mono(D,h); |
|
1123 !!X. [| h(X) \isasymsubseteq X; X \isasymsubseteq D |] ==> A \isasymsubseteq X |
|
1124 |] ==> A \isasymsubseteq lfp(D,h) |
|
1125 |
|
1126 \tdx{lfp_Tarski}: bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h)) |
|
1127 |
|
1128 \tdx{induct}: [| a \isasymin lfp(D,h); bnd_mono(D,h); |
|
1129 !!x. x \isasymin h(Collect(lfp(D,h),P)) ==> P(x) |
|
1130 |] ==> P(a) |
|
1131 |
|
1132 \tdx{lfp_mono}: [| bnd_mono(D,h); bnd_mono(E,i); |
|
1133 !!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X) |
|
1134 |] ==> lfp(D,h) \isasymsubseteq lfp(E,i) |
|
1135 |
|
1136 \tdx{gfp_upperbound}: [| A \isasymsubseteq h(A); A \isasymsubseteq D |] ==> A \isasymsubseteq gfp(D,h) |
|
1137 |
|
1138 \tdx{gfp_subset}: gfp(D,h) \isasymsubseteq D |
|
1139 |
|
1140 \tdx{gfp_least}: [| bnd_mono(D,h); |
|
1141 !!X. [| X \isasymsubseteq h(X); X \isasymsubseteq D |] ==> X \isasymsubseteq A |
|
1142 |] ==> gfp(D,h) \isasymsubseteq A |
|
1143 |
|
1144 \tdx{gfp_Tarski}: bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h)) |
|
1145 |
|
1146 \tdx{coinduct}: [| bnd_mono(D,h); a \isasymin X; X \isasymsubseteq h(X \isasymunion gfp(D,h)); X \isasymsubseteq D |
|
1147 |] ==> a \isasymin gfp(D,h) |
|
1148 |
|
1149 \tdx{gfp_mono}: [| bnd_mono(D,h); D \isasymsubseteq E; |
|
1150 !!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X) |
|
1151 |] ==> gfp(D,h) \isasymsubseteq gfp(E,i) |
|
1152 \end{alltt*} |
|
1153 \caption{Least and greatest fixedpoints} \label{zf-fixedpt} |
|
1154 \end{figure} |
|
1155 |
|
1156 |
|
1157 \subsection{Least and greatest fixedpoints} |
|
1158 |
|
1159 The Knaster-Tarski Theorem states that every monotone function over a |
|
1160 complete lattice has a fixedpoint. Theory \thydx{Fixedpt} proves the |
|
1161 Theorem only for a particular lattice, namely the lattice of subsets of a |
|
1162 set (Fig.\ts\ref{zf-fixedpt}). The theory defines least and greatest |
|
1163 fixedpoint operators with corresponding induction and coinduction rules. |
|
1164 These are essential to many definitions that follow, including the natural |
|
1165 numbers and the transitive closure operator. The (co)inductive definition |
|
1166 package also uses the fixedpoint operators~\cite{paulson-CADE}. See |
|
1167 Davey and Priestley~\cite{davey-priestley} for more on the Knaster-Tarski |
|
1168 Theorem and my paper~\cite{paulson-set-II} for discussion of the Isabelle |
|
1169 proofs. |
|
1170 |
|
1171 Monotonicity properties are proved for most of the set-forming operations: |
|
1172 union, intersection, Cartesian product, image, domain, range, etc. These |
|
1173 are useful for applying the Knaster-Tarski Fixedpoint Theorem. The proofs |
|
1174 themselves are trivial applications of Isabelle's classical reasoner. |
|
1175 |
|
1176 |
|
1177 \subsection{Finite sets and lists} |
|
1178 |
|
1179 Theory \texttt{Finite} (Figure~\ref{zf-fin}) defines the finite set operator; |
|
1180 $\isa{Fin}(A)$ is the set of all finite sets over~$A$. The theory employs |
|
1181 Isabelle's inductive definition package, which proves various rules |
|
1182 automatically. The induction rule shown is stronger than the one proved by |
|
1183 the package. The theory also defines the set of all finite functions |
|
1184 between two given sets. |
|
1185 |
|
1186 \begin{figure} |
|
1187 \begin{alltt*}\isastyleminor |
|
1188 \tdx{Fin.emptyI} 0 \isasymin Fin(A) |
|
1189 \tdx{Fin.consI} [| a \isasymin A; b \isasymin Fin(A) |] ==> cons(a,b) \isasymin Fin(A) |
|
1190 |
|
1191 \tdx{Fin_induct} |
|
1192 [| b \isasymin Fin(A); |
|
1193 P(0); |
|
1194 !!x y. [| x\isasymin{}A; y\isasymin{}Fin(A); x\isasymnotin{}y; P(y) |] ==> P(cons(x,y)) |
|
1195 |] ==> P(b) |
|
1196 |
|
1197 \tdx{Fin_mono}: A \isasymsubseteq B ==> Fin(A) \isasymsubseteq Fin(B) |
|
1198 \tdx{Fin_UnI}: [| b \isasymin Fin(A); c \isasymin Fin(A) |] ==> b \isasymunion c \isasymin Fin(A) |
|
1199 \tdx{Fin_UnionI}: C \isasymin Fin(Fin(A)) ==> Union(C) \isasymin Fin(A) |
|
1200 \tdx{Fin_subset}: [| c \isasymsubseteq b; b \isasymin Fin(A) |] ==> c \isasymin Fin(A) |
|
1201 \end{alltt*} |
|
1202 \caption{The finite set operator} \label{zf-fin} |
|
1203 \end{figure} |
|
1204 |
|
1205 \begin{figure} |
|
1206 \begin{constants} |
|
1207 \it symbol & \it meta-type & \it priority & \it description \\ |
|
1208 \cdx{list} & $i\To i$ && lists over some set\\ |
|
1209 \cdx{list_case} & $[i, [i,i]\To i, i] \To i$ && conditional for $list(A)$ \\ |
|
1210 \cdx{map} & $[i\To i, i] \To i$ & & mapping functional\\ |
|
1211 \cdx{length} & $i\To i$ & & length of a list\\ |
|
1212 \cdx{rev} & $i\To i$ & & reverse of a list\\ |
|
1213 \tt \at & $[i,i]\To i$ & Right 60 & append for lists\\ |
|
1214 \cdx{flat} & $i\To i$ & & append of list of lists |
|
1215 \end{constants} |
|
1216 |
|
1217 \underscoreon %%because @ is used here |
|
1218 \begin{alltt*}\isastyleminor |
|
1219 \tdx{NilI}: Nil \isasymin list(A) |
|
1220 \tdx{ConsI}: [| a \isasymin A; l \isasymin list(A) |] ==> Cons(a,l) \isasymin list(A) |
|
1221 |
|
1222 \tdx{List.induct} |
|
1223 [| l \isasymin list(A); |
|
1224 P(Nil); |
|
1225 !!x y. [| x \isasymin A; y \isasymin list(A); P(y) |] ==> P(Cons(x,y)) |
|
1226 |] ==> P(l) |
|
1227 |
|
1228 \tdx{Cons_iff}: Cons(a,l)=Cons(a',l') <-> a=a' & l=l' |
|
1229 \tdx{Nil_Cons_iff}: Nil \isasymnoteq Cons(a,l) |
|
1230 |
|
1231 \tdx{list_mono}: A \isasymsubseteq B ==> list(A) \isasymsubseteq list(B) |
|
1232 |
|
1233 \tdx{map_ident}: l\isasymin{}list(A) ==> map(\%u. u, l) = l |
|
1234 \tdx{map_compose}: l\isasymin{}list(A) ==> map(h, map(j,l)) = map(\%u. h(j(u)), l) |
|
1235 \tdx{map_app_distrib}: xs\isasymin{}list(A) ==> map(h, xs@ys) = map(h,xs)@map(h,ys) |
|
1236 \tdx{map_type} |
|
1237 [| l\isasymin{}list(A); !!x. x\isasymin{}A ==> h(x)\isasymin{}B |] ==> map(h,l)\isasymin{}list(B) |
|
1238 \tdx{map_flat} |
|
1239 ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls)) |
|
1240 \end{alltt*} |
|
1241 \caption{Lists} \label{zf-list} |
|
1242 \end{figure} |
|
1243 |
|
1244 |
|
1245 Figure~\ref{zf-list} presents the set of lists over~$A$, $\isa{list}(A)$. The |
|
1246 definition employs Isabelle's datatype package, which defines the introduction |
|
1247 and induction rules automatically, as well as the constructors, case operator |
|
1248 (\isa{list\_case}) and recursion operator. The theory then defines the usual |
|
1249 list functions by primitive recursion. See theory \texttt{List}. |
|
1250 |
|
1251 |
|
1252 \subsection{Miscellaneous} |
|
1253 |
|
1254 \begin{figure} |
|
1255 \begin{constants} |
|
1256 \it symbol & \it meta-type & \it priority & \it description \\ |
|
1257 \sdx{O} & $[i,i]\To i$ & Right 60 & composition ($\circ$) \\ |
|
1258 \cdx{id} & $i\To i$ & & identity function \\ |
|
1259 \cdx{inj} & $[i,i]\To i$ & & injective function space\\ |
|
1260 \cdx{surj} & $[i,i]\To i$ & & surjective function space\\ |
|
1261 \cdx{bij} & $[i,i]\To i$ & & bijective function space |
|
1262 \end{constants} |
|
1263 |
|
1264 \begin{alltt*}\isastyleminor |
|
1265 \tdx{comp_def}: r O s == {\ttlbrace}xz \isasymin domain(s)*range(r) . |
|
1266 {\isasymexists}x y z. xz=<x,z> & <x,y> \isasymin s & <y,z> \isasymin r{\ttrbrace} |
|
1267 \tdx{id_def}: id(A) == (lam x \isasymin A. x) |
|
1268 \tdx{inj_def}: inj(A,B) == {\ttlbrace} f\isasymin{}A->B. {\isasymforall}w\isasymin{}A. {\isasymforall}x\isasymin{}A. f`w=f`x --> w=x {\ttrbrace} |
|
1269 \tdx{surj_def}: surj(A,B) == {\ttlbrace} f\isasymin{}A->B . {\isasymforall}y\isasymin{}B. {\isasymexists}x\isasymin{}A. f`x=y {\ttrbrace} |
|
1270 \tdx{bij_def}: bij(A,B) == inj(A,B) \isasyminter surj(A,B) |
|
1271 |
|
1272 |
|
1273 \tdx{left_inverse}: [| f\isasymin{}inj(A,B); a\isasymin{}A |] ==> converse(f)`(f`a) = a |
|
1274 \tdx{right_inverse}: [| f\isasymin{}inj(A,B); b\isasymin{}range(f) |] ==> |
|
1275 f`(converse(f)`b) = b |
|
1276 |
|
1277 \tdx{inj_converse_inj}: f\isasymin{}inj(A,B) ==> converse(f) \isasymin inj(range(f),A) |
|
1278 \tdx{bij_converse_bij}: f\isasymin{}bij(A,B) ==> converse(f) \isasymin bij(B,A) |
|
1279 |
|
1280 \tdx{comp_type}: [| s \isasymsubseteq A*B; r \isasymsubseteq B*C |] ==> (r O s) \isasymsubseteq A*C |
|
1281 \tdx{comp_assoc}: (r O s) O t = r O (s O t) |
|
1282 |
|
1283 \tdx{left_comp_id}: r \isasymsubseteq A*B ==> id(B) O r = r |
|
1284 \tdx{right_comp_id}: r \isasymsubseteq A*B ==> r O id(A) = r |
|
1285 |
|
1286 \tdx{comp_func}: [| g\isasymin{}A->B; f\isasymin{}B->C |] ==> (f O g) \isasymin A->C |
|
1287 \tdx{comp_func_apply}: [| g\isasymin{}A->B; f\isasymin{}B->C; a\isasymin{}A |] ==> (f O g)`a = f`(g`a) |
|
1288 |
|
1289 \tdx{comp_inj}: [| g\isasymin{}inj(A,B); f\isasymin{}inj(B,C) |] ==> (f O g)\isasymin{}inj(A,C) |
|
1290 \tdx{comp_surj}: [| g\isasymin{}surj(A,B); f\isasymin{}surj(B,C) |] ==> (f O g)\isasymin{}surj(A,C) |
|
1291 \tdx{comp_bij}: [| g\isasymin{}bij(A,B); f\isasymin{}bij(B,C) |] ==> (f O g)\isasymin{}bij(A,C) |
|
1292 |
|
1293 \tdx{left_comp_inverse}: f\isasymin{}inj(A,B) ==> converse(f) O f = id(A) |
|
1294 \tdx{right_comp_inverse}: f\isasymin{}surj(A,B) ==> f O converse(f) = id(B) |
|
1295 |
|
1296 \tdx{bij_disjoint_Un}: |
|
1297 [| f\isasymin{}bij(A,B); g\isasymin{}bij(C,D); A \isasyminter C = 0; B \isasyminter D = 0 |] ==> |
|
1298 (f \isasymunion g)\isasymin{}bij(A \isasymunion C, B \isasymunion D) |
|
1299 |
|
1300 \tdx{restrict_bij}: [| f\isasymin{}inj(A,B); C\isasymsubseteq{}A |] ==> restrict(f,C)\isasymin{}bij(C, f``C) |
|
1301 \end{alltt*} |
|
1302 \caption{Permutations} \label{zf-perm} |
|
1303 \end{figure} |
|
1304 |
|
1305 The theory \thydx{Perm} is concerned with permutations (bijections) and |
|
1306 related concepts. These include composition of relations, the identity |
|
1307 relation, and three specialized function spaces: injective, surjective and |
|
1308 bijective. Figure~\ref{zf-perm} displays many of their properties that |
|
1309 have been proved. These results are fundamental to a treatment of |
|
1310 equipollence and cardinality. |
|
1311 |
|
1312 Theory \thydx{Univ} defines a `universe' $\isa{univ}(A)$, which is used by |
|
1313 the datatype package. This set contains $A$ and the |
|
1314 natural numbers. Vitally, it is closed under finite products: |
|
1315 $\isa{univ}(A)\times\isa{univ}(A)\subseteq\isa{univ}(A)$. This theory also |
|
1316 defines the cumulative hierarchy of axiomatic set theory, which |
|
1317 traditionally is written $V@\alpha$ for an ordinal~$\alpha$. The |
|
1318 `universe' is a simple generalization of~$V@\omega$. |
