author | hoelzl |
Wed, 23 Feb 2011 11:33:45 +0100 | |
changeset 41830 | 719b0a517c33 |
parent 28871 | 111bbd2b12db |
child 42619 | 9691759a9b3c |
permissions | -rw-r--r-- |
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%% $Id$ |
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\chapter{Zermelo-Fraenkel Set Theory} |
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\index{set theory|(} |
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The theory~\thydx{ZF} implements Zermelo-Fraenkel set |
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theory~\cite{halmos60,suppes72} as an extension of~\texttt{FOL}, classical |
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first-order logic. The theory includes a collection of derived natural |
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deduction rules, for use with Isabelle's classical reasoner. Some |
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of it is based on the work of No\"el~\cite{noel}. |
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|
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A tremendous amount of set theory has been formally developed, including the |
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basic properties of relations, functions, ordinals and cardinals. Significant |
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results have been proved, such as the Schr\"oder-Bernstein Theorem, the |
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Wellordering Theorem and a version of Ramsey's Theorem. \texttt{ZF} provides |
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both the integers and the natural numbers. General methods have been |
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developed for solving recursion equations over monotonic functors; these have |
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been applied to yield constructions of lists, trees, infinite lists, etc. |
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\texttt{ZF} has a flexible package for handling inductive definitions, |
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such as inference systems, and datatype definitions, such as lists and |
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trees. Moreover it handles coinductive definitions, such as |
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bisimulation relations, and codatatype definitions, such as streams. It |
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provides a streamlined syntax for defining primitive recursive functions over |
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datatypes. |
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|
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Published articles~\cite{paulson-set-I,paulson-set-II} describe \texttt{ZF} |
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less formally than this chapter. Isabelle employs a novel treatment of |
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non-well-founded data structures within the standard {\sc zf} axioms including |
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the Axiom of Foundation~\cite{paulson-mscs}. |
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\section{Which version of axiomatic set theory?} |
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The two main axiom systems for set theory are Bernays-G\"odel~({\sc bg}) |
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and Zermelo-Fraenkel~({\sc zf}). Resolution theorem provers can use {\sc |
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bg} because it is finite~\cite{boyer86,quaife92}. {\sc zf} does not |
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have a finite axiom system because of its Axiom Scheme of Replacement. |
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This makes it awkward to use with many theorem provers, since instances |
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of the axiom scheme have to be invoked explicitly. Since Isabelle has no |
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difficulty with axiom schemes, we may adopt either axiom system. |
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These two theories differ in their treatment of {\bf classes}, which are |
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collections that are `too big' to be sets. The class of all sets,~$V$, |
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cannot be a set without admitting Russell's Paradox. In {\sc bg}, both |
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classes and sets are individuals; $x\in V$ expresses that $x$ is a set. In |
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{\sc zf}, all variables denote sets; classes are identified with unary |
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predicates. The two systems define essentially the same sets and classes, |
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with similar properties. In particular, a class cannot belong to another |
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class (let alone a set). |
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Modern set theorists tend to prefer {\sc zf} because they are mainly concerned |
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with sets, rather than classes. {\sc bg} requires tiresome proofs that various |
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collections are sets; for instance, showing $x\in\{x\}$ requires showing that |
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$x$ is a set. |
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\begin{figure} \small |
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\begin{center} |
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\begin{tabular}{rrr} |
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\it name &\it meta-type & \it description \\ |
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\cdx{Let} & $[\alpha,\alpha\To\beta]\To\beta$ & let binder\\ |
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\cdx{0} & $i$ & empty set\\ |
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\cdx{cons} & $[i,i]\To i$ & finite set constructor\\ |
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\cdx{Upair} & $[i,i]\To i$ & unordered pairing\\ |
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\cdx{Pair} & $[i,i]\To i$ & ordered pairing\\ |
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\cdx{Inf} & $i$ & infinite set\\ |
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\cdx{Pow} & $i\To i$ & powerset\\ |
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\cdx{Union} \cdx{Inter} & $i\To i$ & set union/intersection \\ |
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\cdx{split} & $[[i,i]\To i, i] \To i$ & generalized projection\\ |
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\cdx{fst} \cdx{snd} & $i\To i$ & projections\\ |
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\cdx{converse}& $i\To i$ & converse of a relation\\ |
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\cdx{succ} & $i\To i$ & successor\\ |
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\cdx{Collect} & $[i,i\To o]\To i$ & separation\\ |
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\cdx{Replace} & $[i, [i,i]\To o] \To i$ & replacement\\ |
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\cdx{PrimReplace} & $[i, [i,i]\To o] \To i$ & primitive replacement\\ |
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\cdx{RepFun} & $[i, i\To i] \To i$ & functional replacement\\ |
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\cdx{Pi} \cdx{Sigma} & $[i,i\To i]\To i$ & general product/sum\\ |
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\cdx{domain} & $i\To i$ & domain of a relation\\ |
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\cdx{range} & $i\To i$ & range of a relation\\ |
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\cdx{field} & $i\To i$ & field of a relation\\ |
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\cdx{Lambda} & $[i, i\To i]\To i$ & $\lambda$-abstraction\\ |
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\cdx{restrict}& $[i, i] \To i$ & restriction of a function\\ |
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\cdx{The} & $[i\To o]\To i$ & definite description\\ |
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\cdx{if} & $[o,i,i]\To i$ & conditional\\ |
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\cdx{Ball} \cdx{Bex} & $[i, i\To o]\To o$ & bounded quantifiers |
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\end{tabular} |
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\end{center} |
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\subcaption{Constants} |
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|
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\begin{center} |
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\index{*"`"` symbol} |
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\index{*"-"`"` symbol} |
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\index{*"` symbol}\index{function applications} |
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\index{*"- symbol} |
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\index{*": symbol} |
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\index{*"<"= symbol} |
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\begin{tabular}{rrrr} |
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\it symbol & \it meta-type & \it priority & \it description \\ |
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\tt `` & $[i,i]\To i$ & Left 90 & image \\ |
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\tt -`` & $[i,i]\To i$ & Left 90 & inverse image \\ |
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\tt ` & $[i,i]\To i$ & Left 90 & application \\ |
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\sdx{Int} & $[i,i]\To i$ & Left 70 & intersection ($\int$) \\ |
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\sdx{Un} & $[i,i]\To i$ & Left 65 & union ($\un$) \\ |
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\tt - & $[i,i]\To i$ & Left 65 & set difference ($-$) \\[1ex] |
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\tt: & $[i,i]\To o$ & Left 50 & membership ($\in$) \\ |
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\tt <= & $[i,i]\To o$ & Left 50 & subset ($\subseteq$) |
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\end{tabular} |
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\end{center} |
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\subcaption{Infixes} |
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\caption{Constants of ZF} \label{zf-constants} |
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\end{figure} |
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|
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\section{The syntax of set theory} |
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The language of set theory, as studied by logicians, has no constants. The |
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traditional axioms merely assert the existence of empty sets, unions, |
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powersets, etc.; this would be intolerable for practical reasoning. The |
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Isabelle theory declares constants for primitive sets. It also extends |
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\texttt{FOL} with additional syntax for finite sets, ordered pairs, |
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comprehension, general union/intersection, general sums/products, and |
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bounded quantifiers. In most other respects, Isabelle implements precisely |
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Zermelo-Fraenkel set theory. |
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|
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Figure~\ref{zf-constants} lists the constants and infixes of~ZF, while |
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Figure~\ref{zf-trans} presents the syntax translations. Finally, |
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Figure~\ref{zf-syntax} presents the full grammar for set theory, including the |
126 |
constructs of FOL. |
|
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|
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Local abbreviations can be introduced by a \isa{let} construct whose |
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syntax appears in Fig.\ts\ref{zf-syntax}. Internally it is translated into |
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the constant~\cdx{Let}. It can be expanded by rewriting with its |
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definition, \tdx{Let_def}. |
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|
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Apart from \isa{let}, set theory does not use polymorphism. All terms in |
134 |
ZF have type~\tydx{i}, which is the type of individuals and has |
|
135 |
class~\cldx{term}. The type of first-order formulae, remember, |
|
136 |
is~\tydx{o}. |
|
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137 |
|
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Infix operators include binary union and intersection ($A\un B$ and |
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$A\int B$), set difference ($A-B$), and the subset and membership |
14154 | 140 |
relations. Note that $a$\verb|~:|$b$ is translated to $\lnot(a\in b)$, |
141 |
which is equivalent to $a\notin b$. The |
|
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union and intersection operators ($\bigcup A$ and $\bigcap A$) form the |
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union or intersection of a set of sets; $\bigcup A$ means the same as |
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$\bigcup@{x\in A}x$. Of these operators, only $\bigcup A$ is primitive. |
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145 |
|
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The constant \cdx{Upair} constructs unordered pairs; thus \isa{Upair($A$,$B$)} denotes the set~$\{A,B\}$ and |
147 |
\isa{Upair($A$,$A$)} denotes the singleton~$\{A\}$. General union is |
|
148 |
used to define binary union. The Isabelle version goes on to define |
|
149 |
the constant |
|
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\cdx{cons}: |
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\begin{eqnarray*} |
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A\cup B & \equiv & \bigcup(\isa{Upair}(A,B)) \\ |
153 |
\isa{cons}(a,B) & \equiv & \isa{Upair}(a,a) \un B |
|
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\end{eqnarray*} |
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The $\{a@1, \ldots\}$ notation abbreviates finite sets constructed in the |
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obvious manner using~\isa{cons} and~$\emptyset$ (the empty set) \isasymin \begin{eqnarray*} |
157 |
\{a,b,c\} & \equiv & \isa{cons}(a,\isa{cons}(b,\isa{cons}(c,\emptyset))) |
|
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\end{eqnarray*} |
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|
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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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as {\tt<$a$,$b$>}. The $n$-tuple {\tt<$a@1$,\ldots,$a@{n-1}$,$a@n$>} |
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abbreviates the nest of pairs\par\nobreak |
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\centerline{\isa{Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots).}} |
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164 |
|
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In ZF, a function is a set of pairs. A ZF function~$f$ is simply an |
166 |
individual as far as Isabelle is concerned: its Isabelle type is~$i$, not say |
|
167 |
$i\To i$. The infix operator~{\tt`} denotes the application of a function set |
|
168 |
to its argument; we must write~$f{\tt`}x$, not~$f(x)$. The syntax for image |
|
169 |
is~$f{\tt``}A$ and that for inverse image is~$f{\tt-``}A$. |
|
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170 |
|
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171 |
|
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172 |
\begin{figure} |
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\index{lambda abs@$\lambda$-abstractions} |
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\index{*"-"> symbol} |
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\index{*"* symbol} |
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\begin{center} \footnotesize\tt\frenchspacing |
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\begin{tabular}{rrr} |
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\it external & \it internal & \it description \\ |
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$a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm negated membership\\ |
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\ttlbrace$a@1$, $\ldots$, $a@n$\ttrbrace & cons($a@1$,$\ldots$,cons($a@n$,0)) & |
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\rm finite set \\ |
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<$a@1$, $\ldots$, $a@{n-1}$, $a@n$> & |
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Pair($a@1$,\ldots,Pair($a@{n-1}$,$a@n$)\ldots) & |
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\rm ordered $n$-tuple \\ |
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\ttlbrace$x$:$A . P[x]$\ttrbrace & Collect($A$,$\lambda x. P[x]$) & |
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\rm separation \\ |
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187 |
\ttlbrace$y . x$:$A$, $Q[x,y]$\ttrbrace & Replace($A$,$\lambda x\,y. Q[x,y]$) & |
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188 |
\rm replacement \\ |
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189 |
\ttlbrace$b[x] . x$:$A$\ttrbrace & RepFun($A$,$\lambda x. b[x]$) & |
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\rm functional replacement \\ |
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\sdx{INT} $x$:$A . B[x]$ & Inter(\ttlbrace$B[x] . x$:$A$\ttrbrace) & |
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192 |
\rm general intersection \\ |
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\sdx{UN} $x$:$A . B[x]$ & Union(\ttlbrace$B[x] . x$:$A$\ttrbrace) & |
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194 |
\rm general union \\ |
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195 |
\sdx{PROD} $x$:$A . B[x]$ & Pi($A$,$\lambda x. B[x]$) & |
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\rm general product \\ |
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\sdx{SUM} $x$:$A . B[x]$ & Sigma($A$,$\lambda x. B[x]$) & |
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\rm general sum \\ |
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$A$ -> $B$ & Pi($A$,$\lambda x. B$) & |
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\rm function space \\ |
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$A$ * $B$ & Sigma($A$,$\lambda x. B$) & |
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\rm binary product \\ |
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\sdx{THE} $x . P[x]$ & The($\lambda x. P[x]$) & |
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\rm definite description \\ |
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205 |
\sdx{lam} $x$:$A . b[x]$ & Lambda($A$,$\lambda x. b[x]$) & |
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\rm $\lambda$-abstraction\\[1ex] |
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\sdx{ALL} $x$:$A . P[x]$ & Ball($A$,$\lambda x. P[x]$) & |
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\rm bounded $\forall$ \\ |
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\sdx{EX} $x$:$A . P[x]$ & Bex($A$,$\lambda x. P[x]$) & |
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\rm bounded $\exists$ |
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\end{tabular} |
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\end{center} |
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\caption{Translations for ZF} \label{zf-trans} |
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\end{figure} |
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|
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216 |
|
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\begin{figure} |
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\index{*let symbol} |
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\index{*in symbol} |
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\dquotes |
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\[\begin{array}{rcl} |
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term & = & \hbox{expression of type~$i$} \\ |
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& | & "let"~id~"="~term";"\dots";"~id~"="~term~"in"~term \\ |
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& | & "if"~term~"then"~term~"else"~term \\ |
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& | & "{\ttlbrace} " term\; ("," term)^* " {\ttrbrace}" \\ |
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& | & "< " term\; ("," term)^* " >" \\ |
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& | & "{\ttlbrace} " id ":" term " . " formula " {\ttrbrace}" \\ |
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& | & "{\ttlbrace} " id " . " id ":" term ", " formula " {\ttrbrace}" \\ |
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& | & "{\ttlbrace} " term " . " id ":" term " {\ttrbrace}" \\ |
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& | & term " `` " term \\ |
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& | & term " -`` " term \\ |
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& | & term " ` " term \\ |
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& | & term " * " term \\ |
14158 | 234 |
& | & term " \isasyminter " term \\ |
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& | & term " \isasymunion " term \\ |
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& | & term " - " term \\ |
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& | & term " -> " term \\ |
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& | & "THE~~" id " . " formula\\ |
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& | & "lam~~" id ":" term " . " term \\ |
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& | & "INT~~" id ":" term " . " term \\ |
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& | & "UN~~~" id ":" term " . " term \\ |
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& | & "PROD~" id ":" term " . " term \\ |
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& | & "SUM~~" id ":" term " . " term \\[2ex] |
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formula & = & \hbox{expression of type~$o$} \\ |
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& | & term " : " term \\ |
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& | & term " \ttilde: " term \\ |
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& | & term " <= " term \\ |
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& | & term " = " term \\ |
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& | & term " \ttilde= " term \\ |
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& | & "\ttilde\ " formula \\ |
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& | & formula " \& " formula \\ |
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& | & formula " | " formula \\ |
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& | & formula " --> " formula \\ |
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& | & formula " <-> " formula \\ |
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& | & "ALL " id ":" term " . " formula \\ |
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& | & "EX~~" id ":" term " . " formula \\ |
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& | & "ALL~" id~id^* " . " formula \\ |
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& | & "EX~~" id~id^* " . " formula \\ |
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& | & "EX!~" id~id^* " . " formula |
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\end{array} |
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\] |
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\caption{Full grammar for ZF} \label{zf-syntax} |
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\end{figure} |
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|
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|
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\section{Binding operators} |
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The constant \cdx{Collect} constructs sets by the principle of {\bf |
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separation}. The syntax for separation is |
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\hbox{\tt\ttlbrace$x$:$A$.\ $P[x]$\ttrbrace}, where $P[x]$ is a formula |
14154 | 270 |
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 |
271 |
satisfy~$P[x]$. Note that \isa{Collect} is an unfortunate choice of |
|
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name: some set theories adopt a set-formation principle, related to |
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replacement, called collection. |
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274 |
|
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The constant \cdx{Replace} constructs sets by the principle of {\bf |
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replacement}. The syntax |
14154 | 277 |
\hbox{\tt\ttlbrace$y$.\ $x$:$A$,$Q[x,y]$\ttrbrace} denotes the set |
278 |
\isa{Replace($A$,$\lambda x\,y. Q[x,y]$)}, which consists of all~$y$ such |
|
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that there exists $x\in A$ satisfying~$Q[x,y]$. The Replacement Axiom |
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has the condition that $Q$ must be single-valued over~$A$: for |
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all~$x\in A$ there exists at most one $y$ satisfying~$Q[x,y]$. A |
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single-valued binary predicate is also called a {\bf class function}. |
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283 |
|
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The constant \cdx{RepFun} expresses a special case of replacement, |
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where $Q[x,y]$ has the form $y=b[x]$. Such a $Q$ is trivially |
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single-valued, since it is just the graph of the meta-level |
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function~$\lambda x. b[x]$. The resulting set consists of all $b[x]$ |
14154 | 288 |
for~$x\in A$. This is analogous to the \ML{} functional \isa{map}, |
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since it applies a function to every element of a set. The syntax is |
14154 | 290 |
\isa{\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to |
291 |
\isa{RepFun($A$,$\lambda x. b[x]$)}. |
|
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|
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\index{*INT symbol}\index{*UN symbol} |
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General unions and intersections of indexed |
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families of sets, namely $\bigcup@{x\in A}B[x]$ and $\bigcap@{x\in A}B[x]$, |
14154 | 296 |
are written \isa{UN $x$:$A$.\ $B[x]$} and \isa{INT $x$:$A$.\ $B[x]$}. |
297 |
Their meaning is expressed using \isa{RepFun} as |
|
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\[ |
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\bigcup(\{B[x]. x\in A\}) \qquad\hbox{and}\qquad |
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\bigcap(\{B[x]. x\in A\}). |
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\] |
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General sums $\sum@{x\in A}B[x]$ and products $\prod@{x\in A}B[x]$ can be |
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constructed in set theory, where $B[x]$ is a family of sets over~$A$. They |
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have as special cases $A\times B$ and $A\to B$, where $B$ is simply a set. |
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This is similar to the situation in Constructive Type Theory (set theory |
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has `dependent sets') and calls for similar syntactic conventions. The |
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constants~\cdx{Sigma} and~\cdx{Pi} construct general sums and |
14154 | 308 |
products. Instead of \isa{Sigma($A$,$B$)} and \isa{Pi($A$,$B$)} we may |
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write |
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\isa{SUM $x$:$A$.\ $B[x]$} and \isa{PROD $x$:$A$.\ $B[x]$}. |
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\index{*SUM symbol}\index{*PROD symbol}% |
