doc-src/ZF/ZF.tex
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log is borel measurable
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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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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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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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\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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\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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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
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constructs of FOL.
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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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Apart from \isa{let}, set theory does not use polymorphism.  All terms in
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ZF have type~\tydx{i}, which is the type of individuals and has
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class~\cldx{term}.  The type of first-order formulae, remember, 
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is~\tydx{o}.
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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
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relations.  Note that $a$\verb|~:|$b$ is translated to $\lnot(a\in b)$,
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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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The constant \cdx{Upair} constructs unordered pairs; thus \isa{Upair($A$,$B$)} denotes the set~$\{A,B\}$ and
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\isa{Upair($A$,$A$)} denotes the singleton~$\{A\}$.  General union is
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used to define binary union.  The Isabelle version goes on to define
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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)) \\
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   \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*}
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 \{a,b,c\} & \equiv & \isa{cons}(a,\isa{cons}(b,\isa{cons}(c,\emptyset)))
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\end{eqnarray*}
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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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In ZF, a function is a set of pairs.  A ZF function~$f$ is simply an
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individual as far as Isabelle is concerned: its Isabelle type is~$i$, not say
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$i\To i$.  The infix operator~{\tt`} denotes the application of a function set
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to its argument; we must write~$f{\tt`}x$, not~$f(x)$.  The syntax for image
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is~$f{\tt``}A$ and that for inverse image is~$f{\tt-``}A$.
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\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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  \ttlbrace$y . x$:$A$, $Q[x,y]$\ttrbrace  &  Replace($A$,$\lambda x\,y. Q[x,y]$) &
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        \rm replacement \\
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  \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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        \rm general intersection \\
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  \sdx{UN}  $x$:$A . B[x]$      & Union(\ttlbrace$B[x] . x$:$A$\ttrbrace) &
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        \rm general union \\
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  \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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  \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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\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 \\
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         & | & 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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\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
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that may contain free occurrences of~$x$.  It abbreviates the set \isa{Collect($A$,$\lambda x. P[x]$)}, which consists of all $x\in A$ that
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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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The constant \cdx{Replace} constructs sets by the principle of {\bf
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  replacement}.  The syntax
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\hbox{\tt\ttlbrace$y$.\ $x$:$A$,$Q[x,y]$\ttrbrace} denotes the set 
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\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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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 $b[x]$
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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
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\isa{\ttlbrace$b[x]$.\ $x$:$A$\ttrbrace}, which expands to 
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\isa{RepFun($A$,$\lambda x. b[x]$)}.
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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 $\bigcap@{x\in A}B[x]$,
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are written \isa{UN $x$:$A$.\ $B[x]$} and \isa{INT $x$:$A$.\ $B[x]$}.
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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
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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
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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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\index{*THE symbol} As mentioned above, whenever the axioms assert the
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existence and uniqueness of a set, Isabelle's set theory declares a constant
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for that set.  These constants can express the {\bf definite description}
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operator~$\iota x. P[x]$, which stands for the unique~$a$ satisfying~$P[a]$,
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if such exists.  Since all terms in ZF denote something, a description is
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always meaningful, but we do not know its value unless $P[x]$ defines it
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uniquely.  Using the constant~\cdx{The}, we may write descriptions as 
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\isa{The($\lambda x. P[x]$)} or use the syntax \isa{THE $x$.\ $P[x]$}.
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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
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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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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
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accordingly.  Instead of \isa{Ball($A$,$P$)} and \isa{Bex($A$,$P$)} we may
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write
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\isa{ALL $x$:$A$.\ $P[x]$} and \isa{EX $x$:$A$.\ $P[x]$}.