|
1319 |
|
1320 Theory \thydx{QUniv} defines a `universe' $\isa{quniv}(A)$, which is used by |
|
1321 the datatype package to construct codatatypes such as streams. It is |
|
1322 analogous to $\isa{univ}(A)$ (and is defined in terms of it) but is closed |
|
1323 under the non-standard product and sum. |
|
1324 |
|
1325 |
|
1326 \section{Automatic Tools} |
|
1327 |
|
1328 ZF provides the simplifier and the classical reasoner. Moreover it supplies a |
|
1329 specialized tool to infer `types' of terms. |
|
1330 |
|
1331 \subsection{Simplification and Classical Reasoning} |
|
1332 |
|
1333 ZF inherits simplification from FOL but adopts it for set theory. The |
|
1334 extraction of rewrite rules takes the ZF primitives into account. It can |
|
1335 strip bounded universal quantifiers from a formula; for example, ${\forall |
|
1336 x\in A. f(x)=g(x)}$ yields the conditional rewrite rule $x\in A \Imp |
|
1337 f(x)=g(x)$. Given $a\in\{x\in A. P(x)\}$ it extracts rewrite rules from $a\in |
|
1338 A$ and~$P(a)$. It can also break down $a\in A\int B$ and $a\in A-B$. |
|
1339 |
|
1340 The default simpset used by \isa{simp} contains congruence rules for all of ZF's |
|
1341 binding operators. It contains all the conversion rules, such as |
|
1342 \isa{fst} and |
|
1343 \isa{snd}, as well as the rewrites shown in Fig.\ts\ref{zf-simpdata}. |
|
1344 |
|
1345 Classical reasoner methods such as \isa{blast} and \isa{auto} refer to |
|
1346 a rich collection of built-in axioms for all the set-theoretic |
|
1347 primitives. |
|
1348 |
|
1349 |
|
1350 \begin{figure} |
|
1351 \begin{eqnarray*} |
|
1352 a\in \emptyset & \bimp & \bot\\ |
|
1353 a \in A \un B & \bimp & a\in A \disj a\in B\\ |
|
1354 a \in A \int B & \bimp & a\in A \conj a\in B\\ |
|
1355 a \in A-B & \bimp & a\in A \conj \lnot (a\in B)\\ |
|
1356 \pair{a,b}\in \isa{Sigma}(A,B) |
|
1357 & \bimp & a\in A \conj b\in B(a)\\ |
|
1358 a \in \isa{Collect}(A,P) & \bimp & a\in A \conj P(a)\\ |
|
1359 (\forall x \in \emptyset. P(x)) & \bimp & \top\\ |
|
1360 (\forall x \in A. \top) & \bimp & \top |
|
1361 \end{eqnarray*} |
|
1362 \caption{Some rewrite rules for set theory} \label{zf-simpdata} |
|
1363 \end{figure} |
|
1364 |
|
1365 |
|
1366 \subsection{Type-Checking Tactics} |
|
1367 \index{type-checking tactics} |
|
1368 |
|
1369 Isabelle/ZF provides simple tactics to help automate those proofs that are |
|
1370 essentially type-checking. Such proofs are built by applying rules such as |
|
1371 these: |
|
1372 \begin{ttbox}\isastyleminor |
|
1373 [| ?P ==> ?a \isasymin ?A; ~?P ==> ?b \isasymin ?A |] |
|
1374 ==> (if ?P then ?a else ?b) \isasymin ?A |
|
1375 |
|
1376 [| ?m \isasymin nat; ?n \isasymin nat |] ==> ?m #+ ?n \isasymin nat |
|
1377 |
|
1378 ?a \isasymin ?A ==> Inl(?a) \isasymin ?A + ?B |
|
1379 \end{ttbox} |
|
1380 In typical applications, the goal has the form $t\in\Var{A}$: in other words, |
|
1381 we have a specific term~$t$ and need to infer its `type' by instantiating the |
|
1382 set variable~$\Var{A}$. Neither the simplifier nor the classical reasoner |
|
1383 does this job well. The if-then-else rule, and many similar ones, can make |
|
1384 the classical reasoner loop. The simplifier refuses (on principle) to |
|
1385 instantiate variables during rewriting, so goals such as \isa{i\#+j \isasymin \ ?A} |
|
1386 are left unsolved. |
|
1387 |
|
1388 The simplifier calls the type-checker to solve rewritten subgoals: this stage |
|
1389 can indeed instantiate variables. If you have defined new constants and |
|
1390 proved type-checking rules for them, then declare the rules using |
|
1391 the attribute \isa{TC} and the rest should be automatic. In |
|
1392 particular, the simplifier will use type-checking to help satisfy |
|
1393 conditional rewrite rules. Call the method \ttindex{typecheck} to |
|
1394 break down all subgoals using type-checking rules. You can add new |
|
1395 type-checking rules temporarily like this: |
|
1396 \begin{isabelle} |
|
1397 \isacommand{apply}\ (typecheck add:\ inj_is_fun) |
|
1398 \end{isabelle} |
|
1399 |
|
1400 |
|
1401 %Though the easiest way to invoke the type-checker is via the simplifier, |
|
1402 %specialized applications may require more detailed knowledge of |
|
1403 %the type-checking primitives. They are modelled on the simplifier's: |
|
1404 %\begin{ttdescription} |
|
1405 %\item[\ttindexbold{tcset}] is the type of tcsets: sets of type-checking rules. |
|
1406 % |
|
1407 %\item[\ttindexbold{addTCs}] is an infix operator to add type-checking rules to |
|
1408 % a tcset. |
|
1409 % |
|
1410 %\item[\ttindexbold{delTCs}] is an infix operator to remove type-checking rules |
|
1411 % from a tcset. |
|
1412 % |
|
1413 %\item[\ttindexbold{typecheck_tac}] is a tactic for attempting to prove all |
|
1414 % subgoals using the rules given in its argument, a tcset. |
|
1415 %\end{ttdescription} |
|
1416 % |
|
1417 %Tcsets, like simpsets, are associated with theories and are merged when |
|
1418 %theories are merged. There are further primitives that use the default tcset. |
|
1419 %\begin{ttdescription} |
|
1420 %\item[\ttindexbold{tcset}] is a function to return the default tcset; use the |
|
1421 % expression \isa{tcset()}. |
|
1422 % |
|
1423 %\item[\ttindexbold{AddTCs}] adds type-checking rules to the default tcset. |
|
1424 % |
|
1425 %\item[\ttindexbold{DelTCs}] removes type-checking rules from the default |
|
1426 % tcset. |
|
1427 % |
|
1428 %\item[\ttindexbold{Typecheck_tac}] calls \isa{typecheck_tac} using the |
|
1429 % default tcset. |
|
1430 %\end{ttdescription} |
|
1431 % |
|
1432 %To supply some type-checking rules temporarily, using \isa{Addrules} and |
|
1433 %later \isa{Delrules} is the simplest way. There is also a high-tech |
|
1434 %approach. Call the simplifier with a new solver expressed using |
|
1435 %\ttindexbold{type_solver_tac} and your temporary type-checking rules. |
|
1436 %\begin{ttbox}\isastyleminor |
|
1437 %by (asm_simp_tac |
|
1438 % (simpset() setSolver type_solver_tac (tcset() addTCs prems)) 2); |
|
1439 %\end{ttbox} |
|
1440 |
|
1441 |
|
1442 \section{Natural number and integer arithmetic} |
|
1443 |
|
1444 \index{arithmetic|(} |
|
1445 |
|
1446 \begin{figure}\small |
|
1447 \index{#*@{\tt\#*} symbol} |
|
1448 \index{*div symbol} |
|
1449 \index{*mod symbol} |
|
1450 \index{#+@{\tt\#+} symbol} |
|
1451 \index{#-@{\tt\#-} symbol} |
|
1452 \begin{constants} |
|
1453 \it symbol & \it meta-type & \it priority & \it description \\ |
|
1454 \cdx{nat} & $i$ & & set of natural numbers \\ |
|
1455 \cdx{nat_case}& $[i,i\To i,i]\To i$ & & conditional for $nat$\\ |
|
1456 \tt \#* & $[i,i]\To i$ & Left 70 & multiplication \\ |
|
1457 \tt div & $[i,i]\To i$ & Left 70 & division\\ |
|
1458 \tt mod & $[i,i]\To i$ & Left 70 & modulus\\ |
|
1459 \tt \#+ & $[i,i]\To i$ & Left 65 & addition\\ |
|
1460 \tt \#- & $[i,i]\To i$ & Left 65 & subtraction |
|
1461 \end{constants} |
|
1462 |
|
1463 \begin{alltt*}\isastyleminor |
|
1464 \tdx{nat_def}: nat == lfp(lam r \isasymin Pow(Inf). {\ttlbrace}0{\ttrbrace} \isasymunion {\ttlbrace}succ(x). x \isasymin r{\ttrbrace} |
|
1465 |
|
1466 \tdx{nat_case_def}: nat_case(a,b,k) == |
|
1467 THE y. k=0 & y=a | ({\isasymexists}x. k=succ(x) & y=b(x)) |
|
1468 |
|
1469 \tdx{nat_0I}: 0 \isasymin nat |
|
1470 \tdx{nat_succI}: n \isasymin nat ==> succ(n) \isasymin nat |
|
1471 |
|
1472 \tdx{nat_induct}: |
|
1473 [| n \isasymin nat; P(0); !!x. [| x \isasymin nat; P(x) |] ==> P(succ(x)) |
|
1474 |] ==> P(n) |
|
1475 |
|
1476 \tdx{nat_case_0}: nat_case(a,b,0) = a |
|
1477 \tdx{nat_case_succ}: nat_case(a,b,succ(m)) = b(m) |
|
1478 |
|
1479 \tdx{add_0_natify}: 0 #+ n = natify(n) |
|
1480 \tdx{add_succ}: succ(m) #+ n = succ(m #+ n) |
|
1481 |
|
1482 \tdx{mult_type}: m #* n \isasymin nat |
|
1483 \tdx{mult_0}: 0 #* n = 0 |
|
1484 \tdx{mult_succ}: succ(m) #* n = n #+ (m #* n) |
|
1485 \tdx{mult_commute}: m #* n = n #* m |
|
1486 \tdx{add_mult_dist}: (m #+ n) #* k = (m #* k) #+ (n #* k) |
|
1487 \tdx{mult_assoc}: (m #* n) #* k = m #* (n #* k) |
|
1488 \tdx{mod_div_equality}: m \isasymin nat ==> (m div n)#*n #+ m mod n = m |
|
1489 \end{alltt*} |
|
1490 \caption{The natural numbers} \label{zf-nat} |
|
1491 \end{figure} |
|
1492 |
|
1493 \index{natural numbers} |
|
1494 |
|
1495 Theory \thydx{Nat} defines the natural numbers and mathematical |
|
1496 induction, along with a case analysis operator. The set of natural |
|
1497 numbers, here called \isa{nat}, is known in set theory as the ordinal~$\omega$. |
|
1498 |
|
1499 Theory \thydx{Arith} develops arithmetic on the natural numbers |
|
1500 (Fig.\ts\ref{zf-nat}). Addition, multiplication and subtraction are defined |
|
1501 by primitive recursion. Division and remainder are defined by repeated |
|
1502 subtraction, which requires well-founded recursion; the termination argument |
|
1503 relies on the divisor's being non-zero. Many properties are proved: |
|
1504 commutative, associative and distributive laws, identity and cancellation |
|
1505 laws, etc. The most interesting result is perhaps the theorem $a \bmod b + |
|
1506 (a/b)\times b = a$. |
|
1507 |
|
1508 To minimize the need for tedious proofs of $t\in\isa{nat}$, the arithmetic |
|
1509 operators coerce their arguments to be natural numbers. The function |
|
1510 \cdx{natify} is defined such that $\isa{natify}(n) = n$ if $n$ is a natural |
|
1511 number, $\isa{natify}(\isa{succ}(x)) = |
|
1512 \isa{succ}(\isa{natify}(x))$ for all $x$, and finally |
|
1513 $\isa{natify}(x)=0$ in all other cases. The benefit is that the addition, |
|
1514 subtraction, multiplication, division and remainder operators always return |
|
1515 natural numbers, regardless of their arguments. Algebraic laws (commutative, |
|
1516 associative, distributive) are unconditional. Occurrences of \isa{natify} |
|
1517 as operands of those operators are simplified away. Any remaining occurrences |
|
1518 can either be tolerated or else eliminated by proving that the argument is a |
|
1519 natural number. |
|
1520 |
|
1521 The simplifier automatically cancels common terms on the opposite sides of |
|
1522 subtraction and of relations ($=$, $<$ and $\le$). Here is an example: |
|
1523 \begin{isabelle} |
|
1524 1. i \#+ j \#+ k \#- j < k \#+ l\isanewline |
|
1525 \isacommand{apply}\ simp\isanewline |
|
1526 1. natify(i) < natify(l) |
|
1527 \end{isabelle} |
|