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The special cases as \hbox{\tt$A$*$B$} and \hbox{\tt$A$->$B$} abbreviate |
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general sums and products over a constant family.\footnote{Unlike normal |
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infix operators, {\tt*} and {\tt->} merely define abbreviations; there are |
14154 | 315 |
no constants~\isa{op~*} and~\isa{op~->}.} Isabelle accepts these |
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abbreviations in parsing and uses them whenever possible for printing. |
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|
9695 | 318 |
\index{*THE symbol} As mentioned above, whenever the axioms assert the |
319 |
existence and uniqueness of a set, Isabelle's set theory declares a constant |
|
320 |
for that set. These constants can express the {\bf definite description} |
|
321 |
operator~$\iota x. P[x]$, which stands for the unique~$a$ satisfying~$P[a]$, |
|
322 |
if such exists. Since all terms in ZF denote something, a description is |
|
323 |
always meaningful, but we do not know its value unless $P[x]$ defines it |
|
14154 | 324 |
uniquely. Using the constant~\cdx{The}, we may write descriptions as |
325 |
\isa{The($\lambda x. P[x]$)} or use the syntax \isa{THE $x$.\ $P[x]$}. |
|
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|
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\index{*lam symbol} |
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Function sets may be written in $\lambda$-notation; $\lambda x\in A. b[x]$ |
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stands for the set of all pairs $\pair{x,b[x]}$ for $x\in A$. In order for |
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this to be a set, the function's domain~$A$ must be given. Using the |
14154 | 331 |
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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|
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Isabelle's set theory defines two {\bf bounded quantifiers}: |
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\begin{eqnarray*} |
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\forall x\in A. P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\ |
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\exists x\in A. P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x] |
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\end{eqnarray*} |
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The constants~\cdx{Ball} and~\cdx{Bex} are defined |
14154 | 339 |
accordingly. Instead of \isa{Ball($A$,$P$)} and \isa{Bex($A$,$P$)} we may |
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write |
14154 | 341 |
\isa{ALL $x$:$A$.\ $P[x]$} and \isa{EX $x$:$A$.\ $P[x]$}. |
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342 |
|
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343 |
|
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%%%% ZF.thy |
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345 |
|
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\begin{figure} |
14154 | 347 |
\begin{alltt*}\isastyleminor |
348 |
\tdx{Let_def}: Let(s, f) == f(s) |
|
349 |
||
350 |
\tdx{Ball_def}: Ball(A,P) == {\isasymforall}x. x \isasymin A --> P(x) |
|
351 |
\tdx{Bex_def}: Bex(A,P) == {\isasymexists}x. x \isasymin A & P(x) |
|
352 |
||
353 |
\tdx{subset_def}: A \isasymsubseteq B == {\isasymforall}x \isasymin A. x \isasymin B |
|
354 |
\tdx{extension}: A = B <-> A \isasymsubseteq B & B \isasymsubseteq A |
|
355 |
||
356 |
\tdx{Union_iff}: A \isasymin Union(C) <-> ({\isasymexists}B \isasymin C. A \isasymin B) |
|
357 |
\tdx{Pow_iff}: A \isasymin Pow(B) <-> A \isasymsubseteq B |
|
358 |
\tdx{foundation}: A=0 | ({\isasymexists}x \isasymin A. {\isasymforall}y \isasymin x. y \isasymnotin A) |
|
359 |
||
360 |
\tdx{replacement}: ({\isasymforall}x \isasymin A. {\isasymforall}y z. P(x,y) & P(x,z) --> y=z) ==> |
|
361 |
b \isasymin PrimReplace(A,P) <-> ({\isasymexists}x{\isasymin}A. P(x,b)) |
|
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\subcaption{The Zermelo-Fraenkel Axioms} |
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|
14154 | 364 |
\tdx{Replace_def}: Replace(A,P) == |
365 |
PrimReplace(A, \%x y. (\isasymexists!z. P(x,z)) & P(x,y)) |
|
366 |
\tdx{RepFun_def}: RepFun(A,f) == {\ttlbrace}y . x \isasymin A, y=f(x)\ttrbrace |
|
367 |
\tdx{the_def}: The(P) == Union({\ttlbrace}y . x \isasymin {\ttlbrace}0{\ttrbrace}, P(y){\ttrbrace}) |
|
368 |
\tdx{if_def}: if(P,a,b) == THE z. P & z=a | ~P & z=b |
|
369 |
\tdx{Collect_def}: Collect(A,P) == {\ttlbrace}y . x \isasymin A, x=y & P(x){\ttrbrace} |
|
370 |
\tdx{Upair_def}: Upair(a,b) == |
|
371 |
{\ttlbrace}y. x\isasymin{}Pow(Pow(0)), x=0 & y=a | x=Pow(0) & y=b{\ttrbrace} |
|
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\subcaption{Consequences of replacement} |
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|
14154 | 374 |
\tdx{Inter_def}: Inter(A) == {\ttlbrace}x \isasymin Union(A) . {\isasymforall}y \isasymin A. x \isasymin y{\ttrbrace} |
375 |
\tdx{Un_def}: A \isasymunion B == Union(Upair(A,B)) |
|
14158 | 376 |
\tdx{Int_def}: A \isasyminter B == Inter(Upair(A,B)) |
14154 | 377 |
\tdx{Diff_def}: A - B == {\ttlbrace}x \isasymin A . x \isasymnotin B{\ttrbrace} |
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\subcaption{Union, intersection, difference} |
14154 | 379 |
\end{alltt*} |
9695 | 380 |
\caption{Rules and axioms of ZF} \label{zf-rules} |
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381 |
\end{figure} |
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382 |
|
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383 |
|
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384 |
\begin{figure} |
14154 | 385 |
\begin{alltt*}\isastyleminor |
386 |
\tdx{cons_def}: cons(a,A) == Upair(a,a) \isasymunion A |
|
387 |
\tdx{succ_def}: succ(i) == cons(i,i) |
|
388 |
\tdx{infinity}: 0 \isasymin Inf & ({\isasymforall}y \isasymin Inf. succ(y) \isasymin Inf) |
|
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389 |
\subcaption{Finite and infinite sets} |
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|
390 |
|
14154 | 391 |
\tdx{Pair_def}: <a,b> == {\ttlbrace}{\ttlbrace}a,a{\ttrbrace}, {\ttlbrace}a,b{\ttrbrace}{\ttrbrace} |
392 |
\tdx{split_def}: split(c,p) == THE y. {\isasymexists}a b. p=<a,b> & y=c(a,b) |
|
393 |
\tdx{fst_def}: fst(A) == split(\%x y. x, p) |
|
394 |
\tdx{snd_def}: snd(A) == split(\%x y. y, p) |
|
395 |
\tdx{Sigma_def}: Sigma(A,B) == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x,y>{\ttrbrace} |
|
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|
396 |
\subcaption{Ordered pairs and Cartesian products} |
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|
397 |
|
14154 | 398 |
\tdx{converse_def}: converse(r) == {\ttlbrace}z. w\isasymin{}r, {\isasymexists}x y. w=<x,y> & z=<y,x>{\ttrbrace} |
399 |
\tdx{domain_def}: domain(r) == {\ttlbrace}x. w \isasymin r, {\isasymexists}y. w=<x,y>{\ttrbrace} |
|
400 |
\tdx{range_def}: range(r) == domain(converse(r)) |
|
401 |
\tdx{field_def}: field(r) == domain(r) \isasymunion range(r) |
|
402 |
\tdx{image_def}: r `` A == {\ttlbrace}y\isasymin{}range(r) . {\isasymexists}x \isasymin A. <x,y> \isasymin r{\ttrbrace} |
|
403 |
\tdx{vimage_def}: r -`` A == converse(r)``A |
|
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|
404 |
\subcaption{Operations on relations} |
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|
405 |
|
14154 | 406 |
\tdx{lam_def}: Lambda(A,b) == {\ttlbrace}<x,b(x)> . x \isasymin A{\ttrbrace} |
407 |
\tdx{apply_def}: f`a == THE y. <a,y> \isasymin f |
|
408 |
\tdx{Pi_def}: Pi(A,B) == {\ttlbrace}f\isasymin{}Pow(Sigma(A,B)). {\isasymforall}x\isasymin{}A. \isasymexists!y. <x,y>\isasymin{}f{\ttrbrace} |
|
409 |
\tdx{restrict_def}: restrict(f,A) == lam x \isasymin A. f`x |
|
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|
410 |
\subcaption{Functions and general product} |
14154 | 411 |
\end{alltt*} |
9695 | 412 |
\caption{Further definitions of ZF} \label{zf-defs} |
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|
413 |
\end{figure} |
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|
414 |
|
5fe77b9b5185
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paulson
parents:
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changeset
|
415 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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changeset
|
416 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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|
417 |
\section{The Zermelo-Fraenkel axioms} |
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|
418 |
The axioms appear in Fig.\ts \ref{zf-rules}. They resemble those |
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|
419 |
presented by Suppes~\cite{suppes72}. Most of the theory consists of |
5fe77b9b5185
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parents:
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|
420 |
definitions. In particular, bounded quantifiers and the subset relation |
5fe77b9b5185
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parents:
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|
421 |
appear in other axioms. Object-level quantifiers and implications have |
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the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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|
422 |
been replaced by meta-level ones wherever possible, to simplify use of the |
14154 | 423 |
axioms. |
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|
424 |
|
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parents:
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|
425 |
The traditional replacement axiom asserts |
14154 | 426 |
\[ y \in \isa{PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y)) \] |
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|
427 |
subject to the condition that $P(x,y)$ is single-valued for all~$x\in A$. |
5fe77b9b5185
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|
428 |
The Isabelle theory defines \cdx{Replace} to apply |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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changeset
|
429 |
\cdx{PrimReplace} to the single-valued part of~$P$, namely |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
430 |
\[ (\exists!z. P(x,z)) \conj P(x,y). \] |
14154 | 431 |
Thus $y\in \isa{Replace}(A,P)$ if and only if there is some~$x$ such that |
6121
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|
432 |
$P(x,-)$ holds uniquely for~$y$. Because the equivalence is unconditional, |
14154 | 433 |
\isa{Replace} is much easier to use than \isa{PrimReplace}; it defines the |
6121
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|
434 |
same set, if $P(x,y)$ is single-valued. The nice syntax for replacement |
14154 | 435 |
expands to \isa{Replace}. |
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|
436 |
|
14158 | 437 |
Other consequences of replacement include replacement for |
438 |
meta-level functions |
|
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|
439 |
(\cdx{RepFun}) and definite descriptions (\cdx{The}). |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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changeset
|
440 |
Axioms for separation (\cdx{Collect}) and unordered pairs |
5fe77b9b5185
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parents:
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|
441 |
(\cdx{Upair}) are traditionally assumed, but they actually follow |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
changeset
|
442 |
from replacement~\cite[pages 237--8]{suppes72}. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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changeset
|
443 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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changeset
|
444 |
The definitions of general intersection, etc., are straightforward. Note |
14154 | 445 |
the definition of \isa{cons}, which underlies the finite set notation. |
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|
446 |
The axiom of infinity gives us a set that contains~0 and is closed under |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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changeset
|
447 |
successor (\cdx{succ}). Although this set is not uniquely defined, |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
changeset
|
448 |
the theory names it (\cdx{Inf}) in order to simplify the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
449 |
construction of the natural numbers. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
450 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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changeset
|
451 |
Further definitions appear in Fig.\ts\ref{zf-defs}. Ordered pairs are |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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|
452 |
defined in the standard way, $\pair{a,b}\equiv\{\{a\},\{a,b\}\}$. Recall |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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|
453 |
that \cdx{Sigma}$(A,B)$ generalizes the Cartesian product of two |
5fe77b9b5185
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parents:
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changeset
|
454 |
sets. It is defined to be the union of all singleton sets |
5fe77b9b5185
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parents:
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changeset
|
455 |
$\{\pair{x,y}\}$, for $x\in A$ and $y\in B(x)$. This is a typical usage of |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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|
456 |
general union. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
457 |
|
5fe77b9b5185
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parents:
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changeset
|
458 |
The projections \cdx{fst} and~\cdx{snd} are defined in terms of the |
5fe77b9b5185
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parents:
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changeset
|
459 |
generalized projection \cdx{split}. The latter has been borrowed from |
5fe77b9b5185
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parents:
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changeset
|
460 |
Martin-L\"of's Type Theory, and is often easier to use than \cdx{fst} |
5fe77b9b5185
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parents:
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changeset
|
461 |
and~\cdx{snd}. |
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|
462 |
|
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parents:
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|
463 |
Operations on relations include converse, domain, range, and image. The |
14154 | 464 |
set $\isa{Pi}(A,B)$ generalizes the space of functions between two sets. |
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|
465 |
Note the simple definitions of $\lambda$-abstraction (using |
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|
466 |
\cdx{RepFun}) and application (using a definite description). The |
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parents:
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|
467 |
function \cdx{restrict}$(f,A)$ has the same values as~$f$, but only |
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parents:
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|
468 |
over the domain~$A$. |
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parents:
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|
469 |
|
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parents:
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|
470 |
|
14154 | 471 |
%%%% zf.thy |
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|
472 |
|
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parents:
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|
473 |
\begin{figure} |
14154 | 474 |
\begin{alltt*}\isastyleminor |
475 |
\tdx{ballI}: [| !!x. x\isasymin{}A ==> P(x) |] ==> {\isasymforall}x\isasymin{}A. P(x) |
|
476 |
\tdx{bspec}: [| {\isasymforall}x\isasymin{}A. P(x); x\isasymin{}A |] ==> P(x) |
|
477 |
\tdx{ballE}: [| {\isasymforall}x\isasymin{}A. P(x); P(x) ==> Q; x \isasymnotin A ==> Q |] ==> Q |
|
478 |
||
479 |
\tdx{ball_cong}: [| A=A'; !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==> |
|
480 |
({\isasymforall}x\isasymin{}A. P(x)) <-> ({\isasymforall}x\isasymin{}A'. P'(x)) |
|
481 |
||
482 |
\tdx{bexI}: [| P(x); x\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x) |
|
483 |
\tdx{bexCI}: [| {\isasymforall}x\isasymin{}A. ~P(x) ==> P(a); a\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x) |
|
484 |
\tdx{bexE}: [| {\isasymexists}x\isasymin{}A. P(x); !!x. [| x\isasymin{}A; P(x) |] ==> Q |] ==> Q |
|
485 |
||
486 |
\tdx{bex_cong}: [| A=A'; !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==> |
|
487 |
({\isasymexists}x\isasymin{}A. P(x)) <-> ({\isasymexists}x\isasymin{}A'. P'(x)) |
|
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|
488 |
\subcaption{Bounded quantifiers} |
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|
489 |
|
14154 | 490 |
\tdx{subsetI}: (!!x. x \isasymin A ==> x \isasymin B) ==> A \isasymsubseteq B |
491 |
\tdx{subsetD}: [| A \isasymsubseteq B; c \isasymin A |] ==> c \isasymin B |
|
492 |
\tdx{subsetCE}: [| A \isasymsubseteq B; c \isasymnotin A ==> P; c \isasymin B ==> P |] ==> P |
|
493 |
\tdx{subset_refl}: A \isasymsubseteq A |
|
494 |
\tdx{subset_trans}: [| A \isasymsubseteq B; B \isasymsubseteq C |] ==> A \isasymsubseteq C |
|
495 |
||
496 |
\tdx{equalityI}: [| A \isasymsubseteq B; B \isasymsubseteq A |] ==> A = B |
|
497 |
\tdx{equalityD1}: A = B ==> A \isasymsubseteq B |
|
498 |
\tdx{equalityD2}: A = B ==> B \isasymsubseteq A |
|
499 |
\tdx{equalityE}: [| A = B; [| A \isasymsubseteq B; B \isasymsubseteq A |] ==> P |] ==> P |
|
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|
500 |
\subcaption{Subsets and extensionality} |
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|
501 |
|
14154 | 502 |
\tdx{emptyE}: a \isasymin 0 ==> P |
503 |
\tdx{empty_subsetI}: 0 \isasymsubseteq A |
|
504 |
\tdx{equals0I}: [| !!y. y \isasymin A ==> False |] ==> A=0 |
|
505 |
\tdx{equals0D}: [| A=0; a \isasymin A |] ==> P |
|
506 |
||
507 |
\tdx{PowI}: A \isasymsubseteq B ==> A \isasymin Pow(B) |
|
508 |
\tdx{PowD}: A \isasymin Pow(B) ==> A \isasymsubseteq B |
|
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|
509 |
\subcaption{The empty set; power sets} |
14154 | 510 |
\end{alltt*} |
9695 | 511 |
\caption{Basic derived rules for ZF} \label{zf-lemmas1} |
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|
512 |
\end{figure} |
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|
513 |
|
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parents:
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changeset
|
514 |
|
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parents:
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changeset
|
515 |
\section{From basic lemmas to function spaces} |
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|
516 |
Faced with so many definitions, it is essential to prove lemmas. Even |
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parents:
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|
517 |
trivial theorems like $A \int B = B \int A$ would be difficult to |
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parents:
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|
518 |
prove from the definitions alone. Isabelle's set theory derives many |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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changeset
|
519 |
rules using a natural deduction style. Ideally, a natural deduction |
5fe77b9b5185
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parents:
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|
520 |
rule should introduce or eliminate just one operator, but this is not |
5fe77b9b5185
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parents:
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|
521 |
always practical. For most operators, we may forget its definition |
5fe77b9b5185
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parents:
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changeset
|
522 |
and use its derived rules instead. |
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parents:
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changeset
|
523 |
|
5fe77b9b5185
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paulson
parents:
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changeset
|
524 |
\subsection{Fundamental lemmas} |
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paulson
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|
525 |
Figure~\ref{zf-lemmas1} presents the derived rules for the most basic |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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changeset
|
526 |
operators. The rules for the bounded quantifiers resemble those for the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
527 |
ordinary quantifiers, but note that \tdx{ballE} uses a negated assumption |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
528 |
in the style of Isabelle's classical reasoner. The \rmindex{congruence |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
529 |
rules} \tdx{ball_cong} and \tdx{bex_cong} are required by Isabelle's |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
changeset
|
530 |
simplifier, but have few other uses. Congruence rules must be specially |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
531 |
derived for all binding operators, and henceforth will not be shown. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
532 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
533 |
Figure~\ref{zf-lemmas1} also shows rules for the subset and equality |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
534 |
relations (proof by extensionality), and rules about the empty set and the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
535 |
power set operator. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
536 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
537 |
Figure~\ref{zf-lemmas2} presents rules for replacement and separation. |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
538 |
The rules for \cdx{Replace} and \cdx{RepFun} are much simpler than |
14154 | 539 |
comparable rules for \isa{PrimReplace} would be. The principle of |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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changeset
|
540 |
separation is proved explicitly, although most proofs should use the |
14154 | 541 |
natural deduction rules for \isa{Collect}. The elimination rule |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
542 |
\tdx{CollectE} is equivalent to the two destruction rules |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
543 |
\tdx{CollectD1} and \tdx{CollectD2}, but each rule is suited to |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
544 |
particular circumstances. Although too many rules can be confusing, there |
14154 | 545 |
is no reason to aim for a minimal set of rules. |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
546 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
547 |
Figure~\ref{zf-lemmas3} presents rules for general union and intersection. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
548 |
The empty intersection should be undefined. We cannot have |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
549 |
$\bigcap(\emptyset)=V$ because $V$, the universal class, is not a set. All |
9695 | 550 |
expressions denote something in ZF set theory; the definition of |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
551 |
intersection implies $\bigcap(\emptyset)=\emptyset$, but this value is |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
552 |
arbitrary. The rule \tdx{InterI} must have a premise to exclude |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
553 |
the empty intersection. Some of the laws governing intersections require |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
554 |
similar premises. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
555 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
556 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
557 |
%the [p] gives better page breaking for the book |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
558 |
\begin{figure}[p] |
14154 | 559 |
\begin{alltt*}\isastyleminor |
560 |
\tdx{ReplaceI}: [| x\isasymin{}A; P(x,b); !!y. P(x,y) ==> y=b |] ==> |
|
561 |
b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace} |
|
562 |
||
563 |
\tdx{ReplaceE}: [| b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace}; |
|
564 |
!!x. [| x\isasymin{}A; P(x,b); {\isasymforall}y. P(x,y)-->y=b |] ==> R |
|
565 |
|] ==> R |
|
566 |
||
567 |
\tdx{RepFunI}: [| a\isasymin{}A |] ==> f(a)\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace} |
|
568 |
\tdx{RepFunE}: [| b\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace}; |
|
569 |
!!x.[| x\isasymin{}A; b=f(x) |] ==> P |] ==> P |
|
570 |
||
571 |
\tdx{separation}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} <-> a\isasymin{}A & P(a) |
|
572 |
\tdx{CollectI}: [| a\isasymin{}A; P(a) |] ==> a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} |
|
573 |
\tdx{CollectE}: [| a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace}; [| a\isasymin{}A; P(a) |] ==> R |] ==> R |
|
574 |
\tdx{CollectD1}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> a\isasymin{}A |
|
575 |
\tdx{CollectD2}: a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> P(a) |
|
576 |
\end{alltt*} |
|
6121
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paulson
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changeset
|
577 |
\caption{Replacement and separation} \label{zf-lemmas2} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
578 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
579 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
580 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
581 |
\begin{figure} |
14154 | 582 |
\begin{alltt*}\isastyleminor |
583 |
\tdx{UnionI}: [| B\isasymin{}C; A\isasymin{}B |] ==> A\isasymin{}Union(C) |
|
584 |
\tdx{UnionE}: [| A\isasymin{}Union(C); !!B.[| A\isasymin{}B; B\isasymin{}C |] ==> R |] ==> R |
|
585 |
||
586 |
\tdx{InterI}: [| !!x. x\isasymin{}C ==> A\isasymin{}x; c\isasymin{}C |] ==> A\isasymin{}Inter(C) |
|
587 |
\tdx{InterD}: [| A\isasymin{}Inter(C); B\isasymin{}C |] ==> A\isasymin{}B |
|
588 |
\tdx{InterE}: [| A\isasymin{}Inter(C); A\isasymin{}B ==> R; B \isasymnotin C ==> R |] ==> R |
|
589 |
||
590 |
\tdx{UN_I}: [| a\isasymin{}A; b\isasymin{}B(a) |] ==> b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x)) |
|
591 |
\tdx{UN_E}: [| b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x)); !!x.[| x\isasymin{}A; b\isasymin{}B(x) |] ==> R |
|
592 |
|] ==> R |
|
593 |
||
594 |
\tdx{INT_I}: [| !!x. x\isasymin{}A ==> b\isasymin{}B(x); a\isasymin{}A |] ==> b\isasymin{}({\isasymInter}x\isasymin{}A. B(x)) |
|
595 |
\tdx{INT_E}: [| b\isasymin{}({\isasymInter}x\isasymin{}A. B(x)); a\isasymin{}A |] ==> b\isasymin{}B(a) |
|
596 |
\end{alltt*} |
|
6121
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paulson
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changeset
|
597 |
\caption{General union and intersection} \label{zf-lemmas3} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
598 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
599 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
600 |
|
14154 | 601 |
%%% upair.thy |
6121
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|
602 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
603 |
\begin{figure} |
14154 | 604 |
\begin{alltt*}\isastyleminor |
605 |
\tdx{pairing}: a\isasymin{}Upair(b,c) <-> (a=b | a=c) |
|
606 |
\tdx{UpairI1}: a\isasymin{}Upair(a,b) |
|
607 |
\tdx{UpairI2}: b\isasymin{}Upair(a,b) |
|
608 |
\tdx{UpairE}: [| a\isasymin{}Upair(b,c); a=b ==> P; a=c ==> P |] ==> P |
|
609 |
\end{alltt*} |
|
6121
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paulson
parents:
diff
changeset
|
610 |
\caption{Unordered pairs} \label{zf-upair1} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
611 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
612 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
613 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
614 |
\begin{figure} |
14154 | 615 |
\begin{alltt*}\isastyleminor |
616 |
\tdx{UnI1}: c\isasymin{}A ==> c\isasymin{}A \isasymunion B |
|
617 |
\tdx{UnI2}: c\isasymin{}B ==> c\isasymin{}A \isasymunion B |
|
618 |
\tdx{UnCI}: (c \isasymnotin B ==> c\isasymin{}A) ==> c\isasymin{}A \isasymunion B |
|
619 |
\tdx{UnE}: [| c\isasymin{}A \isasymunion B; c\isasymin{}A ==> P; c\isasymin{}B ==> P |] ==> P |
|
620 |
||
14158 | 621 |
\tdx{IntI}: [| c\isasymin{}A; c\isasymin{}B |] ==> c\isasymin{}A \isasyminter B |
622 |
\tdx{IntD1}: c\isasymin{}A \isasyminter B ==> c\isasymin{}A |
|
623 |
\tdx{IntD2}: c\isasymin{}A \isasyminter B ==> c\isasymin{}B |
|
624 |
\tdx{IntE}: [| c\isasymin{}A \isasyminter B; [| c\isasymin{}A; c\isasymin{}B |] ==> P |] ==> P |