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%%%% ZF.thy
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\begin{figure}
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\begin{alltt*}\isastyleminor
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\tdx{Let_def}:           Let(s, f) == f(s)
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\tdx{Ball_def}:          Ball(A,P) == {\isasymforall}x. x \isasymin A --> P(x)
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\tdx{Bex_def}:           Bex(A,P)  == {\isasymexists}x. x \isasymin A & P(x)
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\tdx{subset_def}:        A \isasymsubseteq B  == {\isasymforall}x \isasymin A. x \isasymin B
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\tdx{extension}:         A = B  <->  A \isasymsubseteq B & B \isasymsubseteq A
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\tdx{Union_iff}:         A \isasymin Union(C) <-> ({\isasymexists}B \isasymin C. A \isasymin B)
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\tdx{Pow_iff}:           A \isasymin Pow(B) <-> A \isasymsubseteq B
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\tdx{foundation}:        A=0 | ({\isasymexists}x \isasymin A. {\isasymforall}y \isasymin x. y \isasymnotin A)
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\tdx{replacement}:       ({\isasymforall}x \isasymin A. {\isasymforall}y z. P(x,y) & P(x,z) --> y=z) ==>
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                   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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\tdx{Replace_def}: Replace(A,P) == 
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                   PrimReplace(A, \%x y. (\isasymexists!z. P(x,z)) & P(x,y))
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\tdx{RepFun_def}:  RepFun(A,f)  == {\ttlbrace}y . x \isasymin A, y=f(x)\ttrbrace
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\tdx{the_def}:     The(P)       == Union({\ttlbrace}y . x \isasymin {\ttlbrace}0{\ttrbrace}, P(y){\ttrbrace})
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\tdx{if_def}:      if(P,a,b)    == THE z. P & z=a | ~P & z=b
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\tdx{Collect_def}: Collect(A,P) == {\ttlbrace}y . x \isasymin A, x=y & P(x){\ttrbrace}
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\tdx{Upair_def}:   Upair(a,b)   == 
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               {\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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\tdx{Inter_def}:   Inter(A) == {\ttlbrace}x \isasymin Union(A) . {\isasymforall}y \isasymin A. x \isasymin y{\ttrbrace}
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\tdx{Un_def}:      A \isasymunion B  == Union(Upair(A,B))
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\tdx{Int_def}:     A \isasyminter B  == Inter(Upair(A,B))
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parents: 9836
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   377
\tdx{Diff_def}:    A - B    == {\ttlbrace}x \isasymin A . x \isasymnotin B{\ttrbrace}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
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   378
\subcaption{Union, intersection, difference}
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paulson
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   379
\end{alltt*}
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wenzelm
parents: 9584
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   380
\caption{Rules and axioms of ZF} \label{zf-rules}
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5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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   381
\end{figure}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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   382
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   383
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   384
\begin{figure}
14154
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paulson
parents: 9836
diff changeset
   385
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   386
\tdx{cons_def}:    cons(a,A) == Upair(a,a) \isasymunion A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   387
\tdx{succ_def}:    succ(i) == cons(i,i)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   388
\tdx{infinity}:    0 \isasymin Inf & ({\isasymforall}y \isasymin Inf. succ(y) \isasymin Inf)
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   389
\subcaption{Finite and infinite sets}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   390
14154
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paulson
parents: 9836
diff changeset
   391
\tdx{Pair_def}:      <a,b>      == {\ttlbrace}{\ttlbrace}a,a{\ttrbrace}, {\ttlbrace}a,b{\ttrbrace}{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   392
\tdx{split_def}:     split(c,p) == THE y. {\isasymexists}a b. p=<a,b> & y=c(a,b)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   393