1528 Given the assumptions \isa{i \isasymin nat} and \isa{l \isasymin nat}, both occurrences of |
|
1529 \cdx{natify} would be simplified away. |
|
1530 |
|
1531 |
|
1532 \begin{figure}\small |
|
1533 \index{$*@{\tt\$*} symbol} |
|
1534 \index{$+@{\tt\$+} symbol} |
|
1535 \index{$-@{\tt\$-} symbol} |
|
1536 \begin{constants} |
|
1537 \it symbol & \it meta-type & \it priority & \it description \\ |
|
1538 \cdx{int} & $i$ & & set of integers \\ |
|
1539 \tt \$* & $[i,i]\To i$ & Left 70 & multiplication \\ |
|
1540 \tt \$+ & $[i,i]\To i$ & Left 65 & addition\\ |
|
1541 \tt \$- & $[i,i]\To i$ & Left 65 & subtraction\\ |
|
1542 \tt \$< & $[i,i]\To o$ & Left 50 & $<$ on integers\\ |
|
1543 \tt \$<= & $[i,i]\To o$ & Left 50 & $\le$ on integers |
|
1544 \end{constants} |
|
1545 |
|
1546 \begin{alltt*}\isastyleminor |
|
1547 \tdx{zadd_0_intify}: 0 $+ n = intify(n) |
|
1548 |
|
1549 \tdx{zmult_type}: m $* n \isasymin int |
|
1550 \tdx{zmult_0}: 0 $* n = 0 |
|
1551 \tdx{zmult_commute}: m $* n = n $* m |
|
1552 \tdx{zadd_zmult_dist}: (m $+ n) $* k = (m $* k) $+ (n $* k) |
|
1553 \tdx{zmult_assoc}: (m $* n) $* k = m $* (n $* k) |
|
1554 \end{alltt*} |
|
1555 \caption{The integers} \label{zf-int} |
|
1556 \end{figure} |
|
1557 |
|
1558 |
|
1559 \index{integers} |
|
1560 |
|
1561 Theory \thydx{Int} defines the integers, as equivalence classes of natural |
|
1562 numbers. Figure~\ref{zf-int} presents a tidy collection of laws. In |
|
1563 fact, a large library of facts is proved, including monotonicity laws for |
|
1564 addition and multiplication, covering both positive and negative operands. |
|
1565 |
|
1566 As with the natural numbers, the need for typing proofs is minimized. All the |
|
1567 operators defined in Fig.\ts\ref{zf-int} coerce their operands to integers by |
|
1568 applying the function \cdx{intify}. This function is the identity on integers |
|
1569 and maps other operands to zero. |
|
1570 |
|
1571 Decimal notation is provided for the integers. Numbers, written as |
|
1572 \isa{\#$nnn$} or \isa{\#-$nnn$}, are represented internally in |
|
1573 two's-complement binary. Expressions involving addition, subtraction and |
|
1574 multiplication of numeral constants are evaluated (with acceptable efficiency) |
|
1575 by simplification. The simplifier also collects similar terms, multiplying |
|
1576 them by a numerical coefficient. It also cancels occurrences of the same |
|
1577 terms on the other side of the relational operators. Example: |
|
1578 \begin{isabelle} |
|
1579 1. y \$+ z \$+ \#-3 \$* x \$+ y \$<= x \$* \#2 \$+ |
|
1580 z\isanewline |
|
1581 \isacommand{apply}\ simp\isanewline |
|
1582 1. \#2 \$* y \$<= \#5 \$* x |
|
1583 \end{isabelle} |
|
1584 For more information on the integers, please see the theories on directory |
|
1585 \texttt{ZF/Integ}. |
|
1586 |
|
1587 \index{arithmetic|)} |
|
1588 |
|
1589 |
|
1590 \section{Datatype definitions} |
|
1591 \label{sec:ZF:datatype} |
|
1592 \index{*datatype|(} |
|
1593 |
|
1594 The \ttindex{datatype} definition package of ZF constructs inductive datatypes |
|
1595 similar to \ML's. It can also construct coinductive datatypes |
|
1596 (codatatypes), which are non-well-founded structures such as streams. It |
|
1597 defines the set using a fixed-point construction and proves induction rules, |
|
1598 as well as theorems for recursion and case combinators. It supplies |
|
1599 mechanisms for reasoning about freeness. The datatype package can handle both |
|
1600 mutual and indirect recursion. |
|
1601 |
|
1602 |
|
1603 \subsection{Basics} |
|
1604 \label{subsec:datatype:basics} |
|
1605 |
|
1606 A \isa{datatype} definition has the following form: |
|
1607 \[ |
|
1608 \begin{array}{llcl} |
|
1609 \mathtt{datatype} & t@1(A@1,\ldots,A@h) & = & |
|
1610 constructor^1@1 ~\mid~ \ldots ~\mid~ constructor^1@{k@1} \\ |
|
1611 & & \vdots \\ |
|
1612 \mathtt{and} & t@n(A@1,\ldots,A@h) & = & |
|
1613 constructor^n@1~ ~\mid~ \ldots ~\mid~ constructor^n@{k@n} |
|
1614 \end{array} |
|
1615 \] |
|
1616 Here $t@1$, \ldots,~$t@n$ are identifiers and $A@1$, \ldots,~$A@h$ are |
|
1617 variables: the datatype's parameters. Each constructor specification has the |
|
1618 form \dquotesoff |
|
1619 \[ C \hbox{\tt~( } \hbox{\tt"} x@1 \hbox{\tt:} T@1 \hbox{\tt"},\; |
|
1620 \ldots,\; |
|
1621 \hbox{\tt"} x@m \hbox{\tt:} T@m \hbox{\tt"} |
|
1622 \hbox{\tt~)} |
|
1623 \] |
|
1624 Here $C$ is the constructor name, and variables $x@1$, \ldots,~$x@m$ are the |
|
1625 constructor arguments, belonging to the sets $T@1$, \ldots, $T@m$, |
|
1626 respectively. Typically each $T@j$ is either a constant set, a datatype |
|
1627 parameter (one of $A@1$, \ldots, $A@h$) or a recursive occurrence of one of |
|
1628 the datatypes, say $t@i(A@1,\ldots,A@h)$. More complex possibilities exist, |
|
1629 but they are much harder to realize. Often, additional information must be |
|
1630 supplied in the form of theorems. |
|
1631 |
|
1632 A datatype can occur recursively as the argument of some function~$F$. This |
|
1633 is called a {\em nested} (or \emph{indirect}) occurrence. It is only allowed |
|
1634 if the datatype package is given a theorem asserting that $F$ is monotonic. |
|
1635 If the datatype has indirect occurrences, then Isabelle/ZF does not support |
|
1636 recursive function definitions. |
|
1637 |
|
1638 A simple example of a datatype is \isa{list}, which is built-in, and is |
|
1639 defined by |
|
1640 \begin{alltt*}\isastyleminor |
|
1641 consts list :: "i=>i" |
|
1642 datatype "list(A)" = Nil | Cons ("a \isasymin A", "l \isasymin list(A)") |
|
1643 \end{alltt*} |
|
1644 Note that the datatype operator must be declared as a constant first. |
|
1645 However, the package declares the constructors. Here, \isa{Nil} gets type |
|
1646 $i$ and \isa{Cons} gets type $[i,i]\To i$. |
|
1647 |
|
1648 Trees and forests can be modelled by the mutually recursive datatype |
|
1649 definition |
|
1650 \begin{alltt*}\isastyleminor |
|
1651 consts |
|
1652 tree :: "i=>i" |
|
1653 forest :: "i=>i" |
|
1654 tree_forest :: "i=>i" |
|
1655 datatype "tree(A)" = Tcons ("a{\isasymin}A", "f{\isasymin}forest(A)") |
|
1656 and "forest(A)" = Fnil | Fcons ("t{\isasymin}tree(A)", "f{\isasymin}forest(A)") |
|
1657 \end{alltt*} |
|
1658 Here $\isa{tree}(A)$ is the set of trees over $A$, $\isa{forest}(A)$ is |
|
1659 the set of forests over $A$, and $\isa{tree_forest}(A)$ is the union of |
|
1660 the previous two sets. All three operators must be declared first. |
|
1661 |
|
1662 The datatype \isa{term}, which is defined by |
|
1663 \begin{alltt*}\isastyleminor |
|
1664 consts term :: "i=>i" |
|
1665 datatype "term(A)" = Apply ("a \isasymin A", "l \isasymin list(term(A))") |
|
1666 monos list_mono |
|
1667 type_elims list_univ [THEN subsetD, elim_format] |
|
1668 \end{alltt*} |
|
1669 is an example of nested recursion. (The theorem \isa{list_mono} is proved |
|
1670 in theory \isa{List}, and the \isa{term} example is developed in |
|
1671 theory |
|
1672 \thydx{Induct/Term}.) |
|
1673 |
|
1674 \subsubsection{Freeness of the constructors} |
|
1675 |
|
1676 Constructors satisfy {\em freeness} properties. Constructions are distinct, |
|
1677 for example $\isa{Nil}\not=\isa{Cons}(a,l)$, and they are injective, for |
|
1678 example $\isa{Cons}(a,l)=\isa{Cons}(a',l') \bimp a=a' \conj l=l'$. |
|
1679 Because the number of freeness is quadratic in the number of constructors, the |
|
1680 datatype package does not prove them. Instead, it ensures that simplification |
|
1681 will prove them dynamically: when the simplifier encounters a formula |
|
1682 asserting the equality of two datatype constructors, it performs freeness |
|
1683 reasoning. |
|
1684 |
|
1685 Freeness reasoning can also be done using the classical reasoner, but it is |
|
1686 more complicated. You have to add some safe elimination rules rules to the |
|
1687 claset. For the \isa{list} datatype, they are called |
|
1688 \isa{list.free_elims}. Occasionally this exposes the underlying |
|
1689 representation of some constructor, which can be rectified using the command |
|
1690 \isa{unfold list.con_defs [symmetric]}. |
|
1691 |
|
1692 |
|
1693 \subsubsection{Structural induction} |
|
1694 |
|
1695 The datatype package also provides structural induction rules. For datatypes |
|
1696 without mutual or nested recursion, the rule has the form exemplified by |
|
1697 \isa{list.induct} in Fig.\ts\ref{zf-list}. For mutually recursive |
|
1698 datatypes, the induction rule is supplied in two forms. Consider datatype |
|
1699 \isa{TF}. The rule \isa{tree_forest.induct} performs induction over a |
|
1700 single predicate~\isa{P}, which is presumed to be defined for both trees |
|
1701 and forests: |
|
1702 \begin{alltt*}\isastyleminor |
|
1703 [| x \isasymin tree_forest(A); |
|
1704 !!a f. [| a \isasymin A; f \isasymin forest(A); P(f) |] ==> P(Tcons(a, f)); |
|
1705 P(Fnil); |
|
1706 !!f t. [| t \isasymin tree(A); P(t); f \isasymin forest(A); P(f) |] |
|
1707 ==> P(Fcons(t, f)) |
|
1708 |] ==> P(x) |
|
1709 \end{alltt*} |
|
1710 The rule \isa{tree_forest.mutual_induct} performs induction over two |
|
1711 distinct predicates, \isa{P_tree} and \isa{P_forest}. |
|
1712 \begin{alltt*}\isastyleminor |
|
1713 [| !!a f. |
|
1714 [| a{\isasymin}A; f{\isasymin}forest(A); P_forest(f) |] ==> P_tree(Tcons(a,f)); |
|
1715 P_forest(Fnil); |
|
1716 !!f t. [| t{\isasymin}tree(A); P_tree(t); f{\isasymin}forest(A); P_forest(f) |] |
|
1717 ==> P_forest(Fcons(t, f)) |
|
1718 |] ==> ({\isasymforall}za. za \isasymin tree(A) --> P_tree(za)) & |
|
1719 ({\isasymforall}za. za \isasymin forest(A) --> P_forest(za)) |
|
1720 \end{alltt*} |
|
1721 |
|
1722 For datatypes with nested recursion, such as the \isa{term} example from |
|
1723 above, things are a bit more complicated. The rule \isa{term.induct} |
|
1724 refers to the monotonic operator, \isa{list}: |
|
1725 \begin{alltt*}\isastyleminor |
|
1726 [| x \isasymin term(A); |
|
1727 !!a l. [| a\isasymin{}A; l\isasymin{}list(Collect(term(A), P)) |] ==> P(Apply(a,l)) |
|
1728 |] ==> P(x) |
|
1729 \end{alltt*} |
|
1730 The theory \isa{Induct/Term.thy} derives two higher-level induction rules, |
|
1731 one of which is particularly useful for proving equations: |
|
1732 \begin{alltt*}\isastyleminor |
|
1733 [| t \isasymin term(A); |
|
1734 !!x zs. [| x \isasymin A; zs \isasymin list(term(A)); map(f, zs) = map(g, zs) |] |
|
1735 ==> f(Apply(x, zs)) = g(Apply(x, zs)) |
|
1736 |] ==> f(t) = g(t) |