|
14154 | 625 |
|
626 |
\tdx{DiffI}: [| c\isasymin{}A; c \isasymnotin B |] ==> c\isasymin{}A - B |
|
627 |
\tdx{DiffD1}: c\isasymin{}A - B ==> c\isasymin{}A |
|
628 |
\tdx{DiffD2}: c\isasymin{}A - B ==> c \isasymnotin B |
|
629 |
\tdx{DiffE}: [| c\isasymin{}A - B; [| c\isasymin{}A; c \isasymnotin B |] ==> P |] ==> P |
|
630 |
\end{alltt*} |
|
6121
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paulson
parents:
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changeset
|
631 |
\caption{Union, intersection, difference} \label{zf-Un} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
632 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
633 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
634 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
635 |
\begin{figure} |
14154 | 636 |
\begin{alltt*}\isastyleminor |
637 |
\tdx{consI1}: a\isasymin{}cons(a,B) |
|
638 |
\tdx{consI2}: a\isasymin{}B ==> a\isasymin{}cons(b,B) |
|
639 |
\tdx{consCI}: (a \isasymnotin B ==> a=b) ==> a\isasymin{}cons(b,B) |
|
640 |
\tdx{consE}: [| a\isasymin{}cons(b,A); a=b ==> P; a\isasymin{}A ==> P |] ==> P |
|
641 |
||
642 |
\tdx{singletonI}: a\isasymin{}{\ttlbrace}a{\ttrbrace} |
|
643 |
\tdx{singletonE}: [| a\isasymin{}{\ttlbrace}b{\ttrbrace}; a=b ==> P |] ==> P |
|
644 |
\end{alltt*} |
|
6121
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paulson
parents:
diff
changeset
|
645 |
\caption{Finite and singleton sets} \label{zf-upair2} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
646 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
647 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
648 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
649 |
\begin{figure} |
14154 | 650 |
\begin{alltt*}\isastyleminor |
651 |
\tdx{succI1}: i\isasymin{}succ(i) |
|
652 |
\tdx{succI2}: i\isasymin{}j ==> i\isasymin{}succ(j) |
|
653 |
\tdx{succCI}: (i \isasymnotin j ==> i=j) ==> i\isasymin{}succ(j) |
|
654 |
\tdx{succE}: [| i\isasymin{}succ(j); i=j ==> P; i\isasymin{}j ==> P |] ==> P |
|
655 |
\tdx{succ_neq_0}: [| succ(n)=0 |] ==> P |
|
656 |
\tdx{succ_inject}: succ(m) = succ(n) ==> m=n |
|
657 |
\end{alltt*} |
|
6121
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paulson
parents:
diff
changeset
|
658 |
\caption{The successor function} \label{zf-succ} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
659 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
660 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
661 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
662 |
\begin{figure} |
14154 | 663 |
\begin{alltt*}\isastyleminor |
664 |
\tdx{the_equality}: [| P(a); !!x. P(x) ==> x=a |] ==> (THE x. P(x))=a |
|
665 |
\tdx{theI}: \isasymexists! x. P(x) ==> P(THE x. P(x)) |
|
666 |
||
667 |
\tdx{if_P}: P ==> (if P then a else b) = a |
|
668 |
\tdx{if_not_P}: ~P ==> (if P then a else b) = b |
|
669 |
||
670 |
\tdx{mem_asym}: [| a\isasymin{}b; b\isasymin{}a |] ==> P |
|
671 |
\tdx{mem_irrefl}: a\isasymin{}a ==> P |
|
672 |
\end{alltt*} |
|
6121
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paulson
parents:
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changeset
|
673 |
\caption{Descriptions; non-circularity} \label{zf-the} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
674 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
675 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
676 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
677 |
\subsection{Unordered pairs and finite sets} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
678 |
Figure~\ref{zf-upair1} presents the principle of unordered pairing, along |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
679 |
with its derived rules. Binary union and intersection are defined in terms |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
680 |
of ordered pairs (Fig.\ts\ref{zf-Un}). Set difference is also included. The |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
681 |
rule \tdx{UnCI} is useful for classical reasoning about unions, |
14154 | 682 |
like \isa{disjCI}\@; it supersedes \tdx{UnI1} and |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
683 |
\tdx{UnI2}, but these rules are often easier to work with. For |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
684 |
intersection and difference we have both elimination and destruction rules. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
685 |
Again, there is no reason to provide a minimal rule set. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
686 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
687 |
Figure~\ref{zf-upair2} is concerned with finite sets: it presents rules |
14154 | 688 |
for~\isa{cons}, the finite set constructor, and rules for singleton |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
689 |
sets. Figure~\ref{zf-succ} presents derived rules for the successor |
14154 | 690 |
function, which is defined in terms of~\isa{cons}. The proof that |
691 |
\isa{succ} is injective appears to require the Axiom of Foundation. |
|
6121
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the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
692 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
693 |
Definite descriptions (\sdx{THE}) are defined in terms of the singleton |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
694 |
set~$\{0\}$, but their derived rules fortunately hide this |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
695 |
(Fig.\ts\ref{zf-the}). The rule~\tdx{theI} is difficult to apply |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
696 |
because of the two occurrences of~$\Var{P}$. However, |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
697 |
\tdx{the_equality} does not have this problem and the files contain |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
698 |
many examples of its use. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
699 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
700 |
Finally, the impossibility of having both $a\in b$ and $b\in a$ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
701 |
(\tdx{mem_asym}) is proved by applying the Axiom of Foundation to |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
702 |
the set $\{a,b\}$. The impossibility of $a\in a$ is a trivial consequence. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
703 |
|
14154 | 704 |
|
705 |
%%% subset.thy? |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
706 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
707 |
\begin{figure} |
14154 | 708 |
\begin{alltt*}\isastyleminor |
709 |
\tdx{Union_upper}: B\isasymin{}A ==> B \isasymsubseteq Union(A) |
|
710 |
\tdx{Union_least}: [| !!x. x\isasymin{}A ==> x \isasymsubseteq C |] ==> Union(A) \isasymsubseteq C |
|
711 |
||
712 |
\tdx{Inter_lower}: B\isasymin{}A ==> Inter(A) \isasymsubseteq B |
|
713 |
\tdx{Inter_greatest}: [| a\isasymin{}A; !!x. x\isasymin{}A ==> C \isasymsubseteq x |] ==> C\isasymsubseteq{}Inter(A) |
|
714 |
||
715 |
\tdx{Un_upper1}: A \isasymsubseteq A \isasymunion B |
|
716 |
\tdx{Un_upper2}: B \isasymsubseteq A \isasymunion B |
|
717 |
\tdx{Un_least}: [| A \isasymsubseteq C; B \isasymsubseteq C |] ==> A \isasymunion B \isasymsubseteq C |
|
718 |
||
14158 | 719 |
\tdx{Int_lower1}: A \isasyminter B \isasymsubseteq A |
720 |
\tdx{Int_lower2}: A \isasyminter B \isasymsubseteq B |
|
721 |
\tdx{Int_greatest}: [| C \isasymsubseteq A; C \isasymsubseteq B |] ==> C \isasymsubseteq A \isasyminter B |
|
14154 | 722 |
|
723 |
\tdx{Diff_subset}: A-B \isasymsubseteq A |
|
14158 | 724 |
\tdx{Diff_contains}: [| C \isasymsubseteq A; C \isasyminter B = 0 |] ==> C \isasymsubseteq A-B |
14154 | 725 |
|
726 |
\tdx{Collect_subset}: Collect(A,P) \isasymsubseteq A |
|
727 |
\end{alltt*} |
|
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
728 |
\caption{Subset and lattice properties} \label{zf-subset} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
729 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
730 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
731 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
732 |
\subsection{Subset and lattice properties} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
733 |
The subset relation is a complete lattice. Unions form least upper bounds; |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
734 |
non-empty intersections form greatest lower bounds. Figure~\ref{zf-subset} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
735 |
shows the corresponding rules. A few other laws involving subsets are |
14154 | 736 |
included. |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
737 |
Reasoning directly about subsets often yields clearer proofs than |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
738 |
reasoning about the membership relation. Section~\ref{sec:ZF-pow-example} |
14154 | 739 |
below presents an example of this, proving the equation |
740 |
${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$. |
|
741 |
||
742 |
%%% pair.thy |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
743 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
744 |
\begin{figure} |
14154 | 745 |
\begin{alltt*}\isastyleminor |
746 |
\tdx{Pair_inject1}: <a,b> = <c,d> ==> a=c |
|
747 |
\tdx{Pair_inject2}: <a,b> = <c,d> ==> b=d |
|
748 |
\tdx{Pair_inject}: [| <a,b> = <c,d>; [| a=c; b=d |] ==> P |] ==> P |
|
749 |
\tdx{Pair_neq_0}: <a,b>=0 ==> P |
|
750 |
||
751 |
\tdx{fst_conv}: fst(<a,b>) = a |
|
752 |
\tdx{snd_conv}: snd(<a,b>) = b |
|
753 |
\tdx{split}: split(\%x y. c(x,y), <a,b>) = c(a,b) |
|
754 |
||
755 |
\tdx{SigmaI}: [| a\isasymin{}A; b\isasymin{}B(a) |] ==> <a,b>\isasymin{}Sigma(A,B) |
|
756 |
||
757 |
\tdx{SigmaE}: [| c\isasymin{}Sigma(A,B); |
|
758 |
!!x y.[| x\isasymin{}A; y\isasymin{}B(x); c=<x,y> |] ==> P |] ==> P |
|
759 |
||
760 |
\tdx{SigmaE2}: [| <a,b>\isasymin{}Sigma(A,B); |
|
761 |
[| a\isasymin{}A; b\isasymin{}B(a) |] ==> P |] ==> P |
|
762 |
\end{alltt*} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
763 |
\caption{Ordered pairs; projections; general sums} \label{zf-pair} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
764 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
765 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
766 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
767 |
\subsection{Ordered pairs} \label{sec:pairs} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
768 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
769 |
Figure~\ref{zf-pair} presents the rules governing ordered pairs, |
14154 | 770 |
projections and general sums --- in particular, that |
771 |
$\{\{a\},\{a,b\}\}$ functions as an ordered pair. This property is |
|
772 |
expressed as two destruction rules, |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
773 |
\tdx{Pair_inject1} and \tdx{Pair_inject2}, and equivalently |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
774 |
as the elimination rule \tdx{Pair_inject}. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
775 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
776 |
The rule \tdx{Pair_neq_0} asserts $\pair{a,b}\neq\emptyset$. This |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
777 |
is a property of $\{\{a\},\{a,b\}\}$, and need not hold for other |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
778 |
encodings of ordered pairs. The non-standard ordered pairs mentioned below |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
779 |
satisfy $\pair{\emptyset;\emptyset}=\emptyset$. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
780 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
781 |
The natural deduction rules \tdx{SigmaI} and \tdx{SigmaE} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
782 |
assert that \cdx{Sigma}$(A,B)$ consists of all pairs of the form |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
783 |
$\pair{x,y}$, for $x\in A$ and $y\in B(x)$. The rule \tdx{SigmaE2} |
14154 | 784 |
merely states that $\pair{a,b}\in \isa{Sigma}(A,B)$ implies $a\in A$ and |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
785 |
$b\in B(a)$. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
786 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
787 |
In addition, it is possible to use tuples as patterns in abstractions: |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
788 |
\begin{center} |
14154 | 789 |
{\tt\%<$x$,$y$>. $t$} \quad stands for\quad \isa{split(\%$x$ $y$.\ $t$)} |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
790 |
\end{center} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
791 |
Nested patterns are translated recursively: |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
792 |
{\tt\%<$x$,$y$,$z$>. $t$} $\leadsto$ {\tt\%<$x$,<$y$,$z$>>. $t$} $\leadsto$ |
14154 | 793 |
\isa{split(\%$x$.\%<$y$,$z$>. $t$)} $\leadsto$ \isa{split(\%$x$. split(\%$y$ |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
794 |
$z$.\ $t$))}. The reverse translation is performed upon printing. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
795 |
\begin{warn} |
14154 | 796 |
The translation between patterns and \isa{split} is performed automatically |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
797 |
by the parser and printer. Thus the internal and external form of a term |
14154 | 798 |
may differ, which affects proofs. For example the term \isa{(\%<x,y>.<y,x>)<a,b>} requires the theorem \isa{split} to rewrite to |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
799 |
{\tt<b,a>}. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
800 |
\end{warn} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
801 |
In addition to explicit $\lambda$-abstractions, patterns can be used in any |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
802 |
variable binding construct which is internally described by a |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
803 |
$\lambda$-abstraction. Here are some important examples: |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
804 |
\begin{description} |
14154 | 805 |
\item[Let:] \isa{let {\it pattern} = $t$ in $u$} |
806 |
\item[Choice:] \isa{THE~{\it pattern}~.~$P$} |
|
807 |
\item[Set operations:] \isa{\isasymUnion~{\it pattern}:$A$.~$B$} |
|
808 |
\item[Comprehension:] \isa{{\ttlbrace}~{\it pattern}:$A$~.~$P$~{\ttrbrace}} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
809 |
\end{description} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
810 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
811 |
|
14154 | 812 |
%%% domrange.thy? |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
813 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
814 |
\begin{figure} |
14154 | 815 |
\begin{alltt*}\isastyleminor |
816 |
\tdx{domainI}: <a,b>\isasymin{}r ==> a\isasymin{}domain(r) |
|
817 |
\tdx{domainE}: [| a\isasymin{}domain(r); !!y. <a,y>\isasymin{}r ==> P |] ==> P |
|
818 |
\tdx{domain_subset}: domain(Sigma(A,B)) \isasymsubseteq A |
|
819 |
||
820 |
\tdx{rangeI}: <a,b>\isasymin{}r ==> b\isasymin{}range(r) |
|
821 |
\tdx{rangeE}: [| b\isasymin{}range(r); !!x. <x,b>\isasymin{}r ==> P |] ==> P |
|
822 |
\tdx{range_subset}: range(A*B) \isasymsubseteq B |
|
823 |
||
824 |
\tdx{fieldI1}: <a,b>\isasymin{}r ==> a\isasymin{}field(r) |
|
825 |
\tdx{fieldI2}: <a,b>\isasymin{}r ==> b\isasymin{}field(r) |
|
826 |
\tdx{fieldCI}: (<c,a> \isasymnotin r ==> <a,b>\isasymin{}r) ==> a\isasymin{}field(r) |
|
827 |
||
828 |
\tdx{fieldE}: [| a\isasymin{}field(r); |
|
14158 | 829 |
!!x. <a,x>\isasymin{}r ==> P; |
830 |
!!x. <x,a>\isasymin{}r ==> P |
|
831 |
|] ==> P |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
832 |
|
14154 | 833 |
\tdx{field_subset}: field(A*A) \isasymsubseteq A |
834 |
\end{alltt*} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
835 |
\caption{Domain, range and field of a relation} \label{zf-domrange} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
836 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
837 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
838 |
\begin{figure} |
14154 | 839 |
\begin{alltt*}\isastyleminor |
840 |
\tdx{imageI}: [| <a,b>\isasymin{}r; a\isasymin{}A |] ==> b\isasymin{}r``A |
|
841 |
\tdx{imageE}: [| b\isasymin{}r``A; !!x.[| <x,b>\isasymin{}r; x\isasymin{}A |] ==> P |] ==> P |
|
842 |
||
843 |
\tdx{vimageI}: [| <a,b>\isasymin{}r; b\isasymin{}B |] ==> a\isasymin{}r-``B |
|
844 |
\tdx{vimageE}: [| a\isasymin{}r-``B; !!x.[| <a,x>\isasymin{}r; x\isasymin{}B |] ==> P |] ==> P |
|
845 |
\end{alltt*} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
846 |
\caption{Image and inverse image} \label{zf-domrange2} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
847 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
848 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
849 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
850 |
\subsection{Relations} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
851 |
Figure~\ref{zf-domrange} presents rules involving relations, which are sets |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
852 |
of ordered pairs. The converse of a relation~$r$ is the set of all pairs |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
853 |
$\pair{y,x}$ such that $\pair{x,y}\in r$; if $r$ is a function, then |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
854 |
{\cdx{converse}$(r)$} is its inverse. The rules for the domain |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
855 |
operation, namely \tdx{domainI} and~\tdx{domainE}, assert that |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
856 |
\cdx{domain}$(r)$ consists of all~$x$ such that $r$ contains |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
857 |
some pair of the form~$\pair{x,y}$. The range operation is similar, and |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
858 |
the field of a relation is merely the union of its domain and range. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
859 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
860 |
Figure~\ref{zf-domrange2} presents rules for images and inverse images. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
861 |
Note that these operations are generalisations of range and domain, |
14154 | 862 |
respectively. |
863 |
||
864 |
||
865 |
%%% func.thy |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
866 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
867 |
\begin{figure} |
14154 | 868 |
\begin{alltt*}\isastyleminor |
869 |
\tdx{fun_is_rel}: f\isasymin{}Pi(A,B) ==> f \isasymsubseteq Sigma(A,B) |
|
870 |
||
14158 | 871 |
\tdx{apply_equality}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> f`a = b |
14154 | 872 |
\tdx{apply_equality2}: [| <a,b>\isasymin{}f; <a,c>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b=c |
873 |
||
874 |
\tdx{apply_type}: [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> f`a\isasymin{}B(a) |
|
875 |
\tdx{apply_Pair}: [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> <a,f`a>\isasymin{}f |
|
876 |
\tdx{apply_iff}: f\isasymin{}Pi(A,B) ==> <a,b>\isasymin{}f <-> a\isasymin{}A & f`a = b |
|
877 |
||
878 |
\tdx{fun_extension}: [| f\isasymin{}Pi(A,B); g\isasymin{}Pi(A,D); |
|
879 |
!!x. x\isasymin{}A ==> f`x = g`x |] ==> f=g |
|
880 |
||
881 |
\tdx{domain_type}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> a\isasymin{}A |
|
882 |
\tdx{range_type}: [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b\isasymin{}B(a) |
|
883 |
||
884 |
\tdx{Pi_type}: [| f\isasymin{}A->C; !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==> f\isasymin{}Pi(A,B) |
|
885 |
\tdx{domain_of_fun}: f\isasymin{}Pi(A,B) ==> domain(f)=A |
|
886 |
\tdx{range_of_fun}: f\isasymin{}Pi(A,B) ==> f\isasymin{}A->range(f) |
|
887 |
||
888 |
\tdx{restrict}: a\isasymin{}A ==> restrict(f,A) ` a = f`a |
|
889 |
\tdx{restrict_type}: [| !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==> |
|
890 |
restrict(f,A)\isasymin{}Pi(A,B) |
|
891 |
\end{alltt*} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
892 |
\caption{Functions} \label{zf-func1} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
893 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
894 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
895 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
896 |
\begin{figure} |
14154 | 897 |
\begin{alltt*}\isastyleminor |
898 |
\tdx{lamI}: a\isasymin{}A ==> <a,b(a)>\isasymin{}(lam x\isasymin{}A. b(x)) |
|
899 |
\tdx{lamE}: [| p\isasymin{}(lam x\isasymin{}A. b(x)); !!x.[| x\isasymin{}A; p=<x,b(x)> |] ==> P |
|
8249 | 900 |
|] ==> P |
901 |
||
14154 | 902 |
\tdx{lam_type}: [| !!x. x\isasymin{}A ==> b(x)\isasymin{}B(x) |] ==> (lam x\isasymin{}A. b(x))\isasymin{}Pi(A,B) |
903 |
||
904 |
\tdx{beta}: a\isasymin{}A ==> (lam x\isasymin{}A. b(x)) ` a = b(a) |
|
905 |
\tdx{eta}: f\isasymin{}Pi(A,B) ==> (lam x\isasymin{}A. f`x) = f |
|
906 |
\end{alltt*} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
907 |
\caption{$\lambda$-abstraction} \label{zf-lam} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
908 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
909 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
910 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
911 |
\begin{figure} |
14154 | 912 |
\begin{alltt*}\isastyleminor |
913 |
\tdx{fun_empty}: 0\isasymin{}0->0 |
|
914 |
\tdx{fun_single}: {\ttlbrace}<a,b>{\ttrbrace}\isasymin{}{\ttlbrace}a{\ttrbrace} -> {\ttlbrace}b{\ttrbrace} |
|
915 |
||
14158 | 916 |
\tdx{fun_disjoint_Un}: [| f\isasymin{}A->B; g\isasymin{}C->D; A \isasyminter C = 0 |] ==> |
14154 | 917 |
(f \isasymunion g)\isasymin{}(A \isasymunion C) -> (B \isasymunion D) |
918 |
||
919 |
\tdx{fun_disjoint_apply1}: [| a\isasymin{}A; f\isasymin{}A->B; g\isasymin{}C->D; A\isasyminter{}C = 0 |] ==> |
|
920 |
(f \isasymunion g)`a = f`a |
|
921 |
||
922 |
\tdx{fun_disjoint_apply2}: [| c\isasymin{}C; f\isasymin{}A->B; g\isasymin{}C->D; A\isasyminter{}C = 0 |] ==> |
|
923 |
(f \isasymunion g)`c = g`c |
|
924 |
\end{alltt*} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
925 |
\caption{Constructing functions from smaller sets} \label{zf-func2} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
926 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
927 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
928 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
929 |
\subsection{Functions} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
930 |
Functions, represented by graphs, are notoriously difficult to reason |
14154 | 931 |
about. The ZF theory provides many derived rules, which overlap more |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
932 |
than they ought. This section presents the more important rules. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
933 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
934 |
Figure~\ref{zf-func1} presents the basic properties of \cdx{Pi}$(A,B)$, |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
935 |
the generalized function space. For example, if $f$ is a function and |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
936 |
$\pair{a,b}\in f$, then $f`a=b$ (\tdx{apply_equality}). Two functions |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
937 |
are equal provided they have equal domains and deliver equals results |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
938 |
(\tdx{fun_extension}). |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
939 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
940 |
By \tdx{Pi_type}, a function typing of the form $f\in A\to C$ can be |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
941 |
refined to the dependent typing $f\in\prod@{x\in A}B(x)$, given a suitable |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
942 |
family of sets $\{B(x)\}@{x\in A}$. Conversely, by \tdx{range_of_fun}, |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
943 |
any dependent typing can be flattened to yield a function type of the form |
14154 | 944 |
$A\to C$; here, $C=\isa{range}(f)$. |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
945 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
946 |
Among the laws for $\lambda$-abstraction, \tdx{lamI} and \tdx{lamE} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
947 |
describe the graph of the generated function, while \tdx{beta} and |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
948 |
\tdx{eta} are the standard conversions. We essentially have a |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
949 |
dependently-typed $\lambda$-calculus (Fig.\ts\ref{zf-lam}). |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
950 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
951 |
Figure~\ref{zf-func2} presents some rules that can be used to construct |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
952 |
functions explicitly. We start with functions consisting of at most one |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
953 |
pair, and may form the union of two functions provided their domains are |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
954 |
disjoint. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
955 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
956 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
957 |
\begin{figure} |
14154 | 958 |
\begin{alltt*}\isastyleminor |
14158 | 959 |
\tdx{Int_absorb}: A \isasyminter A = A |
960 |
\tdx{Int_commute}: A \isasyminter B = B \isasyminter A |
|
961 |
\tdx{Int_assoc}: (A \isasyminter B) \isasyminter C = A \isasyminter (B \isasyminter C) |
|
962 |
\tdx{Int_Un_distrib}: (A \isasymunion B) \isasyminter C = (A \isasyminter C) \isasymunion (B \isasyminter C) |
|
14154 | 963 |
|
964 |
\tdx{Un_absorb}: A \isasymunion A = A |
|
965 |
\tdx{Un_commute}: A \isasymunion B = B \isasymunion A |
|
966 |
\tdx{Un_assoc}: (A \isasymunion B) \isasymunion C = A \isasymunion (B \isasymunion C) |
|
14158 | 967 |
\tdx{Un_Int_distrib}: (A \isasyminter B) \isasymunion C = (A \isasymunion C) \isasyminter (B \isasymunion C) |
14154 | 968 |
|
969 |
\tdx{Diff_cancel}: A-A = 0 |
|
14158 | 970 |
\tdx{Diff_disjoint}: A \isasyminter (B-A) = 0 |
14154 | 971 |
\tdx{Diff_partition}: A \isasymsubseteq B ==> A \isasymunion (B-A) = B |
972 |
\tdx{double_complement}: [| A \isasymsubseteq B; B \isasymsubseteq C |] ==> (B - (C-A)) = A |
|
14158 | 973 |
\tdx{Diff_Un}: A - (B \isasymunion C) = (A-B) \isasyminter (A-C) |
974 |
\tdx{Diff_Int}: A - (B \isasyminter C) = (A-B) \isasymunion (A-C) |
|
14154 | 975 |
|
976 |
\tdx{Union_Un_distrib}: Union(A \isasymunion B) = Union(A) \isasymunion Union(B) |
|
977 |
\tdx{Inter_Un_distrib}: [| a \isasymin A; b \isasymin B |] ==> |
|
14158 | 978 |
Inter(A \isasymunion B) = Inter(A) \isasyminter Inter(B) |
979 |
||
980 |
\tdx{Int_Union_RepFun}: A \isasyminter Union(B) = ({\isasymUnion}C \isasymin B. A \isasyminter C) |
|
14154 | 981 |
|
982 |
\tdx{Un_Inter_RepFun}: b \isasymin B ==> |
|
983 |
A \isasymunion Inter(B) = ({\isasymInter}C \isasymin B. A \isasymunion C) |
|
984 |
||
985 |
\tdx{SUM_Un_distrib1}: (SUM x \isasymin A \isasymunion B. C(x)) = |
|
986 |
(SUM x \isasymin A. C(x)) \isasymunion (SUM x \isasymin B. C(x)) |
|
987 |
||
988 |
\tdx{SUM_Un_distrib2}: (SUM x \isasymin C. A(x) \isasymunion B(x)) = |
|
989 |
(SUM x \isasymin C. A(x)) \isasymunion (SUM x \isasymin C. B(x)) |
|
990 |
||
14158 | 991 |
\tdx{SUM_Int_distrib1}: (SUM x \isasymin A \isasyminter B. C(x)) = |
992 |
(SUM x \isasymin A. C(x)) \isasyminter (SUM x \isasymin B. C(x)) |
|
993 |
||
994 |
\tdx{SUM_Int_distrib2}: (SUM x \isasymin C. A(x) \isasyminter B(x)) = |
|
995 |
(SUM x \isasymin C. A(x)) \isasyminter (SUM x \isasymin C. B(x)) |
|
14154 | 996 |
\end{alltt*} |
6121
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the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
997 |
\caption{Equalities} \label{zf-equalities} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
998 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
999 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1000 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1001 |
\begin{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1002 |
%\begin{constants} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1003 |
% \cdx{1} & $i$ & & $\{\emptyset\}$ \\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1004 |
% \cdx{bool} & $i$ & & the set $\{\emptyset,1\}$ \\ |
14154 | 1005 |
% \cdx{cond} & $[i,i,i]\To i$ & & conditional for \isa{bool} \\ |
1006 |
% \cdx{not} & $i\To i$ & & negation for \isa{bool} \\ |
|
1007 |
% \sdx{and} & $[i,i]\To i$ & Left 70 & conjunction for \isa{bool} \\ |
|
1008 |
% \sdx{or} & $[i,i]\To i$ & Left 65 & disjunction for \isa{bool} \\ |
|
1009 |
% \sdx{xor} & $[i,i]\To i$ & Left 65 & exclusive-or for \isa{bool} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1010 |
%\end{constants} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1011 |
% |
14154 | 1012 |
\begin{alltt*}\isastyleminor |
1013 |
\tdx{bool_def}: bool == {\ttlbrace}0,1{\ttrbrace} |
|
1014 |
\tdx{cond_def}: cond(b,c,d) == if b=1 then c else d |
|
1015 |
\tdx{not_def}: not(b) == cond(b,0,1) |
|
1016 |
\tdx{and_def}: a and b == cond(a,b,0) |
|
1017 |
\tdx{or_def}: a or b == cond(a,1,b) |
|
1018 |
\tdx{xor_def}: a xor b == cond(a,not(b),b) |
|
1019 |
||
1020 |
\tdx{bool_1I}: 1 \isasymin bool |
|
1021 |
\tdx{bool_0I}: 0 \isasymin bool |
|
1022 |
\tdx{boolE}: [| c \isasymin bool; c=1 ==> P; c=0 ==> P |] ==> P |
|
1023 |
\tdx{cond_1}: cond(1,c,d) = c |
|
1024 |
\tdx{cond_0}: cond(0,c,d) = d |
|
1025 |
\end{alltt*} |
|
6121