\tdx{fst_def}:       fst(A)     == split(\%x y. x, p)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   394
\tdx{snd_def}:       snd(A)     == split(\%x y. y, p)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   395
\tdx{Sigma_def}:     Sigma(A,B) == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x,y>{\ttrbrace}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   396
\subcaption{Ordered pairs and Cartesian products}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   397
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   398
\tdx{converse_def}: converse(r) == {\ttlbrace}z. w\isasymin{}r, {\isasymexists}x y. w=<x,y> & z=<y,x>{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   399
\tdx{domain_def}:   domain(r)   == {\ttlbrace}x. w \isasymin r, {\isasymexists}y. w=<x,y>{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   400
\tdx{range_def}:    range(r)    == domain(converse(r))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   401
\tdx{field_def}:    field(r)    == domain(r) \isasymunion range(r)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   402
\tdx{image_def}:    r `` A      == {\ttlbrace}y\isasymin{}range(r) . {\isasymexists}x \isasymin A. <x,y> \isasymin r{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   403
\tdx{vimage_def}:   r -`` A     == converse(r)``A
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   404
\subcaption{Operations on relations}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   405
14154
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paulson
parents: 9836
diff changeset
   406
\tdx{lam_def}:   Lambda(A,b) == {\ttlbrace}<x,b(x)> . x \isasymin A{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   407
\tdx{apply_def}: f`a         == THE y. <a,y> \isasymin f
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   409
\tdx{restrict_def}:  restrict(f,A) == lam x \isasymin A. f`x
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   410
\subcaption{Functions and general product}
14154
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paulson
parents: 9836
diff changeset
   411
\end{alltt*}
9695
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
   412
\caption{Further definitions of ZF} \label{zf-defs}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   413
\end{figure}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   414
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   415
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   416
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   417
\section{The Zermelo-Fraenkel axioms}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   418
The axioms appear in Fig.\ts \ref{zf-rules}.  They resemble those
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   419
presented by Suppes~\cite{suppes72}.  Most of the theory consists of
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   420
definitions.  In particular, bounded quantifiers and the subset relation
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   421
appear in other axioms.  Object-level quantifiers and implications have
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   422
been replaced by meta-level ones wherever possible, to simplify use of the
14154
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paulson
parents: 9836
diff changeset
   423
axioms.  
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   424
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   425
The traditional replacement axiom asserts
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   426
\[ y \in \isa{PrimReplace}(A,P) \bimp (\exists x\in A. P(x,y)) \]
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   427
subject to the condition that $P(x,y)$ is single-valued for all~$x\in A$.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   428
The Isabelle theory defines \cdx{Replace} to apply
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   429
\cdx{PrimReplace} to the single-valued part of~$P$, namely
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   430
\[ (\exists!z. P(x,z)) \conj P(x,y). \]
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   431
Thus $y\in \isa{Replace}(A,P)$ if and only if there is some~$x$ such that
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   432
$P(x,-)$ holds uniquely for~$y$.  Because the equivalence is unconditional,
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   433
\isa{Replace} is much easier to use than \isa{PrimReplace}; it defines the
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   434
same set, if $P(x,y)$ is single-valued.  The nice syntax for replacement
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   435
expands to \isa{Replace}.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   436
14158
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paulson
parents: 14154
diff changeset
   437
Other consequences of replacement include replacement for 
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   438
meta-level functions
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   439
(\cdx{RepFun}) and definite descriptions (\cdx{The}).