|
1737 \end{alltt*} |
|
1738 How this can be generalized to other nested datatypes is a matter for future |
|
1739 research. |
|
1740 |
|
1741 |
|
1742 \subsubsection{The \isa{case} operator} |
|
1743 |
|
1744 The package defines an operator for performing case analysis over the |
|
1745 datatype. For \isa{list}, it is called \isa{list_case} and satisfies |
|
1746 the equations |
|
1747 \begin{ttbox}\isastyleminor |
|
1748 list_case(f_Nil, f_Cons, []) = f_Nil |
|
1749 list_case(f_Nil, f_Cons, Cons(a, l)) = f_Cons(a, l) |
|
1750 \end{ttbox} |
|
1751 Here \isa{f_Nil} is the value to return if the argument is \isa{Nil} and |
|
1752 \isa{f_Cons} is a function that computes the value to return if the |
|
1753 argument has the form $\isa{Cons}(a,l)$. The function can be expressed as |
|
1754 an abstraction, over patterns if desired (\S\ref{sec:pairs}). |
|
1755 |
|
1756 For mutually recursive datatypes, there is a single \isa{case} operator. |
|
1757 In the tree/forest example, the constant \isa{tree_forest_case} handles all |
|
1758 of the constructors of the two datatypes. |
|
1759 |
|
1760 |
|
1761 \subsection{Defining datatypes} |
|
1762 |
|
1763 The theory syntax for datatype definitions is shown in the |
|
1764 Isabelle/Isar reference manual. In order to be well-formed, a |
|
1765 datatype definition has to obey the rules stated in the previous |
|
1766 section. As a result the theory is extended with the new types, the |
|
1767 constructors, and the theorems listed in the previous section. |
|
1768 |
|
1769 Codatatypes are declared like datatypes and are identical to them in every |
|
1770 respect except that they have a coinduction rule instead of an induction rule. |
|
1771 Note that while an induction rule has the effect of limiting the values |
|
1772 contained in the set, a coinduction rule gives a way of constructing new |
|
1773 values of the set. |
|
1774 |
|
1775 Most of the theorems about datatypes become part of the default simpset. You |
|
1776 never need to see them again because the simplifier applies them |
|
1777 automatically. |
|
1778 |
|
1779 \subsubsection{Specialized methods for datatypes} |
|
1780 |
|
1781 Induction and case-analysis can be invoked using these special-purpose |
|
1782 methods: |
|
1783 \begin{ttdescription} |
|
1784 \item[\methdx{induct_tac} $x$] applies structural |
|
1785 induction on variable $x$ to subgoal~1, provided the type of $x$ is a |
|
1786 datatype. The induction variable should not occur among other assumptions |
|
1787 of the subgoal. |
|
1788 \end{ttdescription} |
|
1789 % |
|
1790 % we also have the ind_cases method, but what does it do? |
|
1791 In some situations, induction is overkill and a case distinction over all |
|
1792 constructors of the datatype suffices. |
|
1793 \begin{ttdescription} |
|
1794 \item[\methdx{case_tac} $x$] |
|
1795 performs a case analysis for the variable~$x$. |
|
1796 \end{ttdescription} |
|
1797 |
|
1798 Both tactics can only be applied to a variable, whose typing must be given in |
|
1799 some assumption, for example the assumption \isa{x \isasymin \ list(A)}. The tactics |
|
1800 also work for the natural numbers (\isa{nat}) and disjoint sums, although |
|
1801 these sets were not defined using the datatype package. (Disjoint sums are |
|
1802 not recursive, so only \isa{case_tac} is available.) |
|
1803 |
|
1804 Structured Isar methods are also available. Below, $t$ |
|
1805 stands for the name of the datatype. |
|
1806 \begin{ttdescription} |
|
1807 \item[\methdx{induct} \isa{set:}\ $t$] is the Isar induction tactic. |
|
1808 \item[\methdx{cases} \isa{set:}\ $t$] is the Isar case-analysis tactic. |
|
1809 \end{ttdescription} |
|
1810 |
|
1811 |
|
1812 \subsubsection{The theorems proved by a datatype declaration} |
|
1813 |
|
1814 Here are some more details for the technically minded. Processing the |
|
1815 datatype declaration of a set~$t$ produces a name space~$t$ containing |
|
1816 the following theorems: |
|
1817 \begin{ttbox}\isastyleminor |
|
1818 intros \textrm{the introduction rules} |
|
1819 cases \textrm{the case analysis rule} |
|
1820 induct \textrm{the standard induction rule} |
|
1821 mutual_induct \textrm{the mutual induction rule, if needed} |
|
1822 case_eqns \textrm{equations for the case operator} |
|
1823 recursor_eqns \textrm{equations for the recursor} |
|
1824 simps \textrm{the union of} case_eqns \textrm{and} recursor_eqns |
|
1825 con_defs \textrm{definitions of the case operator and constructors} |
|
1826 free_iffs \textrm{logical equivalences for proving freeness} |
|
1827 free_elims \textrm{elimination rules for proving freeness} |
|
1828 defs \textrm{datatype definition(s)} |
|
1829 \end{ttbox} |
|
1830 Furthermore there is the theorem $C$ for every constructor~$C$; for |
|
1831 example, the \isa{list} datatype's introduction rules are bound to the |
|
1832 identifiers \isa{Nil} and \isa{Cons}. |
|
1833 |
|
1834 For a codatatype, the component \isa{coinduct} is the coinduction rule, |
|
1835 replacing the \isa{induct} component. |
|
1836 |
|
1837 See the theories \isa{Induct/Ntree} and \isa{Induct/Brouwer} for examples of |
|
1838 infinitely branching datatypes. See theory \isa{Induct/LList} for an example |
|
1839 of a codatatype. Some of these theories illustrate the use of additional, |
|
1840 undocumented features of the datatype package. Datatype definitions are |
|
1841 reduced to inductive definitions, and the advanced features should be |
|
1842 understood in that light. |
|
1843 |
|
1844 |
|
1845 \subsection{Examples} |
|
1846 |
|
1847 \subsubsection{The datatype of binary trees} |
|
1848 |
|
1849 Let us define the set $\isa{bt}(A)$ of binary trees over~$A$. The theory |
|
1850 must contain these lines: |
|
1851 \begin{alltt*}\isastyleminor |
|
1852 consts bt :: "i=>i" |
|
1853 datatype "bt(A)" = Lf | Br ("a\isasymin{}A", "t1\isasymin{}bt(A)", "t2\isasymin{}bt(A)") |
|
1854 \end{alltt*} |
|
1855 After loading the theory, we can prove some theorem. |
|
1856 We begin by declaring the constructor's typechecking rules |
|
1857 as simplification rules: |
|
1858 \begin{isabelle} |
|
1859 \isacommand{declare}\ bt.intros\ [simp]% |
|
1860 \end{isabelle} |
|
1861 |
|
1862 Our first example is the theorem that no tree equals its |
|
1863 left branch. To make the inductive hypothesis strong enough, |
|
1864 the proof requires a quantified induction formula, but |
|
1865 the \isa{rule\_format} attribute will remove the quantifiers |
|
1866 before the theorem is stored. |
|
1867 \begin{isabelle} |
|
1868 \isacommand{lemma}\ Br\_neq\_left\ [rule\_format]:\ "l\isasymin bt(A)\ ==>\ \isasymforall x\ r.\ Br(x,l,r)\isasymnoteq{}l"\isanewline |
|
1869 \ 1.\ l\ \isasymin \ bt(A)\ \isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l% |
|
1870 \end{isabelle} |
|
1871 This can be proved by the structural induction tactic: |
|
1872 \begin{isabelle} |
|
1873 \ \ \isacommand{apply}\ (induct\_tac\ l)\isanewline |
|
1874 \ 1.\ \isasymforall x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline |
|
1875 \ 2.\ \isasymAnd a\ t1\ t2.\isanewline |
|
1876 \isaindent{\ 2.\ \ \ \ }\isasymlbrakk a\ \isasymin \ A;\ t1\ \isasymin \ bt(A);\ \isasymforall x\ r.\ Br(x,\ t1,\ r)\ \isasymnoteq \ t1;\ t2\ \isasymin \ bt(A);\isanewline |
|
1877 \isaindent{\ 2.\ \ \ \ \ \ \ }\isasymforall x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline |
|
1878 \isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2) |
|
1879 \end{isabelle} |
|
1880 Both subgoals are proved using \isa{auto}, which performs the necessary |
|
1881 freeness reasoning. |
|
1882 \begin{isabelle} |
|
1883 \ \ \isacommand{apply}\ auto\isanewline |
|
1884 No\ subgoals!\isanewline |
|
1885 \isacommand{done} |
|
1886 \end{isabelle} |
|
1887 |
|
1888 An alternative proof uses Isar's fancy \isa{induct} method, which |
|
1889 automatically quantifies over all free variables: |
|
1890 |
|
1891 \begin{isabelle} |
|
1892 \isacommand{lemma}\ Br\_neq\_left':\ "l\ \isasymin \ bt(A)\ ==>\ (!!x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l)"\isanewline |
|
1893 \ \ \isacommand{apply}\ (induct\ set:\ bt)\isanewline |
|
1894 \ 1.\ \isasymAnd x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline |
|
1895 \ 2.\ \isasymAnd a\ t1\ t2\ x\ r.\isanewline |
|
1896 \isaindent{\ 2.\ \ \ \ }\isasymlbrakk a\ \isasymin \ A;\ t1\ \isasymin \ bt(A);\ \isasymAnd x\ r.\ Br(x,\ t1,\ r)\ \isasymnoteq \ t1;\ t2\ \isasymin \ bt(A);\isanewline |
|
1897 \isaindent{\ 2.\ \ \ \ \ \ \ }\isasymAnd x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline |
|
1898 \isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2) |
|
1899 \end{isabelle} |
|
1900 Compare the form of the induction hypotheses with the corresponding ones in |
|
1901 the previous proof. As before, to conclude requires only \isa{auto}. |
|
1902 |
|
1903 When there are only a few constructors, we might prefer to prove the freenness |
|
1904 theorems for each constructor. This is simple: |
|
1905 \begin{isabelle} |
|
1906 \isacommand{lemma}\ Br\_iff:\ "Br(a,l,r)\ =\ Br(a',l',r')\ <->\ a=a'\ \&\ l=l'\ \&\ r=r'"\isanewline |
|
1907 \ \ \isacommand{by}\ (blast\ elim!:\ bt.free\_elims) |
|
1908 \end{isabelle} |
|
1909 Here we see a demonstration of freeness reasoning using |
|
1910 \isa{bt.free\_elims}, but simpler still is just to apply \isa{auto}. |
|
1911 |
|
1912 An \ttindex{inductive\_cases} declaration generates instances of the |
|
1913 case analysis rule that have been simplified using freeness |
|
1914 reasoning. |
|
1915 \begin{isabelle} |
|
1916 \isacommand{inductive\_cases}\ Br\_in\_bt:\ "Br(a,\ l,\ r)\ \isasymin \ bt(A)" |
|
1917 \end{isabelle} |
|
1918 The theorem just created is |
|
1919 \begin{isabelle} |
|
1920 \isasymlbrakk Br(a,\ l,\ r)\ \isasymin \ bt(A);\ \isasymlbrakk a\ \isasymin \ A;\ l\ \isasymin \ bt(A);\ r\ \isasymin \ bt(A)\isasymrbrakk \ \isasymLongrightarrow \ Q\isasymrbrakk \ \isasymLongrightarrow \ Q. |
|
1921 \end{isabelle} |
|
1922 It is an elimination rule that from $\isa{Br}(a,l,r)\in\isa{bt}(A)$ |
|
1923 lets us infer $a\in A$, $l\in\isa{bt}(A)$ and |
|
1924 $r\in\isa{bt}(A)$. |
|
1925 |
|
1926 |
|
1927 \subsubsection{Mixfix syntax in datatypes} |
|
1928 |
|
1929 Mixfix syntax is sometimes convenient. The theory \isa{Induct/PropLog} makes a |
|
1930 deep embedding of propositional logic: |
|
1931 \begin{alltt*}\isastyleminor |
|
1932 consts prop :: i |
|
1933 datatype "prop" = Fls |