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the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1026 |
\caption{The booleans} \label{zf-bool} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1027 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1028 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1029 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1030 |
\section{Further developments} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1031 |
The next group of developments is complex and extensive, and only |
14154 | 1032 |
highlights can be covered here. It involves many theories and proofs. |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1033 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1034 |
Figure~\ref{zf-equalities} presents commutative, associative, distributive, |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1035 |
and idempotency laws of union and intersection, along with other equations. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1036 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1037 |
Theory \thydx{Bool} defines $\{0,1\}$ as a set of booleans, with the usual |
9695 | 1038 |
operators including a conditional (Fig.\ts\ref{zf-bool}). Although ZF is a |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1039 |
first-order theory, you can obtain the effect of higher-order logic using |
14154 | 1040 |
\isa{bool}-valued functions, for example. The constant~\isa{1} is |
1041 |
translated to \isa{succ(0)}. |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1042 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1043 |
\begin{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1044 |
\index{*"+ symbol} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1045 |
\begin{constants} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1046 |
\it symbol & \it meta-type & \it priority & \it description \\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1047 |
\tt + & $[i,i]\To i$ & Right 65 & disjoint union operator\\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1048 |
\cdx{Inl}~~\cdx{Inr} & $i\To i$ & & injections\\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1049 |
\cdx{case} & $[i\To i,i\To i, i]\To i$ & & conditional for $A+B$ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1050 |
\end{constants} |
14154 | 1051 |
\begin{alltt*}\isastyleminor |
1052 |
\tdx{sum_def}: A+B == {\ttlbrace}0{\ttrbrace}*A \isasymunion {\ttlbrace}1{\ttrbrace}*B |
|
1053 |
\tdx{Inl_def}: Inl(a) == <0,a> |
|
1054 |
\tdx{Inr_def}: Inr(b) == <1,b> |
|
1055 |
\tdx{case_def}: case(c,d,u) == split(\%y z. cond(y, d(z), c(z)), u) |
|
1056 |
||
1057 |
\tdx{InlI}: a \isasymin A ==> Inl(a) \isasymin A+B |
|
1058 |
\tdx{InrI}: b \isasymin B ==> Inr(b) \isasymin A+B |
|
1059 |
||
1060 |
\tdx{Inl_inject}: Inl(a)=Inl(b) ==> a=b |
|
1061 |
\tdx{Inr_inject}: Inr(a)=Inr(b) ==> a=b |
|
1062 |
\tdx{Inl_neq_Inr}: Inl(a)=Inr(b) ==> P |
|
1063 |
||
1064 |
\tdx{sum_iff}: u \isasymin A+B <-> ({\isasymexists}x\isasymin{}A. u=Inl(x)) | ({\isasymexists}y\isasymin{}B. u=Inr(y)) |
|
1065 |
||
1066 |
\tdx{case_Inl}: case(c,d,Inl(a)) = c(a) |
|
1067 |
\tdx{case_Inr}: case(c,d,Inr(b)) = d(b) |
|
1068 |
\end{alltt*} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1069 |
\caption{Disjoint unions} \label{zf-sum} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1070 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1071 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1072 |
|
9584
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1073 |
\subsection{Disjoint unions} |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1074 |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1075 |
Theory \thydx{Sum} defines the disjoint union of two sets, with |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1076 |
injections and a case analysis operator (Fig.\ts\ref{zf-sum}). Disjoint |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1077 |
unions play a role in datatype definitions, particularly when there is |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1078 |
mutual recursion~\cite{paulson-set-II}. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1079 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1080 |
\begin{figure} |
14154 | 1081 |
\begin{alltt*}\isastyleminor |
1082 |
\tdx{QPair_def}: <a;b> == a+b |
|
1083 |
\tdx{qsplit_def}: qsplit(c,p) == THE y. {\isasymexists}a b. p=<a;b> & y=c(a,b) |
|
1084 |
\tdx{qfsplit_def}: qfsplit(R,z) == {\isasymexists}x y. z=<x;y> & R(x,y) |
|
1085 |
\tdx{qconverse_def}: qconverse(r) == {\ttlbrace}z. w \isasymin r, {\isasymexists}x y. w=<x;y> & z=<y;x>{\ttrbrace} |
|
1086 |
\tdx{QSigma_def}: QSigma(A,B) == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x;y>{\ttrbrace} |
|
1087 |
||
1088 |
\tdx{qsum_def}: A <+> B == ({\ttlbrace}0{\ttrbrace} <*> A) \isasymunion ({\ttlbrace}1{\ttrbrace} <*> B) |
|
1089 |
\tdx{QInl_def}: QInl(a) == <0;a> |
|
1090 |
\tdx{QInr_def}: QInr(b) == <1;b> |
|
1091 |
\tdx{qcase_def}: qcase(c,d) == qsplit(\%y z. cond(y, d(z), c(z))) |
|
1092 |
\end{alltt*} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1093 |
\caption{Non-standard pairs, products and sums} \label{zf-qpair} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1094 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1095 |
|
9584
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1096 |
|
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1097 |
\subsection{Non-standard ordered pairs} |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1098 |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1099 |
Theory \thydx{QPair} defines a notion of ordered pair that admits |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1100 |
non-well-founded tupling (Fig.\ts\ref{zf-qpair}). Such pairs are written |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1101 |
{\tt<$a$;$b$>}. It also defines the eliminator \cdx{qsplit}, the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1102 |
converse operator \cdx{qconverse}, and the summation operator |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1103 |
\cdx{QSigma}. These are completely analogous to the corresponding |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1104 |
versions for standard ordered pairs. The theory goes on to define a |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1105 |
non-standard notion of disjoint sum using non-standard pairs. All of these |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1106 |
concepts satisfy the same properties as their standard counterparts; in |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1107 |
addition, {\tt<$a$;$b$>} is continuous. The theory supports coinductive |
6592 | 1108 |
definitions, for example of infinite lists~\cite{paulson-mscs}. |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1109 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1110 |
\begin{figure} |
14154 | 1111 |
\begin{alltt*}\isastyleminor |
1112 |
\tdx{bnd_mono_def}: bnd_mono(D,h) == |
|
14158 | 1113 |
h(D)\isasymsubseteq{}D & ({\isasymforall}W X. W\isasymsubseteq{}X --> X\isasymsubseteq{}D --> h(W)\isasymsubseteq{}h(X)) |
14154 | 1114 |
|
1115 |
\tdx{lfp_def}: lfp(D,h) == Inter({\ttlbrace}X \isasymin Pow(D). h(X) \isasymsubseteq X{\ttrbrace}) |
|
1116 |
\tdx{gfp_def}: gfp(D,h) == Union({\ttlbrace}X \isasymin Pow(D). X \isasymsubseteq h(X){\ttrbrace}) |
|
1117 |
||
1118 |
||
14158 | 1119 |
\tdx{lfp_lowerbound}: [| h(A) \isasymsubseteq A; A \isasymsubseteq D |] ==> lfp(D,h) \isasymsubseteq A |
14154 | 1120 |
|
1121 |
\tdx{lfp_subset}: lfp(D,h) \isasymsubseteq D |
|
1122 |
||
1123 |
\tdx{lfp_greatest}: [| bnd_mono(D,h); |
|
1124 |
!!X. [| h(X) \isasymsubseteq X; X \isasymsubseteq D |] ==> A \isasymsubseteq X |
|
1125 |
|] ==> A \isasymsubseteq lfp(D,h) |
|
1126 |
||
1127 |
\tdx{lfp_Tarski}: bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h)) |
|
1128 |
||
1129 |
\tdx{induct}: [| a \isasymin lfp(D,h); bnd_mono(D,h); |
|
1130 |
!!x. x \isasymin h(Collect(lfp(D,h),P)) ==> P(x) |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1131 |
|] ==> P(a) |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1132 |
|
14154 | 1133 |
\tdx{lfp_mono}: [| bnd_mono(D,h); bnd_mono(E,i); |
1134 |
!!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X) |
|
1135 |
|] ==> lfp(D,h) \isasymsubseteq lfp(E,i) |
|
1136 |
||
14158 | 1137 |
\tdx{gfp_upperbound}: [| A \isasymsubseteq h(A); A \isasymsubseteq D |] ==> A \isasymsubseteq gfp(D,h) |
14154 | 1138 |
|
1139 |
\tdx{gfp_subset}: gfp(D,h) \isasymsubseteq D |
|
1140 |
||
1141 |
\tdx{gfp_least}: [| bnd_mono(D,h); |
|
1142 |
!!X. [| X \isasymsubseteq h(X); X \isasymsubseteq D |] ==> X \isasymsubseteq A |
|
1143 |
|] ==> gfp(D,h) \isasymsubseteq A |
|
1144 |
||
1145 |
\tdx{gfp_Tarski}: bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h)) |
|
1146 |
||
1147 |
\tdx{coinduct}: [| bnd_mono(D,h); a \isasymin X; X \isasymsubseteq h(X \isasymunion gfp(D,h)); X \isasymsubseteq D |
|
1148 |
|] ==> a \isasymin gfp(D,h) |
|
1149 |
||
1150 |
\tdx{gfp_mono}: [| bnd_mono(D,h); D \isasymsubseteq E; |
|
1151 |
!!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X) |
|
1152 |
|] ==> gfp(D,h) \isasymsubseteq gfp(E,i) |
|
1153 |
\end{alltt*} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1154 |
\caption{Least and greatest fixedpoints} \label{zf-fixedpt} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1155 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1156 |
|
9584
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1157 |
|
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1158 |
\subsection{Least and greatest fixedpoints} |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1159 |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1160 |
The Knaster-Tarski Theorem states that every monotone function over a |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1161 |
complete lattice has a fixedpoint. Theory \thydx{Fixedpt} proves the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1162 |
Theorem only for a particular lattice, namely the lattice of subsets of a |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1163 |
set (Fig.\ts\ref{zf-fixedpt}). The theory defines least and greatest |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1164 |
fixedpoint operators with corresponding induction and coinduction rules. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1165 |
These are essential to many definitions that follow, including the natural |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1166 |
numbers and the transitive closure operator. The (co)inductive definition |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1167 |
package also uses the fixedpoint operators~\cite{paulson-CADE}. See |
6745 | 1168 |
Davey and Priestley~\cite{davey-priestley} for more on the Knaster-Tarski |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1169 |
Theorem and my paper~\cite{paulson-set-II} for discussion of the Isabelle |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1170 |
proofs. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1171 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1172 |
Monotonicity properties are proved for most of the set-forming operations: |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1173 |
union, intersection, Cartesian product, image, domain, range, etc. These |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1174 |
are useful for applying the Knaster-Tarski Fixedpoint Theorem. The proofs |
14154 | 1175 |
themselves are trivial applications of Isabelle's classical reasoner. |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1176 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1177 |
|
9584
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1178 |
\subsection{Finite sets and lists} |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1179 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1180 |
Theory \texttt{Finite} (Figure~\ref{zf-fin}) defines the finite set operator; |
14154 | 1181 |
$\isa{Fin}(A)$ is the set of all finite sets over~$A$. The theory employs |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1182 |
Isabelle's inductive definition package, which proves various rules |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1183 |
automatically. The induction rule shown is stronger than the one proved by |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1184 |
the package. The theory also defines the set of all finite functions |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1185 |
between two given sets. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1186 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1187 |
\begin{figure} |
14154 | 1188 |
\begin{alltt*}\isastyleminor |
1189 |
\tdx{Fin.emptyI} 0 \isasymin Fin(A) |
|
1190 |
\tdx{Fin.consI} [| a \isasymin A; b \isasymin Fin(A) |] ==> cons(a,b) \isasymin Fin(A) |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1191 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1192 |
\tdx{Fin_induct} |
14154 | 1193 |
[| b \isasymin Fin(A); |
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|
1194 |
P(0); |
14158 | 1195 |
!!x y. [| x\isasymin{}A; y\isasymin{}Fin(A); x\isasymnotin{}y; P(y) |] ==> P(cons(x,y)) |
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|
1196 |
|] ==> P(b) |
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|
1197 |
|
14154 | 1198 |
\tdx{Fin_mono}: A \isasymsubseteq B ==> Fin(A) \isasymsubseteq Fin(B) |
1199 |
\tdx{Fin_UnI}: [| b \isasymin Fin(A); c \isasymin Fin(A) |] ==> b \isasymunion c \isasymin Fin(A) |
|
1200 |
\tdx{Fin_UnionI}: C \isasymin Fin(Fin(A)) ==> Union(C) \isasymin Fin(A) |
|
1201 |
\tdx{Fin_subset}: [| c \isasymsubseteq b; b \isasymin Fin(A) |] ==> c \isasymin Fin(A) |
|
1202 |
\end{alltt*} |
|
6121
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|
1203 |
\caption{The finite set operator} \label{zf-fin} |
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|
1204 |
\end{figure} |
5fe77b9b5185
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|
1205 |
|
5fe77b9b5185
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parents:
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changeset
|
1206 |
\begin{figure} |
5fe77b9b5185
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parents:
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|
1207 |
\begin{constants} |
5fe77b9b5185
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parents:
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|
1208 |
\it symbol & \it meta-type & \it priority & \it description \\ |
5fe77b9b5185
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parents:
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changeset
|
1209 |
\cdx{list} & $i\To i$ && lists over some set\\ |
5fe77b9b5185
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parents:
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|
1210 |
\cdx{list_case} & $[i, [i,i]\To i, i] \To i$ && conditional for $list(A)$ \\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1211 |
\cdx{map} & $[i\To i, i] \To i$ & & mapping functional\\ |
5fe77b9b5185
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parents:
diff
changeset
|
1212 |
\cdx{length} & $i\To i$ & & length of a list\\ |
5fe77b9b5185
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parents:
diff
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|
1213 |
\cdx{rev} & $i\To i$ & & reverse of a list\\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
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|
1214 |
\tt \at & $[i,i]\To i$ & Right 60 & append for lists\\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
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|
1215 |
\cdx{flat} & $i\To i$ & & append of list of lists |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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|
1216 |
\end{constants} |
5fe77b9b5185
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diff
changeset
|
1217 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
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|
1218 |
\underscoreon %%because @ is used here |
14154 | 1219 |
\begin{alltt*}\isastyleminor |
14158 | 1220 |
\tdx{NilI}: Nil \isasymin list(A) |
1221 |
\tdx{ConsI}: [| a \isasymin A; l \isasymin list(A) |] ==> Cons(a,l) \isasymin list(A) |
|
6121
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|
1222 |
|
5fe77b9b5185
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parents:
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|
1223 |
\tdx{List.induct} |
14154 | 1224 |
[| l \isasymin list(A); |
6121
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|
1225 |
P(Nil); |
14154 | 1226 |
!!x y. [| x \isasymin A; y \isasymin list(A); P(y) |] ==> P(Cons(x,y)) |
6121
5fe77b9b5185
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|
1227 |
|] ==> P(l) |
5fe77b9b5185
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parents:
diff
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|
1228 |
|
14154 | 1229 |
\tdx{Cons_iff}: Cons(a,l)=Cons(a',l') <-> a=a' & l=l' |
1230 |
\tdx{Nil_Cons_iff}: Nil \isasymnoteq Cons(a,l) |
|
1231 |
||
1232 |
\tdx{list_mono}: A \isasymsubseteq B ==> list(A) \isasymsubseteq list(B) |
|
1233 |
||
14158 | 1234 |
\tdx{map_ident}: l\isasymin{}list(A) ==> map(\%u. u, l) = l |
1235 |
\tdx{map_compose}: l\isasymin{}list(A) ==> map(h, map(j,l)) = map(\%u. h(j(u)), l) |
|
1236 |
\tdx{map_app_distrib}: xs\isasymin{}list(A) ==> map(h, xs@ys) = map(h,xs)@map(h,ys) |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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|
1237 |
\tdx{map_type} |
14158 | 1238 |
[| l\isasymin{}list(A); !!x. x\isasymin{}A ==> h(x)\isasymin{}B |] ==> map(h,l)\isasymin{}list(B) |
6121
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|
1239 |
\tdx{map_flat} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1240 |
ls: list(list(A)) ==> map(h, flat(ls)) = flat(map(map(h),ls)) |
14154 | 1241 |
\end{alltt*} |
6121
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|
1242 |
\caption{Lists} \label{zf-list} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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diff
changeset
|
1243 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
changeset
|
1244 |
|
5fe77b9b5185
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|
1245 |
|
14154 | 1246 |
Figure~\ref{zf-list} presents the set of lists over~$A$, $\isa{list}(A)$. The |
6121
5fe77b9b5185
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|
1247 |
definition employs Isabelle's datatype package, which defines the introduction |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1248 |
and induction rules automatically, as well as the constructors, case operator |
14154 | 1249 |
(\isa{list\_case}) and recursion operator. The theory then defines the usual |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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|
1250 |
list functions by primitive recursion. See theory \texttt{List}. |
5fe77b9b5185
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parents:
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changeset
|
1251 |
|
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parents:
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changeset
|
1252 |
|
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|
1253 |
\subsection{Miscellaneous} |
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|
1254 |
|
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|
1255 |
\begin{figure} |
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|
1256 |
\begin{constants} |
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|
1257 |
\it symbol & \it meta-type & \it priority & \it description \\ |
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|
1258 |
\sdx{O} & $[i,i]\To i$ & Right 60 & composition ($\circ$) \\ |
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|
1259 |
\cdx{id} & $i\To i$ & & identity function \\ |
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|
1260 |
\cdx{inj} & $[i,i]\To i$ & & injective function space\\ |
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|
1261 |
\cdx{surj} & $[i,i]\To i$ & & surjective function space\\ |
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|
1262 |
\cdx{bij} & $[i,i]\To i$ & & bijective function space |
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|
1263 |
\end{constants} |
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|
1264 |
|
14154 | 1265 |
\begin{alltt*}\isastyleminor |
1266 |
\tdx{comp_def}: r O s == {\ttlbrace}xz \isasymin domain(s)*range(r) . |
|
1267 |
{\isasymexists}x y z. xz=<x,z> & <x,y> \isasymin s & <y,z> \isasymin r{\ttrbrace} |
|
1268 |
\tdx{id_def}: id(A) == (lam x \isasymin A. x) |
|
14158 | 1269 |
\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} |
1270 |
\tdx{surj_def}: surj(A,B) == {\ttlbrace} f\isasymin{}A->B . {\isasymforall}y\isasymin{}B. {\isasymexists}x\isasymin{}A. f`x=y {\ttrbrace} |
|
1271 |
\tdx{bij_def}: bij(A,B) == inj(A,B) \isasyminter surj(A,B) |
|
1272 |
||
1273 |
||
1274 |
\tdx{left_inverse}: [| f\isasymin{}inj(A,B); a\isasymin{}A |] ==> converse(f)`(f`a) = a |
|
1275 |
\tdx{right_inverse}: [| f\isasymin{}inj(A,B); b\isasymin{}range(f) |] ==> |
|
9584
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|
1276 |
f`(converse(f)`b) = b |
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|
1277 |
|
14158 | 1278 |
\tdx{inj_converse_inj}: f\isasymin{}inj(A,B) ==> converse(f) \isasymin inj(range(f),A) |
1279 |
\tdx{bij_converse_bij}: f\isasymin{}bij(A,B) ==> converse(f) \isasymin bij(B,A) |
|
1280 |
||
1281 |
\tdx{comp_type}: [| s \isasymsubseteq A*B; r \isasymsubseteq B*C |] ==> (r O s) \isasymsubseteq A*C |
|
1282 |
\tdx{comp_assoc}: (r O s) O t = r O (s O t) |
|
1283 |
||
1284 |
\tdx{left_comp_id}: r \isasymsubseteq A*B ==> id(B) O r = r |
|
1285 |
\tdx{right_comp_id}: r \isasymsubseteq A*B ==> r O id(A) = r |
|
1286 |
||
1287 |
\tdx{comp_func}: [| g\isasymin{}A->B; f\isasymin{}B->C |] ==> (f O g) \isasymin A->C |
|
1288 |
\tdx{comp_func_apply}: [| g\isasymin{}A->B; f\isasymin{}B->C; a\isasymin{}A |] ==> (f O g)`a = f`(g`a) |
|
1289 |
||
1290 |
\tdx{comp_inj}: [| g\isasymin{}inj(A,B); f\isasymin{}inj(B,C) |] ==> (f O g)\isasymin{}inj(A,C) |
|
1291 |
\tdx{comp_surj}: [| g\isasymin{}surj(A,B); f\isasymin{}surj(B,C) |] ==> (f O g)\isasymin{}surj(A,C) |
|
1292 |
\tdx{comp_bij}: [| g\isasymin{}bij(A,B); f\isasymin{}bij(B,C) |] ==> (f O g)\isasymin{}bij(A,C) |
|
1293 |
||
1294 |
\tdx{left_comp_inverse}: f\isasymin{}inj(A,B) ==> converse(f) O f = id(A) |
|
1295 |
\tdx{right_comp_inverse}: f\isasymin{}surj(A,B) ==> f O converse(f) = id(B) |
|
14154 | 1296 |
|
1297 |
\tdx{bij_disjoint_Un}: |
|
14158 | 1298 |
[| f\isasymin{}bij(A,B); g\isasymin{}bij(C,D); A \isasyminter C = 0; B \isasyminter D = 0 |] ==> |
1299 |
(f \isasymunion g)\isasymin{}bij(A \isasymunion C, B \isasymunion D) |
|
1300 |
||
1301 |
\tdx{restrict_bij}: [| f\isasymin{}inj(A,B); C\isasymsubseteq{}A |] ==> restrict(f,C)\isasymin{}bij(C, f``C) |
|
14154 | 1302 |
\end{alltt*} |
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|
1303 |
\caption{Permutations} \label{zf-perm} |
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|
1304 |
\end{figure} |
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|
1305 |
|
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|
1306 |
The theory \thydx{Perm} is concerned with permutations (bijections) and |
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|
1307 |
related concepts. These include composition of relations, the identity |
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|
1308 |
relation, and three specialized function spaces: injective, surjective and |
af21f4364c05
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|
1309 |
bijective. Figure~\ref{zf-perm} displays many of their properties that |
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|
1310 |
have been proved. These results are fundamental to a treatment of |
af21f4364c05
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|
1311 |
equipollence and cardinality. |
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|
1312 |
|
14154 | 1313 |
Theory \thydx{Univ} defines a `universe' $\isa{univ}(A)$, which is used by |
9584
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|
1314 |
the datatype package. This set contains $A$ and the |
14154 | 1315 |
natural numbers. Vitally, it is closed under finite products: |
1316 |
$\isa{univ}(A)\times\isa{univ}(A)\subseteq\isa{univ}(A)$. This theory also |
|
9584
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|
1317 |
defines the cumulative hierarchy of axiomatic set theory, which |
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|
1318 |
traditionally is written $V@\alpha$ for an ordinal~$\alpha$. The |
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|
1319 |
`universe' is a simple generalization of~$V@\omega$. |
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|
1320 |
|
14154 | 1321 |
Theory \thydx{QUniv} defines a `universe' $\isa{quniv}(A)$, which is used by |
9584
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|
1322 |
the datatype package to construct codatatypes such as streams. It is |
14154 | 1323 |
analogous to $\isa{univ}(A)$ (and is defined in terms of it) but is closed |
9584
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|
1324 |
under the non-standard product and sum. |
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|
1325 |
|
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|
1326 |
|
6173 | 1327 |
\section{Automatic Tools} |
1328 |
||
9695 | 1329 |
ZF provides the simplifier and the classical reasoner. Moreover it supplies a |
1330 |
specialized tool to infer `types' of terms. |
|
6173 | 1331 |
|
14154 | 1332 |
\subsection{Simplification and Classical Reasoning} |
6121
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|
1333 |
|
9695 | 1334 |
ZF inherits simplification from FOL but adopts it for set theory. The |
1335 |
extraction of rewrite rules takes the ZF primitives into account. It can |
|
6121
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|
1336 |
strip bounded universal quantifiers from a formula; for example, ${\forall |
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|
1337 |
x\in A. f(x)=g(x)}$ yields the conditional rewrite rule $x\in A \Imp |
5fe77b9b5185
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diff
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|
1338 |
f(x)=g(x)$. Given $a\in\{x\in A. P(x)\}$ it extracts rewrite rules from $a\in |
5fe77b9b5185
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|
1339 |
A$ and~$P(a)$. It can also break down $a\in A\int B$ and $a\in A-B$. |
5fe77b9b5185
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|
1340 |
|
14154 | 1341 |
The default simpset used by \isa{simp} contains congruence rules for all of ZF's |
1342 |
binding operators. It contains all the conversion rules, such as |
|
1343 |
\isa{fst} and |
|
1344 |
\isa{snd}, as well as the rewrites shown in Fig.\ts\ref{zf-simpdata}. |
|
1345 |
||
1346 |
Classical reasoner methods such as \isa{blast} and \isa{auto} refer to |
|
1347 |
a rich collection of built-in axioms for all the set-theoretic |
|
1348 |
primitives. |
|
6121
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|
1349 |
|
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diff
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|
1350 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1351 |
\begin{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1352 |
\begin{eqnarray*} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1353 |
a\in \emptyset & \bimp & \bot\\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1354 |
a \in A \un B & \bimp & a\in A \disj a\in B\\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1355 |
a \in A \int B & \bimp & a\in A \conj a\in B\\ |
14154 | 1356 |
a \in A-B & \bimp & a\in A \conj \lnot (a\in B)\\ |
1357 |
\pair{a,b}\in \isa{Sigma}(A,B) |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1358 |
& \bimp & a\in A \conj b\in B(a)\\ |
14154 | 1359 |
a \in \isa{Collect}(A,P) & \bimp & a\in A \conj P(a)\\ |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1360 |
(\forall x \in \emptyset. P(x)) & \bimp & \top\\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1361 |
(\forall x \in A. \top) & \bimp & \top |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1362 |
\end{eqnarray*} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1363 |
\caption{Some rewrite rules for set theory} \label{zf-simpdata} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1364 |
\end{figure} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1365 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1366 |
|
6173 | 1367 |
\subsection{Type-Checking Tactics} |
1368 |
\index{type-checking tactics} |
|
1369 |
||
9695 | 1370 |
Isabelle/ZF provides simple tactics to help automate those proofs that are |
6173 | 1371 |
essentially type-checking. Such proofs are built by applying rules such as |
1372 |
these: |
|
14154 | 1373 |
\begin{ttbox}\isastyleminor |
14158 | 1374 |
[| ?P ==> ?a \isasymin ?A; ~?P ==> ?b \isasymin ?A |] |
1375 |
==> (if ?P then ?a else ?b) \isasymin ?A |
|
14154 | 1376 |
|
1377 |
[| ?m \isasymin nat; ?n \isasymin nat |] ==> ?m #+ ?n \isasymin nat |
|
1378 |
||
1379 |
?a \isasymin ?A ==> Inl(?a) \isasymin ?A + ?B |
|
6173 | 1380 |
\end{ttbox} |
1381 |
In typical applications, the goal has the form $t\in\Var{A}$: in other words, |