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   440
Axioms for separation (\cdx{Collect}) and unordered pairs
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   441
(\cdx{Upair}) are traditionally assumed, but they actually follow
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
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
paulson
parents:
diff changeset
   443
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   444
The definitions of general intersection, etc., are straightforward.  Note
14154
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paulson
parents: 9836
diff changeset
   445
the definition of \isa{cons}, which underlies the finite set notation.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   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
paulson
parents:
diff 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
paulson
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:
diff 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
paulson
parents:
diff changeset
   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
paulson
parents:
diff changeset
   453
that \cdx{Sigma}$(A,B)$ generalizes the Cartesian product of two
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   454
sets.  It is defined to be the union of all singleton sets
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
paulson
parents:
diff changeset
   456
general union.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   457
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   458
The projections \cdx{fst} and~\cdx{snd} are defined in terms of the
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   459
generalized projection \cdx{split}.  The latter has been borrowed from
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   460
Martin-L\"of's Type Theory, and is often easier to use than \cdx{fst}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   461
and~\cdx{snd}.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   462
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   463
Operations on relations include converse, domain, range, and image.  The
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   464
set $\isa{Pi}(A,B)$ generalizes the space of functions between two sets.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   465
Note the simple definitions of $\lambda$-abstraction (using
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   466
\cdx{RepFun}) and application (using a definite description).  The
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   467
function \cdx{restrict}$(f,A)$ has the same values as~$f$, but only
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   468
over the domain~$A$.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   469
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   470
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   471
%%%% zf.thy
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   472
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   473
\begin{figure}
14154
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paulson
parents: 9836
diff changeset
   474
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   475
\tdx{ballI}:     [| !!x. x\isasymin{}A ==> P(x) |] ==> {\isasymforall}x\isasymin{}A. P(x)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   476
\tdx{bspec}:     [| {\isasymforall}x\isasymin{}A. P(x);  x\isasymin{}A |] ==> P(x)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   477
\tdx{ballE}:     [| {\isasymforall}x\isasymin{}A. P(x);  P(x) ==> Q;  x \isasymnotin A ==> Q |] ==> Q
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   478
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   479
\tdx{ball_cong}:  [| A=A';  !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==> 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   480
             ({\isasymforall}x\isasymin{}A. P(x)) <-> ({\isasymforall}x\isasymin{}A'. P'(x))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   481
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   482
\tdx{bexI}:      [| P(x);  x\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   483
\tdx{bexCI}:     [| {\isasymforall}x\isasymin{}A. ~P(x) ==> P(a);  a\isasymin{}A |] ==> {\isasymexists}x\isasymin{}A. P(x)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   484
\tdx{bexE}:      [| {\isasymexists}x\isasymin{}A. P(x);  !!x. [| x\isasymin{}A; P(x) |] ==> Q |] ==> Q
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   485
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   486
\tdx{bex_cong}:  [| A=A';  !!x. x\isasymin{}A' ==> P(x) <-> P'(x) |] ==> 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   487
             ({\isasymexists}x\isasymin{}A. P(x)) <-> ({\isasymexists}x\isasymin{}A'. P'(x))
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   488
\subcaption{Bounded quantifiers}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   489
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   490
\tdx{subsetI}:     (!!x. x \isasymin A ==> x \isasymin B) ==> A \isasymsubseteq B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   491
\tdx{subsetD}:     [| A \isasymsubseteq B;  c \isasymin A |] ==> c \isasymin B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   492
\tdx{subsetCE}:    [| A \isasymsubseteq B;  c \isasymnotin A ==> P;  c \isasymin B ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   493
\tdx{subset_refl}:  A \isasymsubseteq A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   494
\tdx{subset_trans}: [| A \isasymsubseteq B;  B \isasymsubseteq C |] ==> A \isasymsubseteq C
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   495
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   496
\tdx{equalityI}:   [| A \isasymsubseteq B;  B \isasymsubseteq A |] ==> A = B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   497
\tdx{equalityD1}:  A = B ==> A \isasymsubseteq B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   498
\tdx{equalityD2}:  A = B ==> B \isasymsubseteq A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   499
\tdx{equalityE}:   [| A = B;  [| A \isasymsubseteq B; B \isasymsubseteq A |] ==> P |]  ==>  P
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   500
\subcaption{Subsets and extensionality}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   501
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   502