|
1934 | Var ("n \isasymin nat") ("#_" [100] 100) |
|
1935 | "=>" ("p \isasymin prop", "q \isasymin prop") (infixr 90) |
|
1936 \end{alltt*} |
|
1937 The second constructor has a special $\#n$ syntax, while the third constructor |
|
1938 is an infixed arrow. |
|
1939 |
|
1940 |
|
1941 \subsubsection{A giant enumeration type} |
|
1942 |
|
1943 This example shows a datatype that consists of 60 constructors: |
|
1944 \begin{alltt*}\isastyleminor |
|
1945 consts enum :: i |
|
1946 datatype |
|
1947 "enum" = C00 | C01 | C02 | C03 | C04 | C05 | C06 | C07 | C08 | C09 |
|
1948 | C10 | C11 | C12 | C13 | C14 | C15 | C16 | C17 | C18 | C19 |
|
1949 | C20 | C21 | C22 | C23 | C24 | C25 | C26 | C27 | C28 | C29 |
|
1950 | C30 | C31 | C32 | C33 | C34 | C35 | C36 | C37 | C38 | C39 |
|
1951 | C40 | C41 | C42 | C43 | C44 | C45 | C46 | C47 | C48 | C49 |
|
1952 | C50 | C51 | C52 | C53 | C54 | C55 | C56 | C57 | C58 | C59 |
|
1953 end |
|
1954 \end{alltt*} |
|
1955 The datatype package scales well. Even though all properties are proved |
|
1956 rather than assumed, full processing of this definition takes around two seconds |
|
1957 (on a 1.8GHz machine). The constructors have a balanced representation, |
|
1958 related to binary notation, so freeness properties can be proved fast. |
|
1959 \begin{isabelle} |
|
1960 \isacommand{lemma}\ "C00 \isasymnoteq\ C01"\isanewline |
|
1961 \ \ \isacommand{by}\ simp |
|
1962 \end{isabelle} |
|
1963 You need not derive such inequalities explicitly. The simplifier will |
|
1964 dispose of them automatically. |
|
1965 |
|
1966 \index{*datatype|)} |
|
1967 |
|
1968 |
|
1969 \subsection{Recursive function definitions}\label{sec:ZF:recursive} |
|
1970 \index{recursive functions|see{recursion}} |
|
1971 \index{*primrec|(} |
|
1972 \index{recursion!primitive|(} |
|
1973 |
|
1974 Datatypes come with a uniform way of defining functions, {\bf primitive |
|
1975 recursion}. Such definitions rely on the recursion operator defined by the |
|
1976 datatype package. Isabelle proves the desired recursion equations as |
|
1977 theorems. |
|
1978 |
|
1979 In principle, one could introduce primitive recursive functions by asserting |
|
1980 their reduction rules as axioms. Here is a dangerous way of defining a |
|
1981 recursive function over binary trees: |
|
1982 \begin{isabelle} |
|
1983 \isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline |
|
1984 \isacommand{axioms}\isanewline |
|
1985 \ \ n\_nodes\_Lf:\ "n\_nodes(Lf)\ =\ 0"\isanewline |
|
1986 \ \ n\_nodes\_Br:\ "n\_nodes(Br(a,l,r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))" |
|
1987 \end{isabelle} |
|
1988 Asserting axioms brings the danger of accidentally introducing |
|
1989 contradictions. It should be avoided whenever possible. |
|
1990 |
|
1991 The \ttindex{primrec} declaration is a safe means of defining primitive |
|
1992 recursive functions on datatypes: |
|
1993 \begin{isabelle} |
|
1994 \isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline |
|
1995 \isacommand{primrec}\isanewline |
|
1996 \ \ "n\_nodes(Lf)\ =\ 0"\isanewline |
|
1997 \ \ "n\_nodes(Br(a,\ l,\ r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))" |
|
1998 \end{isabelle} |
|
1999 Isabelle will now derive the two equations from a low-level definition |
|
2000 based upon well-founded recursion. If they do not define a legitimate |
|
2001 recursion, then Isabelle will reject the declaration. |
|
2002 |
|
2003 |
|
2004 \subsubsection{Syntax of recursive definitions} |
|
2005 |
|
2006 The general form of a primitive recursive definition is |
|
2007 \begin{ttbox}\isastyleminor |
|
2008 primrec |
|
2009 {\it reduction rules} |
|
2010 \end{ttbox} |
|
2011 where \textit{reduction rules} specify one or more equations of the form |
|
2012 \[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \, |
|
2013 \dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$ |
|
2014 contains only the free variables on the left-hand side, and all recursive |
|
2015 calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$. |
|
2016 There must be at most one reduction rule for each constructor. The order is |
|
2017 immaterial. For missing constructors, the function is defined to return zero. |
|
2018 |
|
2019 All reduction rules are added to the default simpset. |
|
2020 If you would like to refer to some rule by name, then you must prefix |
|
2021 the rule with an identifier. These identifiers, like those in the |
|
2022 \isa{rules} section of a theory, will be visible in proof scripts. |
|
2023 |
|
2024 The reduction rules become part of the default simpset, which |
|
2025 leads to short proof scripts: |
|
2026 \begin{isabelle} |
|
2027 \isacommand{lemma}\ n\_nodes\_type\ [simp]:\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes(t)\ \isasymin \ nat"\isanewline |
|
2028 \ \ \isacommand{by}\ (induct\_tac\ t,\ auto) |
|
2029 \end{isabelle} |
|
2030 |
|
2031 You can even use the \isa{primrec} form with non-recursive datatypes and |
|
2032 with codatatypes. Recursion is not allowed, but it provides a convenient |
|
2033 syntax for defining functions by cases. |
|
2034 |
|
2035 |
|
2036 \subsubsection{Example: varying arguments} |
|
2037 |
|
2038 All arguments, other than the recursive one, must be the same in each equation |
|
2039 and in each recursive call. To get around this restriction, use explict |
|
2040 $\lambda$-abstraction and function application. For example, let us |
|
2041 define the tail-recursive version of \isa{n\_nodes}, using an |
|
2042 accumulating argument for the counter. The second argument, $k$, varies in |
|
2043 recursive calls. |
|
2044 \begin{isabelle} |
|
2045 \isacommand{consts}\ \ n\_nodes\_aux\ ::\ "i\ =>\ i"\isanewline |
|
2046 \isacommand{primrec}\isanewline |
|
2047 \ \ "n\_nodes\_aux(Lf)\ =\ (\isasymlambda k\ \isasymin \ nat.\ k)"\isanewline |
|
2048 \ \ "n\_nodes\_aux(Br(a,l,r))\ =\ \isanewline |
|
2049 \ \ \ \ \ \ (\isasymlambda k\ \isasymin \ nat.\ n\_nodes\_aux(r)\ `\ \ (n\_nodes\_aux(l)\ `\ succ(k)))" |
|
2050 \end{isabelle} |
|
2051 Now \isa{n\_nodes\_aux(t)\ `\ k} is our function in two arguments. We |
|
2052 can prove a theorem relating it to \isa{n\_nodes}. Note the quantification |
|
2053 over \isa{k\ \isasymin \ nat}: |
|
2054 \begin{isabelle} |
|
2055 \isacommand{lemma}\ n\_nodes\_aux\_eq\ [rule\_format]:\isanewline |
|
2056 \ \ \ \ \ "t\ \isasymin \ bt(A)\ ==>\ \isasymforall k\ \isasymin \ nat.\ n\_nodes\_aux(t)`k\ =\ n\_nodes(t)\ \#+\ k"\isanewline |
|
2057 \ \ \isacommand{by}\ (induct\_tac\ t,\ simp\_all) |
|
2058 \end{isabelle} |
|
2059 |
|
2060 Now, we can use \isa{n\_nodes\_aux} to define a tail-recursive version |
|
2061 of \isa{n\_nodes}: |
|
2062 \begin{isabelle} |
|
2063 \isacommand{constdefs}\isanewline |
|
2064 \ \ n\_nodes\_tail\ ::\ "i\ =>\ i"\isanewline |
|
2065 \ \ \ "n\_nodes\_tail(t)\ ==\ n\_nodes\_aux(t)\ `\ 0" |
|
2066 \end{isabelle} |
|
2067 It is easy to |
|
2068 prove that \isa{n\_nodes\_tail} is equivalent to \isa{n\_nodes}: |
|
2069 \begin{isabelle} |
|
2070 \isacommand{lemma}\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes\_tail(t)\ =\ n\_nodes(t)"\isanewline |
|
2071 \ \isacommand{by}\ (simp\ add:\ n\_nodes\_tail\_def\ n\_nodes\_aux\_eq) |
|
2072 \end{isabelle} |
|
2073 |
|
2074 |
|
2075 |
|
2076 |
|
2077 \index{recursion!primitive|)} |
|
2078 \index{*primrec|)} |
|
2079 |
|
2080 |
|
2081 \section{Inductive and coinductive definitions} |
|
2082 \index{*inductive|(} |
|
2083 \index{*coinductive|(} |
|
2084 |
|
2085 An {\bf inductive definition} specifies the least set~$R$ closed under given |
|
2086 rules. (Applying a rule to elements of~$R$ yields a result within~$R$.) For |
|
2087 example, a structural operational semantics is an inductive definition of an |
|
2088 evaluation relation. Dually, a {\bf coinductive definition} specifies the |
|
2089 greatest set~$R$ consistent with given rules. (Every element of~$R$ can be |
|
2090 seen as arising by applying a rule to elements of~$R$.) An important example |
|
2091 is using bisimulation relations to formalise equivalence of processes and |
|
2092 infinite data structures. |
|
2093 |
|
2094 A theory file may contain any number of inductive and coinductive |
|
2095 definitions. They may be intermixed with other declarations; in |
|
2096 particular, the (co)inductive sets {\bf must} be declared separately as |
|
2097 constants, and may have mixfix syntax or be subject to syntax translations. |
|
2098 |
|
2099 Each (co)inductive definition adds definitions to the theory and also |
|
2100 proves some theorems. It behaves identially to the analogous |
|
2101 inductive definition except that instead of an induction rule there is |
|
2102 a coinduction rule. Its treatment of coinduction is described in |
|
2103 detail in a separate paper,% |
|
2104 \footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is |
|
2105 distributed with Isabelle as \emph{A Fixedpoint Approach to |
|
2106 (Co)Inductive and (Co)Datatype Definitions}.} % |
|
2107 which you might refer to for background information. |
|
2108 |
|
2109 |
|
2110 \subsection{The syntax of a (co)inductive definition} |
|
2111 An inductive definition has the form |
|
2112 \begin{ttbox}\isastyleminor |
|
2113 inductive |
|
2114 domains {\it domain declarations} |
|
2115 intros {\it introduction rules} |
|
2116 monos {\it monotonicity theorems} |
|
2117 con_defs {\it constructor definitions} |
|
2118 type_intros {\it introduction rules for type-checking} |
|
2119 type_elims {\it elimination rules for type-checking} |
|
2120 \end{ttbox} |
|
2121 A coinductive definition is identical, but starts with the keyword |
|
2122 \isa{co\-inductive}. |
|
2123 |
|
2124 The \isa{monos}, \isa{con\_defs}, \isa{type\_intros} and \isa{type\_elims} |
|
2125 sections are optional. If present, each is specified as a list of |
|
2126 theorems, which may contain Isar attributes as usual. |
|
2127 |
|
2128 \begin{description} |
|
2129 \item[\it domain declarations] are items of the form |
|
2130 {\it string\/}~\isa{\isasymsubseteq }~{\it string}, associating each recursive set with |
|
2131 its domain. (The domain is some existing set that is large enough to |
|
2132 hold the new set being defined.) |
|
2133 |
|
2134 \item[\it introduction rules] specify one or more introduction rules in |
|
2135 the form {\it ident\/}~{\it string}, where the identifier gives the name of |
|
2136 the rule in the result structure. |
|
2137 |