|
1382 |
we have a specific term~$t$ and need to infer its `type' by instantiating the |
|
1383 |
set variable~$\Var{A}$. Neither the simplifier nor the classical reasoner |
|
1384 |
does this job well. The if-then-else rule, and many similar ones, can make |
|
1385 |
the classical reasoner loop. The simplifier refuses (on principle) to |
|
14154 | 1386 |
instantiate variables during rewriting, so goals such as \isa{i\#+j \isasymin \ ?A} |
6173 | 1387 |
are left unsolved. |
1388 |
||
1389 |
The simplifier calls the type-checker to solve rewritten subgoals: this stage |
|
1390 |
can indeed instantiate variables. If you have defined new constants and |
|
14154 | 1391 |
proved type-checking rules for them, then declare the rules using |
1392 |
the attribute \isa{TC} and the rest should be automatic. In |
|
1393 |
particular, the simplifier will use type-checking to help satisfy |
|
1394 |
conditional rewrite rules. Call the method \ttindex{typecheck} to |
|
1395 |
break down all subgoals using type-checking rules. You can add new |
|
1396 |
type-checking rules temporarily like this: |
|
1397 |
\begin{isabelle} |
|
14158 | 1398 |
\isacommand{apply}\ (typecheck add:\ inj_is_fun) |
14154 | 1399 |
\end{isabelle} |
1400 |
||
1401 |
||
1402 |
%Though the easiest way to invoke the type-checker is via the simplifier, |
|
1403 |
%specialized applications may require more detailed knowledge of |
|
1404 |
%the type-checking primitives. They are modelled on the simplifier's: |
|
1405 |
%\begin{ttdescription} |
|
1406 |
%\item[\ttindexbold{tcset}] is the type of tcsets: sets of type-checking rules. |
|
1407 |
% |
|
1408 |
%\item[\ttindexbold{addTCs}] is an infix operator to add type-checking rules to |
|
1409 |
% a tcset. |
|
1410 |
% |
|
1411 |
%\item[\ttindexbold{delTCs}] is an infix operator to remove type-checking rules |
|
1412 |
% from a tcset. |
|
1413 |
% |
|
1414 |
%\item[\ttindexbold{typecheck_tac}] is a tactic for attempting to prove all |
|
1415 |
% subgoals using the rules given in its argument, a tcset. |
|
1416 |
%\end{ttdescription} |
|
1417 |
% |
|
1418 |
%Tcsets, like simpsets, are associated with theories and are merged when |
|
1419 |
%theories are merged. There are further primitives that use the default tcset. |
|
1420 |
%\begin{ttdescription} |
|
1421 |
%\item[\ttindexbold{tcset}] is a function to return the default tcset; use the |
|
1422 |
% expression \isa{tcset()}. |
|
1423 |
% |
|
1424 |
%\item[\ttindexbold{AddTCs}] adds type-checking rules to the default tcset. |
|
1425 |
% |
|
1426 |
%\item[\ttindexbold{DelTCs}] removes type-checking rules from the default |
|
1427 |
% tcset. |
|
1428 |
% |
|
1429 |
%\item[\ttindexbold{Typecheck_tac}] calls \isa{typecheck_tac} using the |
|
1430 |
% default tcset. |
|
1431 |
%\end{ttdescription} |
|
1432 |
% |
|
1433 |
%To supply some type-checking rules temporarily, using \isa{Addrules} and |
|
1434 |
%later \isa{Delrules} is the simplest way. There is also a high-tech |
|
1435 |
%approach. Call the simplifier with a new solver expressed using |
|
1436 |
%\ttindexbold{type_solver_tac} and your temporary type-checking rules. |
|
1437 |
%\begin{ttbox}\isastyleminor |
|
1438 |
%by (asm_simp_tac |
|
1439 |
% (simpset() setSolver type_solver_tac (tcset() addTCs prems)) 2); |
|
1440 |
%\end{ttbox} |
|
6173 | 1441 |
|
1442 |
||
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|
1443 |
\section{Natural number and integer arithmetic} |
af21f4364c05
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|
1444 |
|
af21f4364c05
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|
1445 |
\index{arithmetic|(} |
af21f4364c05
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|
1446 |
|
af21f4364c05
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|
1447 |
\begin{figure}\small |
af21f4364c05
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diff
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|
1448 |
\index{#*@{\tt\#*} symbol} |
af21f4364c05
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parents:
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diff
changeset
|
1449 |
\index{*div symbol} |
af21f4364c05
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parents:
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diff
changeset
|
1450 |
\index{*mod symbol} |
af21f4364c05
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parents:
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diff
changeset
|
1451 |
\index{#+@{\tt\#+} symbol} |
af21f4364c05
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paulson
parents:
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diff
changeset
|
1452 |
\index{#-@{\tt\#-} symbol} |
af21f4364c05
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paulson
parents:
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diff
changeset
|
1453 |
\begin{constants} |
af21f4364c05
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parents:
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diff
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|
1454 |
\it symbol & \it meta-type & \it priority & \it description \\ |
af21f4364c05
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|
1455 |
\cdx{nat} & $i$ & & set of natural numbers \\ |
af21f4364c05
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|
1456 |
\cdx{nat_case}& $[i,i\To i,i]\To i$ & & conditional for $nat$\\ |
af21f4364c05
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|
1457 |
\tt \#* & $[i,i]\To i$ & Left 70 & multiplication \\ |
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changeset
|
1458 |
\tt div & $[i,i]\To i$ & Left 70 & division\\ |
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|
1459 |
\tt mod & $[i,i]\To i$ & Left 70 & modulus\\ |
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changeset
|
1460 |
\tt \#+ & $[i,i]\To i$ & Left 65 & addition\\ |
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|
1461 |
\tt \#- & $[i,i]\To i$ & Left 65 & subtraction |
af21f4364c05
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diff
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|
1462 |
\end{constants} |
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|
1463 |
|
14158 | 1464 |
\begin{alltt*}\isastyleminor |
14154 | 1465 |
\tdx{nat_def}: nat == lfp(lam r \isasymin Pow(Inf). {\ttlbrace}0{\ttrbrace} \isasymunion {\ttlbrace}succ(x). x \isasymin r{\ttrbrace} |
1466 |
||
14158 | 1467 |
\tdx{nat_case_def}: nat_case(a,b,k) == |
14154 | 1468 |
THE y. k=0 & y=a | ({\isasymexists}x. k=succ(x) & y=b(x)) |
1469 |
||
14158 | 1470 |
\tdx{nat_0I}: 0 \isasymin nat |
1471 |
\tdx{nat_succI}: n \isasymin nat ==> succ(n) \isasymin nat |
|
1472 |
||
1473 |
\tdx{nat_induct}: |
|
14154 | 1474 |
[| n \isasymin nat; P(0); !!x. [| x \isasymin nat; P(x) |] ==> P(succ(x)) |
9584
af21f4364c05
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diff
changeset
|
1475 |
|] ==> P(n) |
af21f4364c05
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diff
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|
1476 |
|
14158 | 1477 |
\tdx{nat_case_0}: nat_case(a,b,0) = a |
1478 |
\tdx{nat_case_succ}: nat_case(a,b,succ(m)) = b(m) |
|
1479 |
||
1480 |
\tdx{add_0_natify}: 0 #+ n = natify(n) |
|
1481 |
\tdx{add_succ}: succ(m) #+ n = succ(m #+ n) |
|
1482 |
||
1483 |
\tdx{mult_type}: m #* n \isasymin nat |
|
1484 |
\tdx{mult_0}: 0 #* n = 0 |
|
1485 |
\tdx{mult_succ}: succ(m) #* n = n #+ (m #* n) |
|
1486 |
\tdx{mult_commute}: m #* n = n #* m |
|
1487 |
\tdx{add_mult_dist}: (m #+ n) #* k = (m #* k) #+ (n #* k) |
|
1488 |
\tdx{mult_assoc}: (m #* n) #* k = m #* (n #* k) |
|
1489 |
\tdx{mod_div_equality}: m \isasymin nat ==> (m div n)#*n #+ m mod n = m |
|
1490 |
\end{alltt*} |
|
9584
af21f4364c05
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|
1491 |
\caption{The natural numbers} \label{zf-nat} |
af21f4364c05
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diff
changeset
|
1492 |
\end{figure} |
af21f4364c05
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diff
changeset
|
1493 |
|
af21f4364c05
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paulson
parents:
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diff
changeset
|
1494 |
\index{natural numbers} |
af21f4364c05
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paulson
parents:
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diff
changeset
|
1495 |
|
af21f4364c05
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paulson
parents:
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diff
changeset
|
1496 |
Theory \thydx{Nat} defines the natural numbers and mathematical |
af21f4364c05
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paulson
parents:
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diff
changeset
|
1497 |
induction, along with a case analysis operator. The set of natural |
14154 | 1498 |
numbers, here called \isa{nat}, is known in set theory as the ordinal~$\omega$. |
9584
af21f4364c05
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diff
changeset
|
1499 |
|
af21f4364c05
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paulson
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diff
changeset
|
1500 |
Theory \thydx{Arith} develops arithmetic on the natural numbers |
af21f4364c05
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parents:
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diff
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|
1501 |
(Fig.\ts\ref{zf-nat}). Addition, multiplication and subtraction are defined |
af21f4364c05
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paulson
parents:
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diff
changeset
|
1502 |
by primitive recursion. Division and remainder are defined by repeated |
af21f4364c05
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parents:
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diff
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|
1503 |
subtraction, which requires well-founded recursion; the termination argument |
af21f4364c05
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parents:
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diff
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|
1504 |
relies on the divisor's being non-zero. Many properties are proved: |
af21f4364c05
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paulson
parents:
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diff
changeset
|
1505 |
commutative, associative and distributive laws, identity and cancellation |
af21f4364c05
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paulson
parents:
8249
diff
changeset
|
1506 |
laws, etc. The most interesting result is perhaps the theorem $a \bmod b + |
af21f4364c05
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paulson
parents:
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diff
changeset
|
1507 |
(a/b)\times b = a$. |
af21f4364c05
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paulson
parents:
8249
diff
changeset
|
1508 |
|
14154 | 1509 |
To minimize the need for tedious proofs of $t\in\isa{nat}$, the arithmetic |
9584
af21f4364c05
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paulson
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8249
diff
changeset
|
1510 |
operators coerce their arguments to be natural numbers. The function |
14154 | 1511 |
\cdx{natify} is defined such that $\isa{natify}(n) = n$ if $n$ is a natural |
1512 |
number, $\isa{natify}(\isa{succ}(x)) = |
|
1513 |
\isa{succ}(\isa{natify}(x))$ for all $x$, and finally |
|
1514 |
$\isa{natify}(x)=0$ in all other cases. The benefit is that the addition, |
|
9584
af21f4364c05
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paulson
parents:
8249
diff
changeset
|
1515 |
subtraction, multiplication, division and remainder operators always return |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1516 |
natural numbers, regardless of their arguments. Algebraic laws (commutative, |
14154 | 1517 |
associative, distributive) are unconditional. Occurrences of \isa{natify} |
9584
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
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8249
diff
changeset
|
1518 |
as operands of those operators are simplified away. Any remaining occurrences |
af21f4364c05
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paulson
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diff
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|
1519 |
can either be tolerated or else eliminated by proving that the argument is a |
af21f4364c05
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diff
changeset
|
1520 |
natural number. |
af21f4364c05
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paulson
parents:
8249
diff
changeset
|
1521 |
|
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1522 |
The simplifier automatically cancels common terms on the opposite sides of |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1523 |
subtraction and of relations ($=$, $<$ and $\le$). Here is an example: |
14154 | 1524 |
\begin{isabelle} |
1525 |
1. i \#+ j \#+ k \#- j < k \#+ l\isanewline |
|
1526 |
\isacommand{apply}\ simp\isanewline |
|
9584
af21f4364c05
documented the integers and updated section on nat arithmetic
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8249
diff
changeset
|
1527 |
1. natify(i) < natify(l) |
14154 | 1528 |
\end{isabelle} |
1529 |
Given the assumptions \isa{i \isasymin nat} and \isa{l \isasymin nat}, both occurrences of |
|
9584
af21f4364c05
documented the integers and updated section on nat arithmetic
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diff
changeset
|
1530 |
\cdx{natify} would be simplified away. |
af21f4364c05
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diff
changeset
|
1531 |
|
af21f4364c05
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diff
changeset
|
1532 |
|
af21f4364c05
documented the integers and updated section on nat arithmetic
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diff
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|
1533 |
\begin{figure}\small |
af21f4364c05
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diff
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|
1534 |
\index{$*@{\tt\$*} symbol} |
af21f4364c05
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diff
changeset
|
1535 |
\index{$+@{\tt\$+} symbol} |
af21f4364c05
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paulson
parents:
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diff
changeset
|
1536 |
\index{$-@{\tt\$-} symbol} |
af21f4364c05
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paulson
parents:
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diff
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|
1537 |
\begin{constants} |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
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diff
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|
1538 |
\it symbol & \it meta-type & \it priority & \it description \\ |
af21f4364c05
documented the integers and updated section on nat arithmetic
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diff
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|
1539 |
\cdx{int} & $i$ & & set of integers \\ |
af21f4364c05
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diff
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|
1540 |
\tt \$* & $[i,i]\To i$ & Left 70 & multiplication \\ |
af21f4364c05
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|
1541 |
\tt \$+ & $[i,i]\To i$ & Left 65 & addition\\ |
af21f4364c05
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|
1542 |
\tt \$- & $[i,i]\To i$ & Left 65 & subtraction\\ |
af21f4364c05
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|
1543 |
\tt \$< & $[i,i]\To o$ & Left 50 & $<$ on integers\\ |
af21f4364c05
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diff
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|
1544 |
\tt \$<= & $[i,i]\To o$ & Left 50 & $\le$ on integers |
af21f4364c05
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|
1545 |
\end{constants} |
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|
1546 |
|
14158 | 1547 |
\begin{alltt*}\isastyleminor |
14154 | 1548 |
\tdx{zadd_0_intify}: 0 $+ n = intify(n) |
1549 |
||
1550 |
\tdx{zmult_type}: m $* n \isasymin int |
|
1551 |
\tdx{zmult_0}: 0 $* n = 0 |
|
1552 |
\tdx{zmult_commute}: m $* n = n $* m |
|
1553 |
\tdx{zadd_zmult_dist}: (m $+ n) $* k = (m $* k) $+ (n $* k) |
|
1554 |
\tdx{zmult_assoc}: (m $* n) $* k = m $* (n $* k) |
|
14158 | 1555 |
\end{alltt*} |
9584
af21f4364c05
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diff
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|
1556 |
\caption{The integers} \label{zf-int} |
af21f4364c05
documented the integers and updated section on nat arithmetic
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parents:
8249
diff
changeset
|
1557 |
\end{figure} |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1558 |
|
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1559 |
|
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1560 |
\index{integers} |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1561 |
|
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1562 |
Theory \thydx{Int} defines the integers, as equivalence classes of natural |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1563 |
numbers. Figure~\ref{zf-int} presents a tidy collection of laws. In |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1564 |
fact, a large library of facts is proved, including monotonicity laws for |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1565 |
addition and multiplication, covering both positive and negative operands. |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1566 |
|
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1567 |
As with the natural numbers, the need for typing proofs is minimized. All the |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1568 |
operators defined in Fig.\ts\ref{zf-int} coerce their operands to integers by |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1569 |
applying the function \cdx{intify}. This function is the identity on integers |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1570 |
and maps other operands to zero. |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1571 |
|
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1572 |
Decimal notation is provided for the integers. Numbers, written as |
14154 | 1573 |
\isa{\#$nnn$} or \isa{\#-$nnn$}, are represented internally in |
9584
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1574 |
two's-complement binary. Expressions involving addition, subtraction and |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1575 |
multiplication of numeral constants are evaluated (with acceptable efficiency) |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1576 |
by simplification. The simplifier also collects similar terms, multiplying |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1577 |
them by a numerical coefficient. It also cancels occurrences of the same |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1578 |
terms on the other side of the relational operators. Example: |
14154 | 1579 |
\begin{isabelle} |
1580 |
1. y \$+ z \$+ \#-3 \$* x \$+ y \$<= x \$* \#2 \$+ |
|
1581 |
z\isanewline |
|
1582 |
\isacommand{apply}\ simp\isanewline |
|
1583 |
1. \#2 \$* y \$<= \#5 \$* x |
|
1584 |
\end{isabelle} |
|
9584
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
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diff
changeset
|
1585 |
For more information on the integers, please see the theories on directory |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1586 |
\texttt{ZF/Integ}. |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1587 |
|
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1588 |
\index{arithmetic|)} |
af21f4364c05
documented the integers and updated section on nat arithmetic
paulson
parents:
8249
diff
changeset
|
1589 |
|
6173 | 1590 |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1591 |
\section{Datatype definitions} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1592 |
\label{sec:ZF:datatype} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1593 |
\index{*datatype|(} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1594 |
|
9695 | 1595 |
The \ttindex{datatype} definition package of ZF constructs inductive datatypes |
14154 | 1596 |
similar to \ML's. It can also construct coinductive datatypes |
9695 | 1597 |
(codatatypes), which are non-well-founded structures such as streams. It |
1598 |
defines the set using a fixed-point construction and proves induction rules, |
|
1599 |
as well as theorems for recursion and case combinators. It supplies |
|
1600 |
mechanisms for reasoning about freeness. The datatype package can handle both |
|
1601 |
mutual and indirect recursion. |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1602 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1603 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1604 |
\subsection{Basics} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1605 |
\label{subsec:datatype:basics} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1606 |
|
14154 | 1607 |
A \isa{datatype} definition has the following form: |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1608 |
\[ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1609 |
\begin{array}{llcl} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1610 |
\mathtt{datatype} & t@1(A@1,\ldots,A@h) & = & |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1611 |
constructor^1@1 ~\mid~ \ldots ~\mid~ constructor^1@{k@1} \\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1612 |
& & \vdots \\ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1613 |
\mathtt{and} & t@n(A@1,\ldots,A@h) & = & |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1614 |
constructor^n@1~ ~\mid~ \ldots ~\mid~ constructor^n@{k@n} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1615 |
\end{array} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1616 |
\] |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1617 |
Here $t@1$, \ldots,~$t@n$ are identifiers and $A@1$, \ldots,~$A@h$ are |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1618 |
variables: the datatype's parameters. Each constructor specification has the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1619 |
form \dquotesoff |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1620 |
\[ C \hbox{\tt~( } \hbox{\tt"} x@1 \hbox{\tt:} T@1 \hbox{\tt"},\; |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1621 |
\ldots,\; |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1622 |
\hbox{\tt"} x@m \hbox{\tt:} T@m \hbox{\tt"} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1623 |
\hbox{\tt~)} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1624 |
\] |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1625 |
Here $C$ is the constructor name, and variables $x@1$, \ldots,~$x@m$ are the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1626 |
constructor arguments, belonging to the sets $T@1$, \ldots, $T@m$, |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1627 |
respectively. Typically each $T@j$ is either a constant set, a datatype |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1628 |
parameter (one of $A@1$, \ldots, $A@h$) or a recursive occurrence of one of |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1629 |
the datatypes, say $t@i(A@1,\ldots,A@h)$. More complex possibilities exist, |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1630 |
but they are much harder to realize. Often, additional information must be |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1631 |
supplied in the form of theorems. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1632 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1633 |
A datatype can occur recursively as the argument of some function~$F$. This |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1634 |
is called a {\em nested} (or \emph{indirect}) occurrence. It is only allowed |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1635 |
if the datatype package is given a theorem asserting that $F$ is monotonic. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1636 |
If the datatype has indirect occurrences, then Isabelle/ZF does not support |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1637 |
recursive function definitions. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1638 |
|
14154 | 1639 |
A simple example of a datatype is \isa{list}, which is built-in, and is |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1640 |
defined by |
14158 | 1641 |
\begin{alltt*}\isastyleminor |
14154 | 1642 |
consts list :: "i=>i" |
1643 |
datatype "list(A)" = Nil | Cons ("a \isasymin A", "l \isasymin list(A)") |
|
14158 | 1644 |
\end{alltt*} |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1645 |
Note that the datatype operator must be declared as a constant first. |
14154 | 1646 |
However, the package declares the constructors. Here, \isa{Nil} gets type |
1647 |
$i$ and \isa{Cons} gets type $[i,i]\To i$. |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1648 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1649 |
Trees and forests can be modelled by the mutually recursive datatype |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1650 |
definition |
14158 | 1651 |
\begin{alltt*}\isastyleminor |
14154 | 1652 |
consts |
1653 |
tree :: "i=>i" |
|
1654 |
forest :: "i=>i" |
|
1655 |
tree_forest :: "i=>i" |
|
1656 |
datatype "tree(A)" = Tcons ("a{\isasymin}A", "f{\isasymin}forest(A)") |
|
1657 |
and "forest(A)" = Fnil | Fcons ("t{\isasymin}tree(A)", "f{\isasymin}forest(A)") |
|
14158 | 1658 |
\end{alltt*} |
14154 | 1659 |
Here $\isa{tree}(A)$ is the set of trees over $A$, $\isa{forest}(A)$ is |
1660 |
the set of forests over $A$, and $\isa{tree_forest}(A)$ is the union of |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1661 |
the previous two sets. All three operators must be declared first. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1662 |
|
14154 | 1663 |
The datatype \isa{term}, which is defined by |
14158 | 1664 |
\begin{alltt*}\isastyleminor |
14154 | 1665 |
consts term :: "i=>i" |
1666 |
datatype "term(A)" = Apply ("a \isasymin A", "l \isasymin list(term(A))") |
|
1667 |
monos list_mono |
|
28871
111bbd2b12db
Added a line that was missing from the definition
paulson
parents:
14202
diff
changeset
|
1668 |
type_elims list_univ [THEN subsetD, elim_format] |
14158 | 1669 |
\end{alltt*} |
14154 | 1670 |
is an example of nested recursion. (The theorem \isa{list_mono} is proved |
1671 |
in theory \isa{List}, and the \isa{term} example is developed in |
|
1672 |
theory |
|
1673 |
\thydx{Induct/Term}.) |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1674 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1675 |
\subsubsection{Freeness of the constructors} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1676 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1677 |
Constructors satisfy {\em freeness} properties. Constructions are distinct, |
14154 | 1678 |
for example $\isa{Nil}\not=\isa{Cons}(a,l)$, and they are injective, for |
1679 |
example $\isa{Cons}(a,l)=\isa{Cons}(a',l') \bimp a=a' \conj l=l'$. |
|
6121
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the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1680 |
Because the number of freeness is quadratic in the number of constructors, the |
6143 | 1681 |
datatype package does not prove them. Instead, it ensures that simplification |
1682 |
will prove them dynamically: when the simplifier encounters a formula |
|
1683 |
asserting the equality of two datatype constructors, it performs freeness |
|
1684 |
reasoning. |
|
1685 |
||
1686 |
Freeness reasoning can also be done using the classical reasoner, but it is |
|
1687 |
more complicated. You have to add some safe elimination rules rules to the |
|
14154 | 1688 |
claset. For the \isa{list} datatype, they are called |
1689 |
\isa{list.free_elims}. Occasionally this exposes the underlying |
|
6143 | 1690 |
representation of some constructor, which can be rectified using the command |
14154 | 1691 |
\isa{unfold list.con_defs [symmetric]}. |
6143 | 1692 |
|
6121
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the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1693 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1694 |
\subsubsection{Structural induction} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1695 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1696 |
The datatype package also provides structural induction rules. For datatypes |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1697 |
without mutual or nested recursion, the rule has the form exemplified by |
14154 | 1698 |
\isa{list.induct} in Fig.\ts\ref{zf-list}. For mutually recursive |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1699 |
datatypes, the induction rule is supplied in two forms. Consider datatype |
14154 | 1700 |
\isa{TF}. The rule \isa{tree_forest.induct} performs induction over a |
1701 |
single predicate~\isa{P}, which is presumed to be defined for both trees |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1702 |
and forests: |
14158 | 1703 |
\begin{alltt*}\isastyleminor |
14154 | 1704 |
[| x \isasymin tree_forest(A); |
1705 |