|
2138 \item[\it monotonicity theorems] are required for each operator applied to |
|
2139 a recursive set in the introduction rules. There \textbf{must} be a theorem |
|
2140 of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each premise $t\in M(R_i)$ |
|
2141 in an introduction rule! |
|
2142 |
|
2143 \item[\it constructor definitions] contain definitions of constants |
|
2144 appearing in the introduction rules. The (co)datatype package supplies |
|
2145 the constructors' definitions here. Most (co)inductive definitions omit |
|
2146 this section; one exception is the primitive recursive functions example; |
|
2147 see theory \isa{Induct/Primrec}. |
|
2148 |
|
2149 \item[\it type\_intros] consists of introduction rules for type-checking the |
|
2150 definition: for demonstrating that the new set is included in its domain. |
|
2151 (The proof uses depth-first search.) |
|
2152 |
|
2153 \item[\it type\_elims] consists of elimination rules for type-checking the |
|
2154 definition. They are presumed to be safe and are applied as often as |
|
2155 possible prior to the \isa{type\_intros} search. |
|
2156 \end{description} |
|
2157 |
|
2158 The package has a few restrictions: |
|
2159 \begin{itemize} |
|
2160 \item The theory must separately declare the recursive sets as |
|
2161 constants. |
|
2162 |
|
2163 \item The names of the recursive sets must be identifiers, not infix |
|
2164 operators. |
|
2165 |
|
2166 \item Side-conditions must not be conjunctions. However, an introduction rule |
|
2167 may contain any number of side-conditions. |
|
2168 |
|
2169 \item Side-conditions of the form $x=t$, where the variable~$x$ does not |
|
2170 occur in~$t$, will be substituted through the rule \isa{mutual\_induct}. |
|
2171 \end{itemize} |
|
2172 |
|
2173 |
|
2174 \subsection{Example of an inductive definition} |
|
2175 |
|
2176 Below, we shall see how Isabelle/ZF defines the finite powerset |
|
2177 operator. The first step is to declare the constant~\isa{Fin}. Then we |
|
2178 must declare it inductively, with two introduction rules: |
|
2179 \begin{isabelle} |
|
2180 \isacommand{consts}\ \ Fin\ ::\ "i=>i"\isanewline |
|
2181 \isacommand{inductive}\isanewline |
|
2182 \ \ \isakeyword{domains}\ \ \ "Fin(A)"\ \isasymsubseteq\ "Pow(A)"\isanewline |
|
2183 \ \ \isakeyword{intros}\isanewline |
|
2184 \ \ \ \ emptyI:\ \ "0\ \isasymin\ Fin(A)"\isanewline |
|
2185 \ \ \ \ consI:\ \ \ "[|\ a\ \isasymin\ A;\ \ b\ \isasymin\ Fin(A)\ |]\ ==>\ cons(a,b)\ \isasymin\ Fin(A)"\isanewline |
|
2186 \ \ \isakeyword{type\_intros}\ \ empty\_subsetI\ cons\_subsetI\ PowI\isanewline |
|
2187 \ \ \isakeyword{type\_elims}\ \ \ PowD\ [THEN\ revcut\_rl]\end{isabelle} |
|
2188 The resulting theory contains a name space, called~\isa{Fin}. |
|
2189 The \isa{Fin}$~A$ introduction rules can be referred to collectively as |
|
2190 \isa{Fin.intros}, and also individually as \isa{Fin.emptyI} and |
|
2191 \isa{Fin.consI}. The induction rule is \isa{Fin.induct}. |
|
2192 |
|
2193 The chief problem with making (co)inductive definitions involves type-checking |
|
2194 the rules. Sometimes, additional theorems need to be supplied under |
|
2195 \isa{type_intros} or \isa{type_elims}. If the package fails when trying |
|
2196 to prove your introduction rules, then set the flag \ttindexbold{trace_induct} |
|
2197 to \isa{true} and try again. (See the manual \emph{A Fixedpoint Approach |
|
2198 \ldots} for more discussion of type-checking.) |
|
2199 |
|
2200 In the example above, $\isa{Pow}(A)$ is given as the domain of |
|
2201 $\isa{Fin}(A)$, for obviously every finite subset of~$A$ is a subset |
|
2202 of~$A$. However, the inductive definition package can only prove that given a |
|
2203 few hints. |
|
2204 Here is the output that results (with the flag set) when the |
|
2205 \isa{type_intros} and \isa{type_elims} are omitted from the inductive |
|
2206 definition above: |
|
2207 \begin{alltt*}\isastyleminor |
|
2208 Inductive definition Finite.Fin |
|
2209 Fin(A) == |
|
2210 lfp(Pow(A), |
|
2211 \%X. {z\isasymin{}Pow(A) . z = 0 | ({\isasymexists}a b. z = cons(a,b) & a\isasymin{}A & b\isasymin{}X)}) |
|
2212 Proving monotonicity... |
|
2213 \ttbreak |
|
2214 Proving the introduction rules... |
|
2215 The type-checking subgoal: |
|
2216 0 \isasymin Fin(A) |
|
2217 1. 0 \isasymin Pow(A) |
|
2218 \ttbreak |
|
2219 The subgoal after monos, type_elims: |
|
2220 0 \isasymin Fin(A) |
|
2221 1. 0 \isasymin Pow(A) |
|
2222 *** prove_goal: tactic failed |
|
2223 \end{alltt*} |
|
2224 We see the need to supply theorems to let the package prove |
|
2225 $\emptyset\in\isa{Pow}(A)$. Restoring the \isa{type_intros} but not the |
|
2226 \isa{type_elims}, we again get an error message: |
|
2227 \begin{alltt*}\isastyleminor |
|
2228 The type-checking subgoal: |
|
2229 0 \isasymin Fin(A) |
|
2230 1. 0 \isasymin Pow(A) |
|
2231 \ttbreak |
|
2232 The subgoal after monos, type_elims: |
|
2233 0 \isasymin Fin(A) |
|
2234 1. 0 \isasymin Pow(A) |
|
2235 \ttbreak |
|
2236 The type-checking subgoal: |
|
2237 cons(a, b) \isasymin Fin(A) |
|
2238 1. [| a \isasymin A; b \isasymin Fin(A) |] ==> cons(a, b) \isasymin Pow(A) |
|
2239 \ttbreak |
|
2240 The subgoal after monos, type_elims: |
|
2241 cons(a, b) \isasymin Fin(A) |
|
2242 1. [| a \isasymin A; b \isasymin Pow(A) |] ==> cons(a, b) \isasymin Pow(A) |
|
2243 *** prove_goal: tactic failed |
|
2244 \end{alltt*} |
|
2245 The first rule has been type-checked, but the second one has failed. The |
|
2246 simplest solution to such problems is to prove the failed subgoal separately |
|
2247 and to supply it under \isa{type_intros}. The solution actually used is |
|
2248 to supply, under \isa{type_elims}, a rule that changes |
|
2249 $b\in\isa{Pow}(A)$ to $b\subseteq A$; together with \isa{cons_subsetI} |
|
2250 and \isa{PowI}, it is enough to complete the type-checking. |
|
2251 |
|
2252 |
|
2253 |
|
2254 \subsection{Further examples} |
|
2255 |
|
2256 An inductive definition may involve arbitrary monotonic operators. Here is a |
|
2257 standard example: the accessible part of a relation. Note the use |
|
2258 of~\isa{Pow} in the introduction rule and the corresponding mention of the |
|
2259 rule \isa{Pow\_mono} in the \isa{monos} list. If the desired rule has a |
|
2260 universally quantified premise, usually the effect can be obtained using |
|
2261 \isa{Pow}. |
|
2262 \begin{isabelle} |
|
2263 \isacommand{consts}\ \ acc\ ::\ "i\ =>\ i"\isanewline |
|
2264 \isacommand{inductive}\isanewline |
|
2265 \ \ \isakeyword{domains}\ "acc(r)"\ \isasymsubseteq \ "field(r)"\isanewline |
|
2266 \ \ \isakeyword{intros}\isanewline |
|
2267 \ \ \ \ vimage:\ \ "[|\ r-``\isacharbraceleft a\isacharbraceright\ \isasymin\ Pow(acc(r));\ a\ \isasymin \ field(r)\ |] |
|
2268 \isanewline |
|
2269 \ \ \ \ \ \ \ \ \ \ \ \ \ \ ==>\ a\ \isasymin \ acc(r)"\isanewline |
|
2270 \ \ \isakeyword{monos}\ \ Pow\_mono |
|
2271 \end{isabelle} |
|
2272 |
|
2273 Finally, here are some coinductive definitions. We begin by defining |
|
2274 lazy (potentially infinite) lists as a codatatype: |
|
2275 \begin{isabelle} |
|
2276 \isacommand{consts}\ \ llist\ \ ::\ "i=>i"\isanewline |
|
2277 \isacommand{codatatype}\isanewline |
|
2278 \ \ "llist(A)"\ =\ LNil\ |\ LCons\ ("a\ \isasymin \ A",\ "l\ \isasymin \ llist(A)")\isanewline |
|
2279 \end{isabelle} |
|
2280 |
|
2281 The notion of equality on such lists is modelled as a bisimulation: |
|
2282 \begin{isabelle} |
|
2283 \isacommand{consts}\ \ lleq\ ::\ "i=>i"\isanewline |
|
2284 \isacommand{coinductive}\isanewline |
|
2285 \ \ \isakeyword{domains}\ "lleq(A)"\ <=\ "llist(A)\ *\ llist(A)"\isanewline |
|
2286 \ \ \isakeyword{intros}\isanewline |
|
2287 \ \ \ \ LNil:\ \ "<LNil,\ LNil>\ \isasymin \ lleq(A)"\isanewline |
|
2288 \ \ \ \ LCons:\ "[|\ a\ \isasymin \ A;\ <l,l'>\ \isasymin \ lleq(A)\ |]\ \isanewline |
|
2289 \ \ \ \ \ \ \ \ \ \ \ \ ==>\ <LCons(a,l),\ LCons(a,l')>\ \isasymin \ lleq(A)"\isanewline |
|
2290 \ \ \isakeyword{type\_intros}\ \ llist.intros |
|
2291 \end{isabelle} |
|
2292 This use of \isa{type_intros} is typical: the relation concerns the |
|
2293 codatatype \isa{llist}, so naturally the introduction rules for that |
|
2294 codatatype will be required for type-checking the rules. |
|
2295 |
|
2296 The Isabelle distribution contains many other inductive definitions. Simple |
|
2297 examples are collected on subdirectory \isa{ZF/Induct}. The directory |
|
2298 \isa{Coind} and the theory \isa{ZF/Induct/LList} contain coinductive |
|
2299 definitions. Larger examples may be found on other subdirectories of |
|
2300 \isa{ZF}, such as \isa{IMP}, and \isa{Resid}. |
|
2301 |
|
2302 |
|
2303 \subsection{Theorems generated} |
|
2304 |
|
2305 Each (co)inductive set defined in a theory file generates a name space |
|
2306 containing the following elements: |
|
2307 \begin{ttbox}\isastyleminor |
|
2308 intros \textrm{the introduction rules} |
|
2309 elim \textrm{the elimination (case analysis) rule} |
|
2310 induct \textrm{the standard induction rule} |
|
2311 mutual_induct \textrm{the mutual induction rule, if needed} |
|
2312 defs \textrm{definitions of inductive sets} |
|
2313 bnd_mono \textrm{monotonicity property} |
|
2314 dom_subset \textrm{inclusion in `bounding set'} |
|
2315 \end{ttbox} |
|
2316 Furthermore, each introduction rule is available under its declared |
|
2317 name. For a codatatype, the component \isa{coinduct} is the coinduction rule, |
|
2318 replacing the \isa{induct} component. |
|
2319 |
|
2320 Recall that the \ttindex{inductive\_cases} declaration generates |
|
2321 simplified instances of the case analysis rule. It is as useful for |
|
2322 inductive definitions as it is for datatypes. There are many examples |
|
2323 in the theory |
|
2324 \isa{Induct/Comb}, which is discussed at length |
|
2325 elsewhere~\cite{paulson-generic}. The theory first defines the |
|
2326 datatype |
|
2327 \isa{comb} of combinators: |
|
2328 \begin{alltt*}\isastyleminor |
|
2329 consts comb :: i |
|
2330 datatype "comb" = K |
|
2331 | S |
|
2332 | "#" ("p \isasymin comb", "q \isasymin comb") (infixl 90) |
|
2333 \end{alltt*} |
|
2334 The theory goes on to define contraction and parallel contraction |
|
2335 inductively. Then the theory \isa{Induct/Comb.thy} defines special |
|
2336 cases of contraction, such as this one: |
|
2337 \begin{isabelle} |
|
2338 \isacommand{inductive\_cases}\ K\_contractE [elim!]:\ "K -1-> r" |