!!a f. [| a \isasymin A; f \isasymin forest(A); P(f) |] ==> P(Tcons(a, f)); |
|
8249 | 1706 |
P(Fnil); |
14154 | 1707 |
!!f t. [| t \isasymin tree(A); P(t); f \isasymin forest(A); P(f) |] |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1708 |
==> P(Fcons(t, f)) |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1709 |
|] ==> P(x) |
14158 | 1710 |
\end{alltt*} |
14154 | 1711 |
The rule \isa{tree_forest.mutual_induct} performs induction over two |
1712 |
distinct predicates, \isa{P_tree} and \isa{P_forest}. |
|
14158 | 1713 |
\begin{alltt*}\isastyleminor |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1714 |
[| !!a f. |
14154 | 1715 |
[| a{\isasymin}A; f{\isasymin}forest(A); P_forest(f) |] ==> P_tree(Tcons(a,f)); |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1716 |
P_forest(Fnil); |
14154 | 1717 |
!!f t. [| t{\isasymin}tree(A); P_tree(t); f{\isasymin}forest(A); P_forest(f) |] |
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|
1718 |
==> P_forest(Fcons(t, f)) |
14154 | 1719 |
|] ==> ({\isasymforall}za. za \isasymin tree(A) --> P_tree(za)) & |
1720 |
({\isasymforall}za. za \isasymin forest(A) --> P_forest(za)) |
|
14158 | 1721 |
\end{alltt*} |
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|
1722 |
|
14154 | 1723 |
For datatypes with nested recursion, such as the \isa{term} example from |
1724 |
above, things are a bit more complicated. The rule \isa{term.induct} |
|
1725 |
refers to the monotonic operator, \isa{list}: |
|
14158 | 1726 |
\begin{alltt*}\isastyleminor |
14154 | 1727 |
[| x \isasymin term(A); |
14158 | 1728 |
!!a l. [| a\isasymin{}A; l\isasymin{}list(Collect(term(A), P)) |] ==> P(Apply(a,l)) |
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parents:
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|
1729 |
|] ==> P(x) |
14158 | 1730 |
\end{alltt*} |
14154 | 1731 |
The theory \isa{Induct/Term.thy} derives two higher-level induction rules, |
1732 |
one of which is particularly useful for proving equations: |
|
14158 | 1733 |
\begin{alltt*}\isastyleminor |
14154 | 1734 |
[| t \isasymin term(A); |
1735 |
!!x zs. [| x \isasymin A; zs \isasymin list(term(A)); map(f, zs) = map(g, zs) |] |
|
6121
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|
1736 |
==> f(Apply(x, zs)) = g(Apply(x, zs)) |
5fe77b9b5185
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parents:
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|
1737 |
|] ==> f(t) = g(t) |
14158 | 1738 |
\end{alltt*} |
6121
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parents:
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|
1739 |
How this can be generalized to other nested datatypes is a matter for future |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1740 |
research. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1741 |
|
5fe77b9b5185
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paulson
parents:
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changeset
|
1742 |
|
14154 | 1743 |
\subsubsection{The \isa{case} operator} |
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parents:
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|
1744 |
|
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1745 |
The package defines an operator for performing case analysis over the |
14154 | 1746 |
datatype. For \isa{list}, it is called \isa{list_case} and satisfies |
6121
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parents:
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|
1747 |
the equations |
14154 | 1748 |
\begin{ttbox}\isastyleminor |
6121
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parents:
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|
1749 |
list_case(f_Nil, f_Cons, []) = f_Nil |
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parents:
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changeset
|
1750 |
list_case(f_Nil, f_Cons, Cons(a, l)) = f_Cons(a, l) |
5fe77b9b5185
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parents:
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|
1751 |
\end{ttbox} |
14154 | 1752 |
Here \isa{f_Nil} is the value to return if the argument is \isa{Nil} and |
1753 |
\isa{f_Cons} is a function that computes the value to return if the |
|
1754 |
argument has the form $\isa{Cons}(a,l)$. The function can be expressed as |
|
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|
1755 |
an abstraction, over patterns if desired (\S\ref{sec:pairs}). |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1756 |
|
14154 | 1757 |
For mutually recursive datatypes, there is a single \isa{case} operator. |
1758 |
In the tree/forest example, the constant \isa{tree_forest_case} handles all |
|
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parents:
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|
1759 |
of the constructors of the two datatypes. |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1760 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1761 |
|
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1762 |
\subsection{Defining datatypes} |
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parents:
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changeset
|
1763 |
|
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1764 |
The theory syntax for datatype definitions is shown in |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1765 |
Fig.~\ref{datatype-grammar}. In order to be well-formed, a datatype |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1766 |
definition has to obey the rules stated in the previous section. As a result |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1767 |
the theory is extended with the new types, the constructors, and the theorems |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1768 |
listed in the previous section. The quotation marks are necessary because |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1769 |
they enclose general Isabelle formul\ae. |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1770 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1771 |
\begin{figure} |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1772 |
\begin{rail} |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1773 |
datatype : ( 'datatype' | 'codatatype' ) datadecls; |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1774 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1775 |
datadecls: ( '"' id arglist '"' '=' (constructor + '|') ) + 'and' |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1776 |
; |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1777 |
constructor : name ( () | consargs ) ( () | ( '(' mixfix ')' ) ) |
5fe77b9b5185
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parents:
diff
changeset
|
1778 |
; |
14154 | 1779 |
consargs : '(' ('"' var ' : ' term '"' + ',') ')' |
6121
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paulson
parents:
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changeset
|
1780 |
; |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1781 |
\end{rail} |
5fe77b9b5185
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paulson
parents:
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changeset
|
1782 |
\caption{Syntax of datatype declarations} |
5fe77b9b5185
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parents:
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changeset
|
1783 |
\label{datatype-grammar} |
5fe77b9b5185
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parents:
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changeset
|
1784 |
\end{figure} |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1785 |
|
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1786 |
Codatatypes are declared like datatypes and are identical to them in every |
5fe77b9b5185
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parents:
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|
1787 |
respect except that they have a coinduction rule instead of an induction rule. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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changeset
|
1788 |
Note that while an induction rule has the effect of limiting the values |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1789 |
contained in the set, a coinduction rule gives a way of constructing new |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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changeset
|
1790 |
values of the set. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1791 |
|
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1792 |
Most of the theorems about datatypes become part of the default simpset. You |
5fe77b9b5185
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parents:
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|
1793 |
never need to see them again because the simplifier applies them |
14154 | 1794 |
automatically. |
1795 |
||
1796 |
\subsubsection{Specialized methods for datatypes} |
|
1797 |
||
1798 |
Induction and case-analysis can be invoked using these special-purpose |
|
1799 |
methods: |
|
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|
1800 |
\begin{ttdescription} |
14154 | 1801 |
\item[\methdx{induct_tac} $x$] applies structural |
1802 |
induction on variable $x$ to subgoal~1, provided the type of $x$ is a |
|
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parents:
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|
1803 |
datatype. The induction variable should not occur among other assumptions |
5fe77b9b5185
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paulson
parents:
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|
1804 |
of the subgoal. |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1805 |
\end{ttdescription} |
14154 | 1806 |
% |
1807 |
% we also have the ind_cases method, but what does it do? |
|
1808 |
In some situations, induction is overkill and a case distinction over all |
|
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|
1809 |
constructors of the datatype suffices. |
5fe77b9b5185
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parents:
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|
1810 |
\begin{ttdescription} |
14202 | 1811 |
\item[\methdx{case_tac} $x$] |
14154 | 1812 |
performs a case analysis for the variable~$x$. |
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|
1813 |
\end{ttdescription} |
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|
1814 |
|
5fe77b9b5185
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parents:
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|
1815 |
Both tactics can only be applied to a variable, whose typing must be given in |
14154 | 1816 |
some assumption, for example the assumption \isa{x \isasymin \ list(A)}. The tactics |
1817 |
also work for the natural numbers (\isa{nat}) and disjoint sums, although |
|
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|
1818 |
these sets were not defined using the datatype package. (Disjoint sums are |
14154 | 1819 |
not recursive, so only \isa{case_tac} is available.) |
1820 |
||
1821 |
Structured Isar methods are also available. Below, $t$ |
|
1822 |
stands for the name of the datatype. |
|
1823 |
\begin{ttdescription} |
|
1824 |
\item[\methdx{induct} \isa{set:}\ $t$] is the Isar induction tactic. |
|
1825 |
\item[\methdx{cases} \isa{set:}\ $t$] is the Isar case-analysis tactic. |
|
1826 |
\end{ttdescription} |
|
1827 |
||
1828 |
||
1829 |
\subsubsection{The theorems proved by a datatype declaration} |
|
1830 |
||
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|
1831 |
Here are some more details for the technically minded. Processing the |
14154 | 1832 |
datatype declaration of a set~$t$ produces a name space~$t$ containing |
1833 |
the following theorems: |
|
1834 |
\begin{ttbox}\isastyleminor |
|
1835 |
intros \textrm{the introduction rules} |
|
1836 |
cases \textrm{the case analysis rule} |
|
1837 |
induct \textrm{the standard induction rule} |
|
1838 |
mutual_induct \textrm{the mutual induction rule, if needed} |
|
1839 |
case_eqns \textrm{equations for the case operator} |
|
1840 |
recursor_eqns \textrm{equations for the recursor} |
|
1841 |
simps \textrm{the union of} case_eqns \textrm{and} recursor_eqns |
|
1842 |
con_defs \textrm{definitions of the case operator and constructors} |
|
1843 |
free_iffs \textrm{logical equivalences for proving freeness} |
|
1844 |
free_elims \textrm{elimination rules for proving freeness} |
|
1845 |
defs \textrm{datatype definition(s)} |
|
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|
1846 |
\end{ttbox} |
14154 | 1847 |
Furthermore there is the theorem $C$ for every constructor~$C$; for |
1848 |
example, the \isa{list} datatype's introduction rules are bound to the |
|
1849 |
identifiers \isa{Nil} and \isa{Cons}. |
|
1850 |
||
1851 |
For a codatatype, the component \isa{coinduct} is the coinduction rule, |
|
1852 |
replacing the \isa{induct} component. |
|
1853 |
||
1854 |
See the theories \isa{Induct/Ntree} and \isa{Induct/Brouwer} for examples of |
|
1855 |
infinitely branching datatypes. See theory \isa{Induct/LList} for an example |
|
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|
1856 |
of a codatatype. Some of these theories illustrate the use of additional, |
5fe77b9b5185
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parents:
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|
1857 |
undocumented features of the datatype package. Datatype definitions are |
5fe77b9b5185
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parents:
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|
1858 |
reduced to inductive definitions, and the advanced features should be |
5fe77b9b5185
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paulson
parents:
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|
1859 |
understood in that light. |
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parents:
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changeset
|
1860 |
|
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parents:
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changeset
|
1861 |
|
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parents:
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changeset
|
1862 |
\subsection{Examples} |
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|
1863 |
|
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parents:
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changeset
|
1864 |
\subsubsection{The datatype of binary trees} |
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parents:
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|
1865 |
|
14154 | 1866 |
Let us define the set $\isa{bt}(A)$ of binary trees over~$A$. The theory |
6121
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parents:
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|
1867 |
must contain these lines: |
14158 | 1868 |
\begin{alltt*}\isastyleminor |
14154 | 1869 |
consts bt :: "i=>i" |
1870 |
datatype "bt(A)" = Lf | Br ("a\isasymin{}A", "t1\isasymin{}bt(A)", "t2\isasymin{}bt(A)") |
|
14158 | 1871 |
\end{alltt*} |
14154 | 1872 |
After loading the theory, we can prove some theorem. |
1873 |
We begin by declaring the constructor's typechecking rules |
|
1874 |
as simplification rules: |
|
1875 |
\begin{isabelle} |
|
1876 |
\isacommand{declare}\ bt.intros\ [simp]% |
|
1877 |
\end{isabelle} |
|
1878 |
||
1879 |
Our first example is the theorem that no tree equals its |
|
1880 |
left branch. To make the inductive hypothesis strong enough, |
|
1881 |
the proof requires a quantified induction formula, but |
|
1882 |
the \isa{rule\_format} attribute will remove the quantifiers |
|
1883 |
before the theorem is stored. |
|
1884 |
\begin{isabelle} |
|
14158 | 1885 |
\isacommand{lemma}\ Br\_neq\_left\ [rule\_format]:\ "l\isasymin bt(A)\ ==>\ \isasymforall x\ r.\ Br(x,l,r)\isasymnoteq{}l"\isanewline |
14154 | 1886 |
\ 1.\ l\ \isasymin \ bt(A)\ \isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l% |
1887 |
\end{isabelle} |
|
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|
1888 |
This can be proved by the structural induction tactic: |
14154 | 1889 |
\begin{isabelle} |
1890 |
\ \ \isacommand{apply}\ (induct\_tac\ l)\isanewline |
|
1891 |
\ 1.\ \isasymforall x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline |
|
1892 |
\ 2.\ \isasymAnd a\ t1\ t2.\isanewline |
|
1893 |
\isaindent{\ 2.\ \ \ \ }\isasymlbrakk a\ \isasymin \ A;\ t1\ \isasymin \ bt(A);\ \isasymforall x\ r.\ Br(x,\ t1,\ r)\ \isasymnoteq \ t1;\ t2\ \isasymin \ bt(A);\isanewline |
|
1894 |
\isaindent{\ 2.\ \ \ \ \ \ \ }\isasymforall x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline |
|
1895 |
\isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2) |
|
1896 |
\end{isabelle} |
|
1897 |
Both subgoals are proved using \isa{auto}, which performs the necessary |
|
6143 | 1898 |
freeness reasoning. |
14154 | 1899 |
\begin{isabelle} |
1900 |
\ \ \isacommand{apply}\ auto\isanewline |
|
1901 |
No\ subgoals!\isanewline |
|
1902 |
\isacommand{done} |
|
1903 |
\end{isabelle} |
|
1904 |
||
1905 |
An alternative proof uses Isar's fancy \isa{induct} method, which |
|
1906 |
automatically quantifies over all free variables: |
|
1907 |
||
1908 |
\begin{isabelle} |
|
1909 |
\isacommand{lemma}\ Br\_neq\_left':\ "l\ \isasymin \ bt(A)\ ==>\ (!!x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l)"\isanewline |
|
1910 |
\ \ \isacommand{apply}\ (induct\ set:\ bt)\isanewline |
|
1911 |
\ 1.\ \isasymAnd x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline |
|
1912 |
\ 2.\ \isasymAnd a\ t1\ t2\ x\ r.\isanewline |
|
1913 |
\isaindent{\ 2.\ \ \ \ }\isasymlbrakk a\ \isasymin \ A;\ t1\ \isasymin \ bt(A);\ \isasymAnd x\ r.\ Br(x,\ t1,\ r)\ \isasymnoteq \ t1;\ t2\ \isasymin \ bt(A);\isanewline |
|
1914 |
\isaindent{\ 2.\ \ \ \ \ \ \ }\isasymAnd x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline |
|
1915 |
\isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2) |
|
1916 |
\end{isabelle} |
|
1917 |
Compare the form of the induction hypotheses with the corresponding ones in |
|
1918 |
the previous proof. As before, to conclude requires only \isa{auto}. |
|
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|
1919 |
|
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|
1920 |
When there are only a few constructors, we might prefer to prove the freenness |
14154 | 1921 |
theorems for each constructor. This is simple: |
1922 |
\begin{isabelle} |
|
1923 |
\isacommand{lemma}\ Br\_iff:\ "Br(a,l,r)\ =\ Br(a',l',r')\ <->\ a=a'\ \&\ l=l'\ \&\ r=r'"\isanewline |
|
1924 |
\ \ \isacommand{by}\ (blast\ elim!:\ bt.free\_elims) |
|
1925 |
\end{isabelle} |
|
1926 |
Here we see a demonstration of freeness reasoning using |
|
1927 |
\isa{bt.free\_elims}, but simpler still is just to apply \isa{auto}. |
|
1928 |
||
1929 |
An \ttindex{inductive\_cases} declaration generates instances of the |
|
1930 |
case analysis rule that have been simplified using freeness |
|
1931 |
reasoning. |
|
1932 |
\begin{isabelle} |
|
1933 |
\isacommand{inductive\_cases}\ Br\_in\_bt:\ "Br(a,\ l,\ r)\ \isasymin \ bt(A)" |
|
1934 |
\end{isabelle} |
|
1935 |
The theorem just created is |
|
1936 |
\begin{isabelle} |
|
1937 |
\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. |
|
1938 |
\end{isabelle} |
|
1939 |
It is an elimination rule that from $\isa{Br}(a,l,r)\in\isa{bt}(A)$ |
|
1940 |
lets us infer $a\in A$, $l\in\isa{bt}(A)$ and |
|
1941 |
$r\in\isa{bt}(A)$. |
|
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|
1942 |
|
5fe77b9b5185
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parents:
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changeset
|
1943 |
|
5fe77b9b5185
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parents:
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|
1944 |
\subsubsection{Mixfix syntax in datatypes} |
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|
1945 |
|
14154 | 1946 |
Mixfix syntax is sometimes convenient. The theory \isa{Induct/PropLog} makes a |
6121
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|
1947 |
deep embedding of propositional logic: |
14158 | 1948 |
\begin{alltt*}\isastyleminor |
6121
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parents:
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|
1949 |
consts prop :: i |
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parents:
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|
1950 |
datatype "prop" = Fls |
14154 | 1951 |
| Var ("n \isasymin nat") ("#_" [100] 100) |
1952 |
| "=>" ("p \isasymin prop", "q \isasymin prop") (infixr 90) |
|
14158 | 1953 |
\end{alltt*} |
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|
1954 |
The second constructor has a special $\#n$ syntax, while the third constructor |
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parents:
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changeset
|
1955 |
is an infixed arrow. |
5fe77b9b5185
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parents:
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changeset
|
1956 |
|
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parents:
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changeset
|
1957 |
|
5fe77b9b5185
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parents:
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changeset
|
1958 |
\subsubsection{A giant enumeration type} |
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parents:
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|
1959 |
|
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parents:
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|
1960 |
This example shows a datatype that consists of 60 constructors: |
14158 | 1961 |
\begin{alltt*}\isastyleminor |
6121
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|
1962 |
consts enum :: i |
5fe77b9b5185
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parents:
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changeset
|
1963 |
datatype |
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parents:
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|
1964 |
"enum" = C00 | C01 | C02 | C03 | C04 | C05 | C06 | C07 | C08 | C09 |
5fe77b9b5185
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parents:
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changeset
|
1965 |
| C10 | C11 | C12 | C13 | C14 | C15 | C16 | C17 | C18 | C19 |
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parents:
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|
1966 |
| C20 | C21 | C22 | C23 | C24 | C25 | C26 | C27 | C28 | C29 |
5fe77b9b5185
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parents:
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|
1967 |
| C30 | C31 | C32 | C33 | C34 | C35 | C36 | C37 | C38 | C39 |
5fe77b9b5185
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parents:
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changeset
|
1968 |
| C40 | C41 | C42 | C43 | C44 | C45 | C46 | C47 | C48 | C49 |
5fe77b9b5185
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parents:
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changeset
|
1969 |
| C50 | C51 | C52 | C53 | C54 | C55 | C56 | C57 | C58 | C59 |
5fe77b9b5185
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|
1970 |
end |
14158 | 1971 |
\end{alltt*} |
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|
1972 |
The datatype package scales well. Even though all properties are proved |
14154 | 1973 |
rather than assumed, full processing of this definition takes around two seconds |
1974 |
(on a 1.8GHz machine). The constructors have a balanced representation, |
|
1975 |
related to binary notation, so freeness properties can be proved fast. |
|
1976 |
\begin{isabelle} |
|
1977 |
\isacommand{lemma}\ "C00 \isasymnoteq\ C01"\isanewline |
|
1978 |
\ \ \isacommand{by}\ simp |
|
1979 |
\end{isabelle} |
|
1980 |
You need not derive such inequalities explicitly. The simplifier will |
|
1981 |
dispose of them automatically. |
|
6121
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parents:
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|
1982 |
|
5fe77b9b5185
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parents:
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changeset
|
1983 |
\index{*datatype|)} |
5fe77b9b5185
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parents:
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changeset
|
1984 |
|
5fe77b9b5185
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parents:
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changeset
|
1985 |
|
5fe77b9b5185
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parents:
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changeset
|
1986 |
\subsection{Recursive function definitions}\label{sec:ZF:recursive} |
5fe77b9b5185
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parents:
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changeset
|
1987 |
\index{recursive functions|see{recursion}} |
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parents:
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changeset
|
1988 |
\index{*primrec|(} |
6173 | 1989 |
\index{recursion!primitive|(} |
6121
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parents:
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changeset
|
1990 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1991 |
Datatypes come with a uniform way of defining functions, {\bf primitive |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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changeset
|
1992 |
recursion}. Such definitions rely on the recursion operator defined by the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1993 |
datatype package. Isabelle proves the desired recursion equations as |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
1994 |
theorems. |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
1995 |
|
5fe77b9b5185
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parents:
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changeset
|
1996 |
In principle, one could introduce primitive recursive functions by asserting |
14154 | 1997 |
their reduction rules as axioms. Here is a dangerous way of defining a |
1998 |
recursive function over binary trees: |
|
1999 |
\begin{isabelle} |
|
2000 |
\isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline |
|
2001 |
\isacommand{axioms}\isanewline |
|
2002 |
\ \ n\_nodes\_Lf:\ "n\_nodes(Lf)\ =\ 0"\isanewline |
|
2003 |
\ \ n\_nodes\_Br:\ "n\_nodes(Br(a,l,r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))" |
|
2004 |
\end{isabelle} |
|
2005 |
Asserting axioms brings the danger of accidentally introducing |
|
2006 |
contradictions. It should be avoided whenever possible. |
|
6121
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|
2007 |
|
5fe77b9b5185
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paulson
parents:
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changeset
|
2008 |
The \ttindex{primrec} declaration is a safe means of defining primitive |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2009 |
recursive functions on datatypes: |
14154 | 2010 |
\begin{isabelle} |
2011 |
\isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline |
|
2012 |
\isacommand{primrec}\isanewline |
|
2013 |
\ \ "n\_nodes(Lf)\ =\ 0"\isanewline |
|
2014 |
\ \ "n\_nodes(Br(a,\ l,\ r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))" |
|
2015 |
\end{isabelle} |
|
2016 |
Isabelle will now derive the two equations from a low-level definition |
|
2017 |
based upon well-founded recursion. If they do not define a legitimate |
|
2018 |
recursion, then Isabelle will reject the declaration. |
|
6121
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parents:
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|
2019 |
|
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2020 |
|
5fe77b9b5185
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parents:
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changeset
|
2021 |
\subsubsection{Syntax of recursive definitions} |
5fe77b9b5185
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parents:
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|
2022 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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changeset
|
2023 |
The general form of a primitive recursive definition is |
14154 | 2024 |
\begin{ttbox}\isastyleminor |
6121
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parents:
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|
2025 |
primrec |
5fe77b9b5185
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paulson
parents:
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changeset
|
2026 |
{\it reduction rules} |
5fe77b9b5185
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paulson
parents:
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changeset
|
2027 |
\end{ttbox} |
5fe77b9b5185
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parents:
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|
2028 |
where \textit{reduction rules} specify one or more equations of the form |
5fe77b9b5185
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parents:
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changeset
|
2029 |
\[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \, |
5fe77b9b5185
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parents:
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|
2030 |
\dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$ |
5fe77b9b5185
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parents:
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changeset
|
2031 |
contains only the free variables on the left-hand side, and all recursive |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
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changeset
|
2032 |
calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$. |
5fe77b9b5185
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parents:
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changeset
|
2033 |
There must be at most one reduction rule for each constructor. The order is |
5fe77b9b5185
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parents:
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changeset
|
2034 |
immaterial. For missing constructors, the function is defined to return zero. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2035 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2036 |
All reduction rules are added to the default simpset. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2037 |
If you would like to refer to some rule by name, then you must prefix |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2038 |
the rule with an identifier. These identifiers, like those in the |
14154 | 2039 |
\isa{rules} section of a theory, will be visible in proof scripts. |
2040 |
||
2041 |
The reduction rules become part of the default simpset, which |
|
6121
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parents:
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|
2042 |
leads to short proof scripts: |
14154 | 2043 |
\begin{isabelle} |
2044 |
\isacommand{lemma}\ n\_nodes\_type\ [simp]:\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes(t)\ \isasymin \ nat"\isanewline |
|
2045 |
\ \ \isacommand{by}\ (induct\_tac\ t,\ auto) |
|
2046 |
\end{isabelle} |
|
2047 |
||
2048 |
You can even use the \isa{primrec} form with non-recursive datatypes and |
|
6121
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|
2049 |
with codatatypes. Recursion is not allowed, but it provides a convenient |
5fe77b9b5185
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parents:
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changeset
|
2050 |
syntax for defining functions by cases. |
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parents:
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changeset
|
2051 |
|
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2052 |
|
5fe77b9b5185
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parents:
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changeset
|
2053 |
\subsubsection{Example: varying arguments} |
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parents:
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changeset
|
2054 |
|
5fe77b9b5185
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parents:
diff
changeset
|
2055 |
All arguments, other than the recursive one, must be the same in each equation |
5fe77b9b5185
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parents:
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changeset
|
2056 |
and in each recursive call. To get around this restriction, use explict |
14154 | 2057 |
$\lambda$-abstraction and function application. For example, let us |
2058 |
define the tail-recursive version of \isa{n\_nodes}, using an |
|
2059 |
accumulating argument for the counter. The second argument, $k$, varies in |
|
6121
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parents:
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|
2060 |
recursive calls. |
14154 | 2061 |
\begin{isabelle} |
2062 |
\isacommand{consts}\ \ n\_nodes\_aux\ ::\ "i\ =>\ i"\isanewline |
|
2063 |
\isacommand{primrec}\isanewline |
|
2064 |
\ \ "n\_nodes\_aux(Lf)\ =\ (\isasymlambda k\ \isasymin \ nat.\ k)"\isanewline |
|
2065 |
\ \ "n\_nodes\_aux(Br(a,l,r))\ =\ \isanewline |
|
2066 |
\ \ \ \ \ \ (\isasymlambda k\ \isasymin \ nat.\ n\_nodes\_aux(r)\ `\ \ (n\_nodes\_aux(l)\ `\ succ(k)))" |
|
2067 |
\end{isabelle} |
|
2068 |
Now \isa{n\_nodes\_aux(t)\ `\ k} is our function in two arguments. We |
|
2069 |
can prove a theorem relating it to \isa{n\_nodes}. Note the quantification |
|
2070 |
over \isa{k\ \isasymin \ nat}: |
|
2071 |
\begin{isabelle} |
|
2072 |
\isacommand{lemma}\ n\_nodes\_aux\_eq\ [rule\_format]:\isanewline |
|
2073 |
\ \ \ \ \ "t\ \isasymin \ bt(A)\ ==>\ \isasymforall k\ \isasymin \ nat.\ n\_nodes\_aux(t)`k\ =\ n\_nodes(t)\ \#+\ k"\isanewline |
|
2074 |
\ \ \isacommand{by}\ (induct\_tac\ t,\ simp\_all) |
|
2075 |
\end{isabelle} |
|
2076 |
||
2077 |
Now, we can use \isa{n\_nodes\_aux} to define a tail-recursive version |
|
2078 |
of \isa{n\_nodes}: |
|
2079 |
\begin{isabelle} |
|
2080 |
\isacommand{constdefs}\isanewline |
|
2081 |
\ \ n\_nodes\_tail\ ::\ "i\ =>\ i"\isanewline |
|
2082 |
\ \ \ "n\_nodes\_tail(t)\ ==\ n\_nodes\_aux(t)\ `\ 0" |
|
2083 |
\end{isabelle} |
|
2084 |
It is easy to |
|
2085 |
prove that \isa{n\_nodes\_tail} is equivalent to \isa{n\_nodes}: |
|
2086 |
\begin{isabelle} |
|
2087 |
\isacommand{lemma}\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes\_tail(t)\ =\ n\_nodes(t)"\isanewline |
|
2088 |
\ \isacommand{by}\ (simp\ add:\ n\_nodes\_tail\_def\ n\_nodes\_aux\_eq) |
|
2089 |
\end{isabelle} |
|
2090 |
||
2091 |
||
2092 |
||
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2093 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2094 |
\index{recursion!primitive|)} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2095 |
\index{*primrec|)} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2096 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2097 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2098 |
\section{Inductive and coinductive definitions} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2099 |
\index{*inductive|(} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2100 |
\index{*coinductive|(} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2101 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2102 |
An {\bf inductive definition} specifies the least set~$R$ closed under given |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2103 |
rules. (Applying a rule to elements of~$R$ yields a result within~$R$.) For |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2104 |
example, a structural operational semantics is an inductive definition of an |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2105 |
evaluation relation. Dually, a {\bf coinductive definition} specifies the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2106 |
greatest set~$R$ consistent with given rules. (Every element of~$R$ can be |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2107 |
seen as arising by applying a rule to elements of~$R$.) An important example |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2108 |
is using bisimulation relations to formalise equivalence of processes and |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2109 |
infinite data structures. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2110 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2111 |
A theory file may contain any number of inductive and coinductive |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2112 |
definitions. They may be intermixed with other declarations; in |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2113 |
particular, the (co)inductive sets {\bf must} be declared separately as |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2114 |
constants, and may have mixfix syntax or be subject to syntax translations. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2115 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2116 |
Each (co)inductive definition adds definitions to the theory and also |
14154 | 2117 |
proves some theorems. It behaves identially to the analogous |
2118 |
inductive definition except that instead of an induction rule there is |
|
2119 |
a coinduction rule. Its treatment of coinduction is described in |
|
2120 |
detail in a separate paper,% |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2121 |
\footnote{It appeared in CADE~\cite{paulson-CADE}; a longer version is |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2122 |
distributed with Isabelle as \emph{A Fixedpoint Approach to |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2123 |
(Co)Inductive and (Co)Datatype Definitions}.} % |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2124 |
which you might refer to for background information. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2125 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2126 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2127 |
\subsection{The syntax of a (co)inductive definition} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2128 |
An inductive definition has the form |
14154 | 2129 |
\begin{ttbox}\isastyleminor |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2130 |
inductive |
14154 | 2131 |
domains {\it domain declarations} |
2132 |
intros {\it introduction rules} |
|
2133 |
monos {\it monotonicity theorems} |
|
2134 |
con_defs {\it constructor definitions} |
|
2135 |
type_intros {\it introduction rules for type-checking} |
|
2136 |
type_elims {\it elimination rules for type-checking} |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2137 |
\end{ttbox} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2138 |
A coinductive definition is identical, but starts with the keyword |
14154 | 2139 |
\isa{co\-inductive}. |
2140 |
||
2141 |
The \isa{monos}, \isa{con\_defs}, \isa{type\_intros} and \isa{type\_elims} |
|
2142 |
sections are optional. If present, each is specified as a list of |
|
2143 |
theorems, which may contain Isar attributes as usual. |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2144 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2145 |
\begin{description} |
8249 | 2146 |
\item[\it domain declarations] are items of the form |
14154 | 2147 |
{\it string\/}~\isa{\isasymsubseteq }~{\it string}, associating each recursive set with |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2148 |
its domain. (The domain is some existing set that is large enough to |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2149 |
hold the new set being defined.) |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2150 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2151 |
\item[\it introduction rules] specify one or more introduction rules in |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2152 |
the form {\it ident\/}~{\it string}, where the identifier gives the name of |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2153 |
the rule in the result structure. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2154 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2155 |
\item[\it monotonicity theorems] are required for each operator applied to |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2156 |
a recursive set in the introduction rules. There \textbf{must} be a theorem |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2157 |
of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each premise $t\in M(R_i)$ |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2158 |
in an introduction rule! |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2159 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2160 |
\item[\it constructor definitions] contain definitions of constants |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2161 |
appearing in the introduction rules. The (co)datatype package supplies |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2162 |
the constructors' definitions here. Most (co)inductive definitions omit |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2163 |
this section; one exception is the primitive recursive functions example; |
14154 | 2164 |
see theory \isa{Induct/Primrec}. |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2165 |
|
14154 | 2166 |
\item[\it type\_intros] consists of introduction rules for type-checking the |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2167 |
definition: for demonstrating that the new set is included in its domain. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2168 |
(The proof uses depth-first search.) |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2169 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2170 |
\item[\it type\_elims] consists of elimination rules for type-checking the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2171 |
definition. They are presumed to be safe and are applied as often as |
14154 | 2172 |
possible prior to the \isa{type\_intros} search. |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2173 |
\end{description} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2174 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2175 |
The package has a few restrictions: |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2176 |
\begin{itemize} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2177 |
\item The theory must separately declare the recursive sets as |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2178 |
constants. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2179 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2180 |
\item The names of the recursive sets must be identifiers, not infix |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2181 |
operators. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2182 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2183 |
\item Side-conditions must not be conjunctions. However, an introduction rule |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2184 |
may contain any number of side-conditions. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2185 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2186 |
\item Side-conditions of the form $x=t$, where the variable~$x$ does not |
14154 | 2187 |
occur in~$t$, will be substituted through the rule \isa{mutual\_induct}. |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2188 |
\end{itemize} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2189 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2190 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2191 |
\subsection{Example of an inductive definition} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2192 |
|
14154 | 2193 |
Below, we shall see how Isabelle/ZF defines the finite powerset |
2194 |
operator. The first step is to declare the constant~\isa{Fin}. Then we |
|
2195 |
must declare it inductively, with two introduction rules: |
|
2196 |
\begin{isabelle} |
|
2197 |
\isacommand{consts}\ \ Fin\ ::\ "i=>i"\isanewline |
|
2198 |
\isacommand{inductive}\isanewline |
|
2199 |
\ \ \isakeyword{domains}\ \ \ "Fin(A)"\ \isasymsubseteq\ "Pow(A)"\isanewline |
|
2200 |
\ \ \isakeyword{intros}\isanewline |
|
2201 |
\ \ \ \ emptyI:\ \ "0\ \isasymin\ Fin(A)"\isanewline |
|
2202 |
\ \ \ \ consI:\ \ \ "[|\ a\ \isasymin\ A;\ \ b\ \isasymin\ Fin(A)\ |]\ ==>\ cons(a,b)\ \isasymin\ Fin(A)"\isanewline |
|
2203 |
\ \ \isakeyword{type\_intros}\ \ empty\_subsetI\ cons\_subsetI\ PowI\isanewline |
|
2204 |
\ \ \isakeyword{type\_elims}\ \ \ PowD\ [THEN\ revcut\_rl]\end{isabelle} |
|
2205 |
The resulting theory contains a name space, called~\isa{Fin}. |
|
2206 |
The \isa{Fin}$~A$ introduction rules can be referred to collectively as |
|
2207 |
\isa{Fin.intros}, and also individually as \isa{Fin.emptyI} and |
|
2208 |
\isa{Fin.consI}. The induction rule is \isa{Fin.induct}. |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2209 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2210 |
The chief problem with making (co)inductive definitions involves type-checking |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2211 |
the rules. Sometimes, additional theorems need to be supplied under |
14154 | 2212 |
\isa{type_intros} or \isa{type_elims}. If the package fails when trying |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2213 |
to prove your introduction rules, then set the flag \ttindexbold{trace_induct} |
14154 | 2214 |
to \isa{true} and try again. (See the manual \emph{A Fixedpoint Approach |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2215 |
\ldots} for more discussion of type-checking.) |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2216 |
|
14154 | 2217 |
In the example above, $\isa{Pow}(A)$ is given as the domain of |
2218 |
$\isa{Fin}(A)$, for obviously every finite subset of~$A$ is a subset |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2219 |
of~$A$. However, the inductive definition package can only prove that given a |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2220 |
few hints. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2221 |
Here is the output that results (with the flag set) when the |
14154 | 2222 |
\isa{type_intros} and \isa{type_elims} are omitted from the inductive |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2223 |
definition above: |
14158 | 2224 |
\begin{alltt*}\isastyleminor |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2225 |
Inductive definition Finite.Fin |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2226 |
Fin(A) == |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2227 |
lfp(Pow(A), |
14158 | 2228 |
\%X. {z\isasymin{}Pow(A) . z = 0 | ({\isasymexists}a b. z = cons(a,b) & a\isasymin{}A & b\isasymin{}X)}) |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2229 |
Proving monotonicity... |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2230 |
\ttbreak |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2231 |
Proving the introduction rules... |
6173 | 2232 |
The type-checking subgoal: |
14154 | 2233 |
0 \isasymin Fin(A) |
2234 |
1. 0 \isasymin Pow(A) |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2235 |
\ttbreak |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2236 |
The subgoal after monos, type_elims: |
14154 | 2237 |
0 \isasymin Fin(A) |
2238 |
1. 0 \isasymin Pow(A) |
|
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2239 |
*** prove_goal: tactic failed |
14158 | 2240 |
\end{alltt*} |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2241 |
We see the need to supply theorems to let the package prove |
14154 | 2242 |
$\emptyset\in\isa{Pow}(A)$. Restoring the \isa{type_intros} but not the |
2243 |
\isa{type_elims}, we again get an error message: |
|
14158 | 2244 |
\begin{alltt*}\isastyleminor |
6173 | 2245 |
The type-checking subgoal: |
14154 | 2246 |
0 \isasymin Fin(A) |
2247 |
1. 0 \isasymin Pow(A) |
|
6121
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|
2248 |
\ttbreak |
5fe77b9b5185
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parents:
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|
2249 |
The subgoal after monos, type_elims: |
14154 | 2250 |
0 \isasymin Fin(A) |
2251 |
1. 0 \isasymin Pow(A) |
|
6121
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|
2252 |
\ttbreak |
6173 | 2253 |
The type-checking subgoal: |
14154 | 2254 |
cons(a, b) \isasymin Fin(A) |
2255 |
1. [| a \isasymin A; b \isasymin Fin(A) |] ==> cons(a, b) \isasymin Pow(A) |
|
6121
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|
2256 |
\ttbreak |
5fe77b9b5185
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paulson
parents:
diff
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|
2257 |
The subgoal after monos, type_elims: |
14154 | 2258 |
cons(a, b) \isasymin Fin(A) |
2259 |
1. [| a \isasymin A; b \isasymin Pow(A) |] ==> cons(a, b) \isasymin Pow(A) |
|
6121
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|
2260 |
*** prove_goal: tactic failed |
14158 | 2261 |
\end{alltt*} |
6121
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|
2262 |
The first rule has been type-checked, but the second one has failed. The |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
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|
2263 |
simplest solution to such problems is to prove the failed subgoal separately |
14154 | 2264 |
and to supply it under \isa{type_intros}. The solution actually used is |
2265 |
to supply, under \isa{type_elims}, a rule that changes |
|
2266 |
$b\in\isa{Pow}(A)$ to $b\subseteq A$; together with \isa{cons_subsetI} |
|
2267 |
and \isa{PowI}, it is enough to complete the type-checking. |
|
6121
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|
2268 |
|
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2269 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2270 |
|
5fe77b9b5185
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paulson
parents:
diff
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|
2271 |
\subsection{Further examples} |
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|
2272 |
|
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parents:
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|
2273 |
An inductive definition may involve arbitrary monotonic operators. Here is a |
5fe77b9b5185
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parents:
diff
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|
2274 |
standard example: the accessible part of a relation. Note the use |
14154 | 2275 |
of~\isa{Pow} in the introduction rule and the corresponding mention of the |
2276 |
rule \isa{Pow\_mono} in the \isa{monos} list. If the desired rule has a |
|
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|
2277 |
universally quantified premise, usually the effect can be obtained using |
14154 | 2278 |
\isa{Pow}. |
2279 |
\begin{isabelle} |
|
2280 |
\isacommand{consts}\ \ acc\ ::\ "i\ =>\ i"\isanewline |
|
2281 |
\isacommand{inductive}\isanewline |
|
2282 |
\ \ \isakeyword{domains}\ "acc(r)"\ \isasymsubseteq \ "field(r)"\isanewline |
|
2283 |
\ \ \isakeyword{intros}\isanewline |
|
2284 |
\ \ \ \ vimage:\ \ "[|\ r-``\isacharbraceleft a\isacharbraceright\ \isasymin\ Pow(acc(r));\ a\ \isasymin \ field(r)\ |] |
|
2285 |
\isanewline |
|
2286 |
\ \ \ \ \ \ \ \ \ \ \ \ \ \ ==>\ a\ \isasymin \ acc(r)"\isanewline |
|
2287 |
\ \ \isakeyword{monos}\ \ Pow\_mono |
|
2288 |
\end{isabelle} |
|
2289 |
||
2290 |
Finally, here are some coinductive definitions. We begin by defining |
|
2291 |
lazy (potentially infinite) lists as a codatatype: |
|
2292 |
\begin{isabelle} |
|
2293 |
\isacommand{consts}\ \ llist\ \ ::\ "i=>i"\isanewline |
|
2294 |
\isacommand{codatatype}\isanewline |
|
2295 |
\ \ "llist(A)"\ =\ LNil\ |\ LCons\ ("a\ \isasymin \ A",\ "l\ \isasymin \ llist(A)")\isanewline |
|
2296 |
\end{isabelle} |
|
2297 |
||
2298 |
The notion of equality on such lists is modelled as a bisimulation: |
|
2299 |
\begin{isabelle} |
|
2300 |
\isacommand{consts}\ \ lleq\ ::\ "i=>i"\isanewline |
|
2301 |
\isacommand{coinductive}\isanewline |
|
2302 |
\ \ \isakeyword{domains}\ "lleq(A)"\ <=\ "llist(A)\ *\ llist(A)"\isanewline |
|
2303 |
\ \ \isakeyword{intros}\isanewline |
|
2304 |
\ \ \ \ LNil:\ \ "<LNil,\ LNil>\ \isasymin \ lleq(A)"\isanewline |
|
2305 |
\ \ \ \ LCons:\ "[|\ a\ \isasymin \ A;\ <l,l'>\ \isasymin \ lleq(A)\ |]\ \isanewline |
|
2306 |
\ \ \ \ \ \ \ \ \ \ \ \ ==>\ <LCons(a,l),\ LCons(a,l')>\ \isasymin \ lleq(A)"\isanewline |
|
2307 |
\ \ \isakeyword{type\_intros}\ \ llist.intros |
|
2308 |
\end{isabelle} |
|
2309 |
This use of \isa{type_intros} is typical: the relation concerns the |
|
2310 |
codatatype \isa{llist}, so naturally the introduction rules for that |
|
6121
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parents:
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|
2311 |
codatatype will be required for type-checking the rules. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2312 |
|
5fe77b9b5185
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paulson
parents:
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changeset
|
2313 |
The Isabelle distribution contains many other inductive definitions. Simple |
14154 | 2314 |
examples are collected on subdirectory \isa{ZF/Induct}. The directory |
2315 |
\isa{Coind} and the theory \isa{ZF/Induct/LList} contain coinductive |
|
6121
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parents:
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|
2316 |
definitions. Larger examples may be found on other subdirectories of |
14154 | 2317 |
\isa{ZF}, such as \isa{IMP}, and \isa{Resid}. |
2318 |
||
2319 |
||
2320 |
\subsection{Theorems generated} |
|
2321 |
||
2322 |
Each (co)inductive set defined in a theory file generates a name space |
|
2323 |
containing the following elements: |
|
2324 |
\begin{ttbox}\isastyleminor |
|
2325 |
intros \textrm{the introduction rules} |
|
2326 |
elim \textrm{the elimination (case analysis) rule} |
|
2327 |
induct \textrm{the standard induction rule} |
|
2328 |
mutual_induct \textrm{the mutual induction rule, if needed} |
|
2329 |
defs \textrm{definitions of inductive sets} |
|
2330 |
bnd_mono \textrm{monotonicity property} |
|
2331 |
dom_subset \textrm{inclusion in `bounding set'} |
|
6121
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|
2332 |
\end{ttbox} |
14154 | 2333 |
Furthermore, each introduction rule is available under its declared |
2334 |
name. For a codatatype, the component \isa{coinduct} is the coinduction rule, |
|
2335 |
replacing the \isa{induct} component. |
|
2336 |
||
2337 |
Recall that the \ttindex{inductive\_cases} declaration generates |
|
2338 |
simplified instances of the case analysis rule. It is as useful for |
|
2339 |
inductive definitions as it is for datatypes. There are many examples |
|
2340 |
in the theory |
|
2341 |
\isa{Induct/Comb}, which is discussed at length |
|
2342 |
elsewhere~\cite{paulson-generic}. The theory first defines the |
|
2343 |
datatype |
|
2344 |
\isa{comb} of combinators: |
|
14158 | 2345 |
\begin{alltt*}\isastyleminor |
6121
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|
2346 |
consts comb :: i |
5fe77b9b5185
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parents:
diff
changeset
|
2347 |
datatype "comb" = K |
5fe77b9b5185
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parents:
diff
changeset
|
2348 |
| S |
14154 | 2349 |
| "#" ("p \isasymin comb", "q \isasymin comb") (infixl 90) |
14158 | 2350 |
\end{alltt*} |
6121
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|
2351 |
The theory goes on to define contraction and parallel contraction |
14154 | 2352 |
inductively. Then the theory \isa{Induct/Comb.thy} defines special |
2353 |
cases of contraction, such as this one: |
|
2354 |
\begin{isabelle} |
|
2355 |
\isacommand{inductive\_cases}\ K\_contractE [elim!]:\ "K -1-> r" |
|
2356 |
\end{isabelle} |
|
2357 |
The theorem just created is \isa{K -1-> r \ \isasymLongrightarrow \ Q}, |
|
2358 |
which expresses that the combinator \isa{K} cannot reduce to |
|
2359 |
anything. (From the assumption \isa{K-1->r}, we can conclude any desired |
|
2360 |
formula \isa{Q}\@.) Similar elimination rules for \isa{S} and application are also |
|
2361 |
generated. The attribute \isa{elim!}\ shown above supplies the generated |
|
2362 |
theorem to the classical reasoner. This mode of working allows |
|
2363 |
effective reasoniung about operational semantics. |
|
6121
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|
2364 |
|
5fe77b9b5185
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parents:
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changeset
|
2365 |
\index{*coinductive|)} \index{*inductive|)} |
5fe77b9b5185
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parents:
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|
2366 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2367 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2368 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
changeset
|
2369 |
\section{The outer reaches of set theory} |
5fe77b9b5185
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parents:
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changeset
|
2370 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2371 |
The constructions of the natural numbers and lists use a suite of |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2372 |
operators for handling recursive function definitions. I have described |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2373 |
the developments in detail elsewhere~\cite{paulson-set-II}. Here is a brief |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2374 |
summary: |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
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|
2375 |
\begin{itemize} |
14154 | 2376 |
\item Theory \isa{Trancl} defines the transitive closure of a relation |
6121
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parents:
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|
2377 |
(as a least fixedpoint). |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2378 |
|
14154 | 2379 |
\item Theory \isa{WF} proves the well-founded recursion theorem, using an |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2380 |
elegant approach of Tobias Nipkow. This theorem permits general |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2381 |
recursive definitions within set theory. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2382 |
|
14154 | 2383 |
\item Theory \isa{Ord} defines the notions of transitive set and ordinal |
6121
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parents:
diff
changeset
|
2384 |
number. It derives transfinite induction. A key definition is {\bf |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2385 |