|
2339 \end{isabelle} |
|
2340 The theorem just created is \isa{K -1-> r \ \isasymLongrightarrow \ Q}, |
|
2341 which expresses that the combinator \isa{K} cannot reduce to |
|
2342 anything. (From the assumption \isa{K-1->r}, we can conclude any desired |
|
2343 formula \isa{Q}\@.) Similar elimination rules for \isa{S} and application are also |
|
2344 generated. The attribute \isa{elim!}\ shown above supplies the generated |
|
2345 theorem to the classical reasoner. This mode of working allows |
|
2346 effective reasoniung about operational semantics. |
|
2347 |
|
2348 \index{*coinductive|)} \index{*inductive|)} |
|
2349 |
|
2350 |
|
2351 |
|
2352 \section{The outer reaches of set theory} |
|
2353 |
|
2354 The constructions of the natural numbers and lists use a suite of |
|
2355 operators for handling recursive function definitions. I have described |
|
2356 the developments in detail elsewhere~\cite{paulson-set-II}. Here is a brief |
|
2357 summary: |
|
2358 \begin{itemize} |
|
2359 \item Theory \isa{Trancl} defines the transitive closure of a relation |
|
2360 (as a least fixedpoint). |
|
2361 |
|
2362 \item Theory \isa{WF} proves the well-founded recursion theorem, using an |
|
2363 elegant approach of Tobias Nipkow. This theorem permits general |
|
2364 recursive definitions within set theory. |
|
2365 |
|
2366 \item Theory \isa{Ord} defines the notions of transitive set and ordinal |
|
2367 number. It derives transfinite induction. A key definition is {\bf |
|
2368 less than}: $i<j$ if and only if $i$ and $j$ are both ordinals and |
|
2369 $i\in j$. As a special case, it includes less than on the natural |
|
2370 numbers. |
|
2371 |
|
2372 \item Theory \isa{Epsilon} derives $\varepsilon$-induction and |
|
2373 $\varepsilon$-recursion, which are generalisations of transfinite |
|
2374 induction and recursion. It also defines \cdx{rank}$(x)$, which is the |
|
2375 least ordinal $\alpha$ such that $x$ is constructed at stage $\alpha$ of |
|
2376 the cumulative hierarchy (thus $x\in V@{\alpha+1}$). |
|
2377 \end{itemize} |
|
2378 |
|
2379 Other important theories lead to a theory of cardinal numbers. They have |
|
2380 not yet been written up anywhere. Here is a summary: |
|
2381 \begin{itemize} |
|
2382 \item Theory \isa{Rel} defines the basic properties of relations, such as |
|
2383 reflexivity, symmetry and transitivity. |
|
2384 |
|
2385 \item Theory \isa{EquivClass} develops a theory of equivalence |
|
2386 classes, not using the Axiom of Choice. |
|
2387 |
|
2388 \item Theory \isa{Order} defines partial orderings, total orderings and |
|
2389 wellorderings. |
|
2390 |
|
2391 \item Theory \isa{OrderArith} defines orderings on sum and product sets. |
|
2392 These can be used to define ordinal arithmetic and have applications to |
|
2393 cardinal arithmetic. |
|
2394 |
|
2395 \item Theory \isa{OrderType} defines order types. Every wellordering is |
|
2396 equivalent to a unique ordinal, which is its order type. |
|
2397 |
|
2398 \item Theory \isa{Cardinal} defines equipollence and cardinal numbers. |
|
2399 |
|
2400 \item Theory \isa{CardinalArith} defines cardinal addition and |
|
2401 multiplication, and proves their elementary laws. It proves that there |
|
2402 is no greatest cardinal. It also proves a deep result, namely |
|
2403 $\kappa\otimes\kappa=\kappa$ for every infinite cardinal~$\kappa$; see |
|
2404 Kunen~\cite[page 29]{kunen80}. None of these results assume the Axiom of |
|
2405 Choice, which complicates their proofs considerably. |
|
2406 \end{itemize} |
|
2407 |
|
2408 The following developments involve the Axiom of Choice (AC): |
|
2409 \begin{itemize} |
|
2410 \item Theory \isa{AC} asserts the Axiom of Choice and proves some simple |
|
2411 equivalent forms. |
|
2412 |
|
2413 \item Theory \isa{Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma |
|
2414 and the Wellordering Theorem, following Abrial and |
|
2415 Laffitte~\cite{abrial93}. |
|
2416 |
|
2417 \item Theory \isa{Cardinal\_AC} uses AC to prove simplified theorems about |
|
2418 the cardinals. It also proves a theorem needed to justify |
|
2419 infinitely branching datatype declarations: if $\kappa$ is an infinite |
|
2420 cardinal and $|X(\alpha)| \le \kappa$ for all $\alpha<\kappa$ then |
|
2421 $|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$. |
|
2422 |
|
2423 \item Theory \isa{InfDatatype} proves theorems to justify infinitely |
|
2424 branching datatypes. Arbitrary index sets are allowed, provided their |
|
2425 cardinalities have an upper bound. The theory also justifies some |
|
2426 unusual cases of finite branching, involving the finite powerset operator |
|
2427 and the finite function space operator. |
|
2428 \end{itemize} |
|
2429 |
|
2430 |
|
2431 |
|
2432 \section{The examples directories} |
|
2433 Directory \isa{HOL/IMP} contains a mechanised version of a semantic |
|
2434 equivalence proof taken from Winskel~\cite{winskel93}. It formalises the |
|
2435 denotational and operational semantics of a simple while-language, then |
|
2436 proves the two equivalent. It contains several datatype and inductive |
|
2437 definitions, and demonstrates their use. |
|
2438 |
|
2439 The directory \isa{ZF/ex} contains further developments in ZF set theory. |
|
2440 Here is an overview; see the files themselves for more details. I describe |
|
2441 much of this material in other |
|
2442 publications~\cite{paulson-set-I,paulson-set-II,paulson-fixedpt-milner}. |
|
2443 \begin{itemize} |
|
2444 \item File \isa{misc.ML} contains miscellaneous examples such as |
|
2445 Cantor's Theorem, the Schr\"oder-Bernstein Theorem and the `Composition |
|
2446 of homomorphisms' challenge~\cite{boyer86}. |
|
2447 |
|
2448 \item Theory \isa{Ramsey} proves the finite exponent 2 version of |
|
2449 Ramsey's Theorem, following Basin and Kaufmann's |
|
2450 presentation~\cite{basin91}. |
|
2451 |
|
2452 \item Theory \isa{Integ} develops a theory of the integers as |
|
2453 equivalence classes of pairs of natural numbers. |
|
2454 |
|
2455 \item Theory \isa{Primrec} develops some computation theory. It |
|
2456 inductively defines the set of primitive recursive functions and presents a |
|
2457 proof that Ackermann's function is not primitive recursive. |
|
2458 |
|
2459 \item Theory \isa{Primes} defines the Greatest Common Divisor of two |
|
2460 natural numbers and and the ``divides'' relation. |
|
2461 |
|
2462 \item Theory \isa{Bin} defines a datatype for two's complement binary |
|
2463 integers, then proves rewrite rules to perform binary arithmetic. For |
|
2464 instance, $1359\times {-}2468 = {-}3354012$ takes 0.3 seconds. |
|
2465 |
|
2466 \item Theory \isa{BT} defines the recursive data structure $\isa{bt}(A)$, labelled binary trees. |
|
2467 |
|
2468 \item Theory \isa{Term} defines a recursive data structure for terms |
|
2469 and term lists. These are simply finite branching trees. |
|
2470 |
|
2471 \item Theory \isa{TF} defines primitives for solving mutually |
|
2472 recursive equations over sets. It constructs sets of trees and forests |
|
2473 as an example, including induction and recursion rules that handle the |
|
2474 mutual recursion. |
|
2475 |
|
2476 \item Theory \isa{Prop} proves soundness and completeness of |
|
2477 propositional logic~\cite{paulson-set-II}. This illustrates datatype |
|
2478 definitions, inductive definitions, structural induction and rule |
|
2479 induction. |
|
2480 |
|
2481 \item Theory \isa{ListN} inductively defines the lists of $n$ |
|
2482 elements~\cite{paulin-tlca}. |
|
2483 |
|
2484 \item Theory \isa{Acc} inductively defines the accessible part of a |
|
2485 relation~\cite{paulin-tlca}. |
|
2486 |
|
2487 \item Theory \isa{Comb} defines the datatype of combinators and |
|
2488 inductively defines contraction and parallel contraction. It goes on to |
|
2489 prove the Church-Rosser Theorem. This case study follows Camilleri and |
|
2490 Melham~\cite{camilleri92}. |
|
2491 |
|
2492 \item Theory \isa{LList} defines lazy lists and a coinduction |
|
2493 principle for proving equations between them. |
|
2494 \end{itemize} |
|
2495 |
|
2496 |
|
2497 \section{A proof about powersets}\label{sec:ZF-pow-example} |
|
2498 To demonstrate high-level reasoning about subsets, let us prove the |
|
2499 equation ${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$. Compared |
|
2500 with first-order logic, set theory involves a maze of rules, and theorems |
|
2501 have many different proofs. Attempting other proofs of the theorem might |
|
2502 be instructive. This proof exploits the lattice properties of |
|
2503 intersection. It also uses the monotonicity of the powerset operation, |
|
2504 from \isa{ZF/mono.ML}: |
|
2505 \begin{isabelle} |
|
2506 \tdx{Pow_mono}: A \isasymsubseteq B ==> Pow(A) \isasymsubseteq Pow(B) |
|
2507 \end{isabelle} |
|
2508 We enter the goal and make the first step, which breaks the equation into |
|
2509 two inclusions by extensionality:\index{*equalityI theorem} |
|
2510 \begin{isabelle} |
|
2511 \isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline |
|
2512 \ 1.\ Pow(A\ \isasyminter \ B)\ =\ Pow(A)\ \isasyminter \ Pow(B)\isanewline |
|
2513 \isacommand{apply}\ (rule\ equalityI)\isanewline |
|
2514 \ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\ \isasyminter \ Pow(B)\isanewline |
|
2515 \ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B) |
|
2516 \end{isabelle} |
|
2517 Both inclusions could be tackled straightforwardly using \isa{subsetI}. |
|
2518 A shorter proof results from noting that intersection forms the greatest |
|
2519 lower bound:\index{*Int_greatest theorem} |
|
2520 \begin{isabelle} |
|
2521 \isacommand{apply}\ (rule\ Int\_greatest)\isanewline |
|
2522 \ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\isanewline |
|
2523 \ 2.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline |
|
2524 \ 3.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B) |
|
2525 \end{isabelle} |
|
2526 Subgoal~1 follows by applying the monotonicity of \isa{Pow} to $A\int |
|
2527 B\subseteq A$; subgoal~2 follows similarly: |
|
2528 \index{*Int_lower1 theorem}\index{*Int_lower2 theorem} |
|
2529 \begin{isabelle} |
|
2530 \isacommand{apply}\ (rule\ Int\_lower1\ [THEN\ Pow\_mono])\isanewline |
|
2531 \ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline |
|
2532 \ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B) |
|
2533 \isanewline |
|