less than}: $i<j$ if and only if $i$ and $j$ are both ordinals and |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2386 |
$i\in j$. As a special case, it includes less than on the natural |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2387 |
numbers. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2388 |
|
14154 | 2389 |
\item Theory \isa{Epsilon} derives $\varepsilon$-induction and |
6121
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paulson
parents:
diff
changeset
|
2390 |
$\varepsilon$-recursion, which are generalisations of transfinite |
14154 | 2391 |
induction and recursion. It also defines \cdx{rank}$(x)$, which is the |
2392 |
least ordinal $\alpha$ such that $x$ is constructed at stage $\alpha$ of |
|
2393 |
the cumulative hierarchy (thus $x\in V@{\alpha+1}$). |
|
6121
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paulson
parents:
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|
2394 |
\end{itemize} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2395 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2396 |
Other important theories lead to a theory of cardinal numbers. They have |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2397 |
not yet been written up anywhere. Here is a summary: |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2398 |
\begin{itemize} |
14154 | 2399 |
\item Theory \isa{Rel} defines the basic properties of relations, such as |
14158 | 2400 |
reflexivity, symmetry and transitivity. |
6121
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parents:
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changeset
|
2401 |
|
14154 | 2402 |
\item Theory \isa{EquivClass} develops a theory of equivalence |
6121
5fe77b9b5185
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paulson
parents:
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changeset
|
2403 |
classes, not using the Axiom of Choice. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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changeset
|
2404 |
|
14154 | 2405 |
\item Theory \isa{Order} defines partial orderings, total orderings and |
6121
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paulson
parents:
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changeset
|
2406 |
wellorderings. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2407 |
|
14154 | 2408 |
\item Theory \isa{OrderArith} defines orderings on sum and product sets. |
6121
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paulson
parents:
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changeset
|
2409 |
These can be used to define ordinal arithmetic and have applications to |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2410 |
cardinal arithmetic. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2411 |
|
14154 | 2412 |
\item Theory \isa{OrderType} defines order types. Every wellordering is |
6121
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parents:
diff
changeset
|
2413 |
equivalent to a unique ordinal, which is its order type. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2414 |
|
14154 | 2415 |
\item Theory \isa{Cardinal} defines equipollence and cardinal numbers. |
6121
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parents:
diff
changeset
|
2416 |
|
14154 | 2417 |
\item Theory \isa{CardinalArith} defines cardinal addition and |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2418 |
multiplication, and proves their elementary laws. It proves that there |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2419 |
is no greatest cardinal. It also proves a deep result, namely |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2420 |
$\kappa\otimes\kappa=\kappa$ for every infinite cardinal~$\kappa$; see |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2421 |
Kunen~\cite[page 29]{kunen80}. None of these results assume the Axiom of |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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changeset
|
2422 |
Choice, which complicates their proofs considerably. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2423 |
\end{itemize} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
changeset
|
2424 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2425 |
The following developments involve the Axiom of Choice (AC): |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2426 |
\begin{itemize} |
14154 | 2427 |
\item Theory \isa{AC} asserts the Axiom of Choice and proves some simple |
6121
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paulson
parents:
diff
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|
2428 |
equivalent forms. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2429 |
|
14154 | 2430 |
\item Theory \isa{Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma |
6121
5fe77b9b5185
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parents:
diff
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|
2431 |
and the Wellordering Theorem, following Abrial and |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2432 |
Laffitte~\cite{abrial93}. |
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2433 |
|
14154 | 2434 |
\item Theory \isa{Cardinal\_AC} uses AC to prove simplified theorems about |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2435 |
the cardinals. It also proves a theorem needed to justify |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2436 |
infinitely branching datatype declarations: if $\kappa$ is an infinite |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2437 |
cardinal and $|X(\alpha)| \le \kappa$ for all $\alpha<\kappa$ then |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2438 |
$|\union\sb{\alpha<\kappa} X(\alpha)| \le \kappa$. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2439 |
|
14154 | 2440 |
\item Theory \isa{InfDatatype} proves theorems to justify infinitely |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2441 |
branching datatypes. Arbitrary index sets are allowed, provided their |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2442 |
cardinalities have an upper bound. The theory also justifies some |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2443 |
unusual cases of finite branching, involving the finite powerset operator |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2444 |
and the finite function space operator. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2445 |
\end{itemize} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2446 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2447 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2448 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2449 |
\section{The examples directories} |
14154 | 2450 |
Directory \isa{HOL/IMP} contains a mechanised version of a semantic |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2451 |
equivalence proof taken from Winskel~\cite{winskel93}. It formalises the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2452 |
denotational and operational semantics of a simple while-language, then |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2453 |
proves the two equivalent. It contains several datatype and inductive |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2454 |
definitions, and demonstrates their use. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2455 |
|
14154 | 2456 |
The directory \isa{ZF/ex} contains further developments in ZF set theory. |
9695 | 2457 |
Here is an overview; see the files themselves for more details. I describe |
2458 |
much of this material in other |
|
14154 | 2459 |
publications~\cite{paulson-set-I,paulson-set-II,paulson-fixedpt-milner}. |
6121
5fe77b9b5185
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diff
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|
2460 |
\begin{itemize} |
14154 | 2461 |
\item File \isa{misc.ML} contains miscellaneous examples such as |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2462 |
Cantor's Theorem, the Schr\"oder-Bernstein Theorem and the `Composition |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2463 |
of homomorphisms' challenge~\cite{boyer86}. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2464 |
|
14154 | 2465 |
\item Theory \isa{Ramsey} proves the finite exponent 2 version of |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2466 |
Ramsey's Theorem, following Basin and Kaufmann's |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2467 |
presentation~\cite{basin91}. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2468 |
|
14154 | 2469 |
\item Theory \isa{Integ} develops a theory of the integers as |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2470 |
equivalence classes of pairs of natural numbers. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2471 |
|
14154 | 2472 |
\item Theory \isa{Primrec} develops some computation theory. It |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2473 |
inductively defines the set of primitive recursive functions and presents a |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2474 |
proof that Ackermann's function is not primitive recursive. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2475 |
|
14154 | 2476 |
\item Theory \isa{Primes} defines the Greatest Common Divisor of two |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2477 |
natural numbers and and the ``divides'' relation. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2478 |
|
14154 | 2479 |
\item Theory \isa{Bin} defines a datatype for two's complement binary |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2480 |
integers, then proves rewrite rules to perform binary arithmetic. For |
14154 | 2481 |
instance, $1359\times {-}2468 = {-}3354012$ takes 0.3 seconds. |
2482 |
||
2483 |
\item Theory \isa{BT} defines the recursive data structure $\isa{bt}(A)$, labelled binary trees. |
|
2484 |
||
2485 |
\item Theory \isa{Term} defines a recursive data structure for terms |
|
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2486 |
and term lists. These are simply finite branching trees. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2487 |
|
14154 | 2488 |
\item Theory \isa{TF} defines primitives for solving mutually |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2489 |
recursive equations over sets. It constructs sets of trees and forests |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2490 |
as an example, including induction and recursion rules that handle the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2491 |
mutual recursion. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2492 |
|
14154 | 2493 |
\item Theory \isa{Prop} proves soundness and completeness of |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2494 |
propositional logic~\cite{paulson-set-II}. This illustrates datatype |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2495 |
definitions, inductive definitions, structural induction and rule |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2496 |
induction. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2497 |
|
14154 | 2498 |
\item Theory \isa{ListN} inductively defines the lists of $n$ |
6592 | 2499 |
elements~\cite{paulin-tlca}. |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2500 |
|
14154 | 2501 |
\item Theory \isa{Acc} inductively defines the accessible part of a |
6592 | 2502 |
relation~\cite{paulin-tlca}. |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2503 |
|
14154 | 2504 |
\item Theory \isa{Comb} defines the datatype of combinators and |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2505 |
inductively defines contraction and parallel contraction. It goes on to |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2506 |
prove the Church-Rosser Theorem. This case study follows Camilleri and |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2507 |
Melham~\cite{camilleri92}. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2508 |
|
14154 | 2509 |
\item Theory \isa{LList} defines lazy lists and a coinduction |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2510 |
principle for proving equations between them. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2511 |
\end{itemize} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2512 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2513 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2514 |
\section{A proof about powersets}\label{sec:ZF-pow-example} |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2515 |
To demonstrate high-level reasoning about subsets, let us prove the |
14154 | 2516 |
equation ${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$. Compared |
6121
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2517 |
with first-order logic, set theory involves a maze of rules, and theorems |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2518 |
have many different proofs. Attempting other proofs of the theorem might |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2519 |
be instructive. This proof exploits the lattice properties of |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2520 |
intersection. It also uses the monotonicity of the powerset operation, |
14154 | 2521 |
from \isa{ZF/mono.ML}: |
2522 |
\begin{isabelle} |
|
2523 |
\tdx{Pow_mono}: A \isasymsubseteq B ==> Pow(A) \isasymsubseteq Pow(B) |
|
2524 |
\end{isabelle} |
|
6121
5fe77b9b5185
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parents:
diff
changeset
|
2525 |
We enter the goal and make the first step, which breaks the equation into |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2526 |
two inclusions by extensionality:\index{*equalityI theorem} |
14154 | 2527 |
\begin{isabelle} |
2528 |
\isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline |
|
2529 |
\ 1.\ Pow(A\ \isasyminter \ B)\ =\ Pow(A)\ \isasyminter \ Pow(B)\isanewline |
|
2530 |
\isacommand{apply}\ (rule\ equalityI)\isanewline |
|
2531 |
\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\ \isasyminter \ Pow(B)\isanewline |
|
2532 |
\ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B) |
|
2533 |
\end{isabelle} |
|
2534 |
Both inclusions could be tackled straightforwardly using \isa{subsetI}. |
|
6121
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parents:
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changeset
|
2535 |
A shorter proof results from noting that intersection forms the greatest |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2536 |
lower bound:\index{*Int_greatest theorem} |
14154 | 2537 |
\begin{isabelle} |
2538 |
\isacommand{apply}\ (rule\ Int\_greatest)\isanewline |
|
2539 |
\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\isanewline |
|
2540 |
\ 2.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline |
|
2541 |
\ 3.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B) |
|
2542 |
\end{isabelle} |
|
2543 |
Subgoal~1 follows by applying the monotonicity of \isa{Pow} to $A\int |
|
6121
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parents:
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|
2544 |
B\subseteq A$; subgoal~2 follows similarly: |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2545 |
\index{*Int_lower1 theorem}\index{*Int_lower2 theorem} |
14154 | 2546 |
\begin{isabelle} |
2547 |
\isacommand{apply}\ (rule\ Int\_lower1\ [THEN\ Pow\_mono])\isanewline |
|
2548 |
\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline |
|
2549 |
\ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B) |
|
2550 |
\isanewline |
|
2551 |
\isacommand{apply}\ (rule\ Int\_lower2\ [THEN\ Pow\_mono])\isanewline |
|
2552 |
\ 1.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B) |
|
2553 |
\end{isabelle} |
|
6121
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parents:
diff
changeset
|
2554 |
We are left with the opposite inclusion, which we tackle in the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2555 |
straightforward way:\index{*subsetI theorem} |
14154 | 2556 |
\begin{isabelle} |
2557 |
\isacommand{apply}\ (rule\ subsetI)\isanewline |
|
2558 |
\ 1.\ \isasymAnd x.\ x\ \isasymin \ Pow(A)\ \isasyminter \ Pow(B)\ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B) |
|
2559 |
\end{isabelle} |
|
2560 |
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 |
|
6121
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parents:
diff
changeset
|
2561 |
subgoals. The rule \tdx{IntE} treats the intersection like a conjunction |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2562 |
instead of unfolding its definition. |
14154 | 2563 |
\begin{isabelle} |
2564 |
\isacommand{apply}\ (erule\ IntE)\isanewline |
|
2565 |
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B) |
|
2566 |
\end{isabelle} |
|
2567 |
The next step replaces the \isa{Pow} by the subset |
|
6121
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the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2568 |
relation~($\subseteq$).\index{*PowI theorem} |
14154 | 2569 |
\begin{isabelle} |
2570 |
\isacommand{apply}\ (rule\ PowI)\isanewline |
|
2571 |
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B% |
|
2572 |
\end{isabelle} |
|
6121
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parents:
diff
changeset
|
2573 |
We perform the same replacement in the assumptions. This is a good |
14154 | 2574 |
demonstration of the tactic \ttindex{drule}:\index{*PowD theorem} |
2575 |
\begin{isabelle} |
|
2576 |
\isacommand{apply}\ (drule\ PowD)+\isanewline |
|
2577 |
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B% |
|
2578 |
\end{isabelle} |
|
6121
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parents:
diff
changeset
|
2579 |
The assumptions are that $x$ is a lower bound of both $A$ and~$B$, but |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2580 |
$A\int B$ is the greatest lower bound:\index{*Int_greatest theorem} |
14154 | 2581 |
\begin{isabelle} |
2582 |
\isacommand{apply}\ (rule\ Int\_greatest)\isanewline |
|
2583 |
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\isanewline |
|
2584 |
\ 2.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ B% |
|
2585 |
\end{isabelle} |
|
6121
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the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2586 |
To conclude the proof, we clear up the trivial subgoals: |
14154 | 2587 |
\begin{isabelle} |
2588 |
\isacommand{apply}\ (assumption+)\isanewline |
|
2589 |
\isacommand{done}% |
|
2590 |
\end{isabelle} |
|
2591 |
||
2592 |
We could have performed this proof instantly by calling |
|
2593 |
\ttindex{blast}: |
|
2594 |
\begin{isabelle} |
|
2595 |
\isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline |
|
2596 |
\isacommand{by} |
|
2597 |
\end{isabelle} |
|
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|
2598 |
Past researchers regarded this as a difficult proof, as indeed it is if all |
5fe77b9b5185
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|
2599 |
the symbols are replaced by their definitions. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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diff
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|
2600 |
\goodbreak |
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diff
changeset
|
2601 |
|
5fe77b9b5185
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|
2602 |
\section{Monotonicity of the union operator} |
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changeset
|
2603 |
For another example, we prove that general union is monotonic: |
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parents:
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|
2604 |
${C\subseteq D}$ implies $\bigcup(C)\subseteq \bigcup(D)$. To begin, we |
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parents:
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|
2605 |
tackle the inclusion using \tdx{subsetI}: |
14154 | 2606 |
\begin{isabelle} |
2607 |
\isacommand{lemma}\ "C\isasymsubseteq D\ ==>\ Union(C)\ |
|
2608 |
\isasymsubseteq \ Union(D)"\isanewline |
|
2609 |
\isacommand{apply}\ (rule\ subsetI)\isanewline |
|
2610 |
\ 1.\ \isasymAnd x.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ \isasymUnion C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D% |
|
2611 |
\end{isabelle} |
|
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|
2612 |
Big union is like an existential quantifier --- the occurrence in the |
5fe77b9b5185
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parents:
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|
2613 |
assumptions must be eliminated early, since it creates parameters. |
5fe77b9b5185
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parents:
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|
2614 |
\index{*UnionE theorem} |
14154 | 2615 |
\begin{isabelle} |
2616 |
\isacommand{apply}\ (erule\ UnionE)\isanewline |
|
2617 |
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D% |
|
2618 |
\end{isabelle} |
|
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|
2619 |
Now we may apply \tdx{UnionI}, which creates an unknown involving the |
14154 | 2620 |
parameters. To show \isa{x\ \isasymin \ \isasymUnion D} it suffices to show that~\isa{x} belongs |
2621 |
to some element, say~\isa{?B2(x,B)}, of~\isa{D}\@. |
|
2622 |
\begin{isabelle} |
|
2623 |
\isacommand{apply}\ (rule\ UnionI)\isanewline |
|
2624 |
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ D\isanewline |
|
2625 |
\ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B) |
|
2626 |
\end{isabelle} |
|
2627 |
Combining the rule \tdx{subsetD} with the assumption \isa{C\ \isasymsubseteq \ D} yields |
|
6121
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|
2628 |
$\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1. Note that |
14154 | 2629 |
\isa{erule} removes the subset assumption. |
2630 |
\begin{isabelle} |
|
2631 |
\isacommand{apply}\ (erule\ subsetD)\isanewline |
|
2632 |
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ C\isanewline |
|
2633 |
\ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B) |
|
2634 |
\end{isabelle} |
|
2635 |
The rest is routine. Observe how the first call to \isa{assumption} |
|
2636 |
instantiates \isa{?B2(x,B)} to~\isa{B}\@. |
|
2637 |
\begin{isabelle} |
|
2638 |
\isacommand{apply}\ assumption\ \isanewline |
|
2639 |
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ B% |
|
2640 |
\isanewline |
|
2641 |
\isacommand{apply}\ assumption\ \isanewline |
|
2642 |
No\ subgoals!\isanewline |
|
2643 |
\isacommand{done}% |
|
2644 |
\end{isabelle} |
|
2645 |
Again, \isa{blast} can prove this theorem in one step. |
|
2646 |
||
2647 |
The theory \isa{ZF/equalities.thy} has many similar proofs. Reasoning about |
|
6121
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|
2648 |
general intersection can be difficult because of its anomalous behaviour on |
14154 | 2649 |
the empty set. However, \isa{blast} copes well with these. Here is |
6121
5fe77b9b5185
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paulson
parents:
diff
changeset
|
2650 |
a typical example, borrowed from Devlin~\cite[page 12]{devlin79}: |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2651 |
\[ a\in C \,\Imp\, \inter@{x\in C} \Bigl(A(x) \int B(x)\Bigr) = |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
changeset
|
2652 |
\Bigl(\inter@{x\in C} A(x)\Bigr) \int |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2653 |
\Bigl(\inter@{x\in C} B(x)\Bigr) \] |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2654 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2655 |
\section{Low-level reasoning about functions} |
14154 | 2656 |
The derived rules \isa{lamI}, \isa{lamE}, \isa{lam_type}, \isa{beta} |
2657 |
and \isa{eta} support reasoning about functions in a |
|
6121
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parents:
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|
2658 |
$\lambda$-calculus style. This is generally easier than regarding |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2659 |
functions as sets of ordered pairs. But sometimes we must look at the |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2660 |
underlying representation, as in the following proof |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2661 |
of~\tdx{fun_disjoint_apply1}. This states that if $f$ and~$g$ are |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2662 |
functions with disjoint domains~$A$ and~$C$, and if $a\in A$, then |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2663 |
$(f\un g)`a = f`a$: |
14154 | 2664 |
\begin{isabelle} |
2665 |
\isacommand{lemma}\ "[|\ a\ \isasymin \ A;\ \ f\ \isasymin \ A->B;\ \ g\ \isasymin \ C->D;\ \ A\ \isasyminter \ C\ =\ 0\ |] |
|
2666 |
\isanewline |
|
2667 |
\ \ \ \ \ \ \ \ ==>\ (f\ \isasymunion \ g)`a\ =\ f`a" |
|
2668 |
\end{isabelle} |
|
6121
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diff
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|
2669 |
Using \tdx{apply_equality}, we reduce the equality to reasoning about |
14154 | 2670 |
ordered pairs. The second subgoal is to verify that \isa{f\ \isasymunion \ g} is a function, since |
2671 |
\isa{Pi(?A,?B)} denotes a dependent function space. |
|
2672 |
\begin{isabelle} |
|
2673 |
\isacommand{apply}\ (rule\ apply\_equality)\isanewline |
|
2674 |
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2675 |
\isaindent{\ 1.\ }\isasymLongrightarrow \ \isasymlangle a,\ f\ `\ a\isasymrangle \ \isasymin \ f\ \isasymunion \ g\isanewline |
|
2676 |
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2677 |
\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B) |
|
2678 |
\end{isabelle} |
|
6121
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|
2679 |
We must show that the pair belongs to~$f$ or~$g$; by~\tdx{UnI1} we |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2680 |
choose~$f$: |
14154 | 2681 |
\begin{isabelle} |
2682 |
\isacommand{apply}\ (rule\ UnI1)\isanewline |
|
2683 |
\ 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 |
|
2684 |
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2685 |
\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B) |
|
2686 |
\end{isabelle} |
|
6121
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diff
changeset
|
2687 |
To show $\pair{a,f`a}\in f$ we use \tdx{apply_Pair}, which is |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2688 |
essentially the converse of \tdx{apply_equality}: |
14154 | 2689 |
\begin{isabelle} |
2690 |
\isacommand{apply}\ (rule\ apply\_Pair)\isanewline |
|
2691 |
\ 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 |
|
2692 |
\ 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 |
|
2693 |
\ 3.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2694 |
\isaindent{\ 3.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B) |
|
2695 |
\end{isabelle} |
|
6121
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|
2696 |
Using the assumptions $f\in A\to B$ and $a\in A$, we solve the two subgoals |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
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parents:
diff
changeset
|
2697 |
from \tdx{apply_Pair}. Recall that a $\Pi$-set is merely a generalized |
14154 | 2698 |
function space, and observe that~{\tt?A2} gets instantiated to~\isa{A}. |
2699 |
\begin{isabelle} |
|
2700 |
\isacommand{apply}\ assumption\ \isanewline |
|
2701 |
\ 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 |
|
2702 |
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2703 |
\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B) |
|
2704 |
\isanewline |
|
2705 |
\isacommand{apply}\ assumption\ \isanewline |
|
2706 |
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline |
|
2707 |
\isaindent{\ 1.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B) |
|
2708 |
\end{isabelle} |
|
6121
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changeset
|
2709 |
To construct functions of the form $f\un g$, we apply |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2710 |
\tdx{fun_disjoint_Un}: |
14154 | 2711 |
\begin{isabelle} |
2712 |
\isacommand{apply}\ (rule\ fun\_disjoint\_Un)\isanewline |
|
2713 |
\ 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 |
|
2714 |
\ 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 |
|
2715 |
\ 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 |
|
2716 |
\end{isabelle} |
|
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paulson
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diff
changeset
|
2717 |
The remaining subgoals are instances of the assumptions. Again, observe how |
14154 | 2718 |
unknowns become instantiated: |
2719 |
\begin{isabelle} |
|
2720 |
\isacommand{apply}\ assumption\ \isanewline |
|
2721 |
\ 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 |
|
2722 |
\ 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 |
|
2723 |
\isanewline |
|
2724 |
\isacommand{apply}\ assumption\ \isanewline |
|
2725 |
\ 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 |
|
2726 |
\isanewline |
|
2727 |
\isacommand{apply}\ assumption\ \isanewline |
|
2728 |
No\ subgoals!\isanewline |
|
2729 |
\isacommand{done} |
|
2730 |
\end{isabelle} |
|
2731 |
See the theories \isa{ZF/func.thy} and \isa{ZF/WF.thy} for more |
|
6121
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diff
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|
2732 |
examples of reasoning about functions. |
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2733 |
|
5fe77b9b5185
the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff
changeset
|
2734 |
\index{set theory|)} |