2534 \isacommand{apply}\ (rule\ Int\_lower2\ [THEN\ Pow\_mono])\isanewline |
|
2535 \ 1.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B) |
|
2536 \end{isabelle} |
|
2537 We are left with the opposite inclusion, which we tackle in the |
|
2538 straightforward way:\index{*subsetI theorem} |
|
2539 \begin{isabelle} |
|
2540 \isacommand{apply}\ (rule\ subsetI)\isanewline |
|
2541 \ 1.\ \isasymAnd x.\ x\ \isasymin \ Pow(A)\ \isasyminter \ Pow(B)\ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B) |
|
2542 \end{isabelle} |
|
2543 The subgoal is to show $x\in \isa{Pow}(A\cap B)$ assuming $x\in\isa{Pow}(A)\cap \isa{Pow}(B)$; eliminating this assumption produces two |
|
2544 subgoals. The rule \tdx{IntE} treats the intersection like a conjunction |
|
2545 instead of unfolding its definition. |
|
2546 \begin{isabelle} |
|
2547 \isacommand{apply}\ (erule\ IntE)\isanewline |
|
2548 \ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B) |
|
2549 \end{isabelle} |
|
2550 The next step replaces the \isa{Pow} by the subset |
|
2551 relation~($\subseteq$).\index{*PowI theorem} |
|
2552 \begin{isabelle} |
|
2553 \isacommand{apply}\ (rule\ PowI)\isanewline |
|
2554 \ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B% |
|
2555 \end{isabelle} |
|
2556 We perform the same replacement in the assumptions. This is a good |
|
2557 demonstration of the tactic \ttindex{drule}:\index{*PowD theorem} |
|
2558 \begin{isabelle} |
|
2559 \isacommand{apply}\ (drule\ PowD)+\isanewline |
|
2560 \ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B% |
|
2561 \end{isabelle} |
|
2562 The assumptions are that $x$ is a lower bound of both $A$ and~$B$, but |
|
2563 $A\int B$ is the greatest lower bound:\index{*Int_greatest theorem} |
|
2564 \begin{isabelle} |
|
2565 \isacommand{apply}\ (rule\ Int\_greatest)\isanewline |
|
2566 \ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\isanewline |
|
2567 \ 2.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ B% |
|
2568 \end{isabelle} |
|
2569 To conclude the proof, we clear up the trivial subgoals: |
|
2570 \begin{isabelle} |
|
2571 \isacommand{apply}\ (assumption+)\isanewline |
|
2572 \isacommand{done}% |
|
2573 \end{isabelle} |
|
2574 |
|
2575 We could have performed this proof instantly by calling |
|
2576 \ttindex{blast}: |
|
2577 \begin{isabelle} |
|
2578 \isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline |
|
2579 \isacommand{by} |
|
2580 \end{isabelle} |
|
2581 Past researchers regarded this as a difficult proof, as indeed it is if all |
|
2582 the symbols are replaced by their definitions. |
|
2583 \goodbreak |
|
2584 |
|
2585 \section{Monotonicity of the union operator} |
|
2586 For another example, we prove that general union is monotonic: |
|
2587 ${C\subseteq D}$ implies $\bigcup(C)\subseteq \bigcup(D)$. To begin, we |
|
2588 tackle the inclusion using \tdx{subsetI}: |
|
2589 \begin{isabelle} |
|
2590 \isacommand{lemma}\ "C\isasymsubseteq D\ ==>\ Union(C)\ |
|
2591 \isasymsubseteq \ Union(D)"\isanewline |
|
2592 \isacommand{apply}\ (rule\ subsetI)\isanewline |
|
2593 \ 1.\ \isasymAnd x.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ \isasymUnion C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D% |
|
2594 \end{isabelle} |
|
2595 Big union is like an existential quantifier --- the occurrence in the |
|
2596 assumptions must be eliminated early, since it creates parameters. |
|
2597 \index{*UnionE theorem} |
|
2598 \begin{isabelle} |
|
2599 \isacommand{apply}\ (erule\ UnionE)\isanewline |
|
2600 \ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D% |
|
2601 \end{isabelle} |
|
2602 Now we may apply \tdx{UnionI}, which creates an unknown involving the |
|
2603 parameters. To show \isa{x\ \isasymin \ \isasymUnion D} it suffices to show that~\isa{x} belongs |
|
2604 to some element, say~\isa{?B2(x,B)}, of~\isa{D}\@. |
|
2605 \begin{isabelle} |
|
2606 \isacommand{apply}\ (rule\ UnionI)\isanewline |
|
2607 \ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ D\isanewline |
|
2608 \ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B) |
|
2609 \end{isabelle} |
|
2610 Combining the rule \tdx{subsetD} with the assumption \isa{C\ \isasymsubseteq \ D} yields |
|
2611 $\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1. Note that |
|
2612 \isa{erule} removes the subset assumption. |
|
2613 \begin{isabelle} |
|
2614 \isacommand{apply}\ (erule\ subsetD)\isanewline |
|
2615 \ 1.\ \isasymAnd x\ B.\ \isasymlbrakk x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ C\isanewline |
|
2616 \ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B) |
|
2617 \end{isabelle} |
|
2618 The rest is routine. Observe how the first call to \isa{assumption} |
|
2619 instantiates \isa{?B2(x,B)} to~\isa{B}\@. |
|
2620 \begin{isabelle} |
|
2621 \isacommand{apply}\ assumption\ \isanewline |
|
2622 \ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ B% |
|
2623 \isanewline |
|
2624 \isacommand{apply}\ assumption\ \isanewline |
|
2625 No\ subgoals!\isanewline |
|
2626 \isacommand{done}% |
|
2627 \end{isabelle} |
|
2628 Again, \isa{blast} can prove this theorem in one step. |
|
2629 |
|
2630 The theory \isa{ZF/equalities.thy} has many similar proofs. Reasoning about |
|
2631 general intersection can be difficult because of its anomalous behaviour on |
|
2632 the empty set. However, \isa{blast} copes well with these. Here is |
|
2633 a typical example, borrowed from Devlin~\cite[page 12]{devlin79}: |
|
2634 \[ a\in C \,\Imp\, \inter@{x\in C} \Bigl(A(x) \int B(x)\Bigr) = |
|
2635 \Bigl(\inter@{x\in C} A(x)\Bigr) \int |
|
2636 \Bigl(\inter@{x\in C} B(x)\Bigr) \] |
|
2637 |
|
2638 \section{Low-level reasoning about functions} |
|
2639 The derived rules \isa{lamI}, \isa{lamE}, \isa{lam_type}, \isa{beta} |
|
2640 and \isa{eta} support reasoning about functions in a |
|
2641 $\lambda$-calculus style. This is generally easier than regarding |
|
2642 functions as sets of ordered pairs. But sometimes we must look at the |
|
2643 underlying representation, as in the following proof |
|
2644 of~\tdx{fun_disjoint_apply1}. This states that if $f$ and~$g$ are |
|
2645 functions with disjoint domains~$A$ and~$C$, and if $a\in A$, then |
|
2646 $(f\un g)`a = f`a$: |
|
2647 \begin{isabelle} |
|
2648 \isacommand{lemma}\ "[|\ a\ \isasymin \ A;\ \ f\ \isasymin \ A->B;\ \ g\ \isasymin \ C->D;\ \ A\ \isasyminter \ C\ =\ 0\ |] |
|
2649 \isanewline |
|
2650 \ \ \ \ \ \ \ \ ==>\ (f\ \isasymunion \ g)`a\ =\ f`a" |
|
2651 \end{isabelle} |
|
2652 Using \tdx{apply_equality}, we reduce the equality to reasoning about |
|
2653 ordered pairs. The second subgoal is to verify that \isa{f\ \isasymunion \ g} is a function, since |
|
2654 \isa{Pi(?A,?B)} denotes a dependent function space. |
|
2655 \begin{isabelle} |
|
2656 \isacommand{apply}\ (rule\ apply\_equality)\isanewline |
|
2657 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2658 \isaindent{\ 1.\ }\isasymLongrightarrow \ \isasymlangle a,\ f\ `\ a\isasymrangle \ \isasymin \ f\ \isasymunion \ g\isanewline |
|
2659 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2660 \isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B) |
|
2661 \end{isabelle} |
|
2662 We must show that the pair belongs to~$f$ or~$g$; by~\tdx{UnI1} we |
|
2663 choose~$f$: |
|
2664 \begin{isabelle} |
|
2665 \isacommand{apply}\ (rule\ UnI1)\isanewline |
|
2666 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ \isasymlangle a,\ f\ `\ a\isasymrangle \ \isasymin \ f\isanewline |
|
2667 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2668 \isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B) |
|
2669 \end{isabelle} |
|
2670 To show $\pair{a,f`a}\in f$ we use \tdx{apply_Pair}, which is |
|
2671 essentially the converse of \tdx{apply_equality}: |
|
2672 \begin{isabelle} |
|
2673 \isacommand{apply}\ (rule\ apply\_Pair)\isanewline |
|
2674 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ f\ \isasymin \ Pi(?A2,?B2)\isanewline |
|
2675 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ a\ \isasymin \ ?A2\isanewline |
|
2676 \ 3.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2677 \isaindent{\ 3.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B) |
|
2678 \end{isabelle} |
|
2679 Using the assumptions $f\in A\to B$ and $a\in A$, we solve the two subgoals |
|
2680 from \tdx{apply_Pair}. Recall that a $\Pi$-set is merely a generalized |
|
2681 function space, and observe that~{\tt?A2} gets instantiated to~\isa{A}. |
|
2682 \begin{isabelle} |
|
2683 \isacommand{apply}\ assumption\ \isanewline |
|
2684 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ a\ \isasymin \ A\isanewline |
|
2685 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2686 \isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B) |
|
2687 \isanewline |
|
2688 \isacommand{apply}\ assumption\ \isanewline |
|
2689 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2690 \isaindent{\ 1.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B) |
|
2691 \end{isabelle} |
|
2692 To construct functions of the form $f\un g$, we apply |
|
2693 \tdx{fun_disjoint_Un}: |
|
2694 \begin{isabelle} |
|
2695 \isacommand{apply}\ (rule\ fun\_disjoint\_Un)\isanewline |
|
2696 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ f\ \isasymin \ ?A3\ \isasymrightarrow \ ?B3\isanewline |
|
2697 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ g\ \isasymin \ ?C3\ \isasymrightarrow \ ?D3\isanewline |
|
2698 \ 3.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ ?A3\ \isasyminter \ ?C3\ =\ 0 |
|
2699 \end{isabelle} |
|
2700 The remaining subgoals are instances of the assumptions. Again, observe how |
|
2701 unknowns become instantiated: |
|
2702 \begin{isabelle} |
|
2703 \isacommand{apply}\ assumption\ \isanewline |
|
2704 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ g\ \isasymin \ ?C3\ \isasymrightarrow \ ?D3\isanewline |
|
2705 \ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ A\ \isasyminter \ ?C3\ =\ 0 |
|
2706 \isanewline |
|
2707 \isacommand{apply}\ assumption\ \isanewline |
|
2708 \ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \ \isasymLongrightarrow \ A\ \isasyminter \ C\ =\ 0 |
|
2709 \isanewline |
|
2710 \isacommand{apply}\ assumption\ \isanewline |
|
2711 No\ subgoals!\isanewline |
|
2712 \isacommand{done} |
|
2713 \end{isabelle} |
|
2714 See the theories \isa{ZF/func.thy} and \isa{ZF/WF.thy} for more |
|
2715 examples of reasoning about functions. |
|
2716 |
|
2717 \index{set theory|)} |