\tdx{emptyE}:        a \isasymin 0 ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   503
\tdx{empty_subsetI}:  0 \isasymsubseteq A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   504
\tdx{equals0I}:      [| !!y. y \isasymin A ==> False |] ==> A=0
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   505
\tdx{equals0D}:      [| A=0;  a \isasymin A |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   506
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   507
\tdx{PowI}:          A \isasymsubseteq B ==> A \isasymin Pow(B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   508
\tdx{PowD}:          A \isasymin Pow(B)  ==>  A \isasymsubseteq B
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   509
\subcaption{The empty set; power sets}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   510
\end{alltt*}
9695
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
   511
\caption{Basic derived rules for ZF} \label{zf-lemmas1}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   512
\end{figure}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   513
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   514
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   515
\section{From basic lemmas to function spaces}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   516
Faced with so many definitions, it is essential to prove lemmas.  Even
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   517
trivial theorems like $A \int B = B \int A$ would be difficult to
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   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
paulson
parents:
diff changeset
   519
rules using a natural deduction style.  Ideally, a natural deduction
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   520
rule should introduce or eliminate just one operator, but this is not
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   521
always practical.  For most operators, we may forget its definition
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   522
and use its derived rules instead.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   523
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   524
\subsection{Fundamental lemmas}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   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:
diff 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 the separate FOL and ZF logics manual, with new material on datatypes and
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
paulson
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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   538
The rules for \cdx{Replace} and \cdx{RepFun} are much simpler than
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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:
diff changeset
   540
separation is proved explicitly, although most proofs should use the
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   559
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   560
\tdx{ReplaceI}:   [| x\isasymin{}A;  P(x,b);  !!y. P(x,y) ==> y=b |] ==> 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   561
            b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   562
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   563
\tdx{ReplaceE}:   [| b\isasymin{}{\ttlbrace}y. x\isasymin{}A, P(x,y){\ttrbrace};  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   564
               !!x. [| x\isasymin{}A; P(x,b); {\isasymforall}y. P(x,y)-->y=b |] ==> R 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   565
            |] ==> R
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   566
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   567
\tdx{RepFunI}:    [| a\isasymin{}A |] ==> f(a)\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   568
\tdx{RepFunE}:    [| b\isasymin{}{\ttlbrace}f(x). x\isasymin{}A{\ttrbrace};  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   569
                !!x.[| x\isasymin{}A;  b=f(x) |] ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   570
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   571
\tdx{separation}:  a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} <-> a\isasymin{}A & P(a)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   572
\tdx{CollectI}:    [| a\isasymin{}A;  P(a) |] ==> a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   573
\tdx{CollectE}:    [| a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace};  [| a\isasymin{}A; P(a) |] ==> R |] ==> R
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   574
\tdx{CollectD1}:   a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> a\isasymin{}A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   575
\tdx{CollectD2}:   a\isasymin{}{\ttlbrace}x\isasymin{}A. P(x){\ttrbrace} ==> P(a)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   576
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   582
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   583
\tdx{UnionI}: [| B\isasymin{}C;  A\isasymin{}B |] ==> A\isasymin{}Union(C)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   584
\tdx{UnionE}: [| A\isasymin{}Union(C);  !!B.[| A\isasymin{}B;  B\isasymin{}C |] ==> R |] ==> R
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   585
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   586
\tdx{InterI}: [| !!x. x\isasymin{}C ==> A\isasymin{}x;  c\isasymin{}C |] ==> A\isasymin{}Inter(C)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   587
\tdx{InterD}: [| A\isasymin{}Inter(C);  B\isasymin{}C |] ==> A\isasymin{}B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   588
\tdx{InterE}: [| A\isasymin{}Inter(C);  A\isasymin{}B ==> R;  B \isasymnotin C ==> R |] ==> R
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   589
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   590
\tdx{UN_I}:   [| a\isasymin{}A;  b\isasymin{}B(a) |] ==> b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   591
\tdx{UN_E}:   [| b\isasymin{}({\isasymUnion}x\isasymin{}A. B(x));  !!x.[| x\isasymin{}A;  b\isasymin{}B(x) |] ==> R 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   592
           |] ==> R
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   593
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   594
\tdx{INT_I}:  [| !!x. x\isasymin{}A ==> b\isasymin{}B(x);  a\isasymin{}A |] ==> b\isasymin{}({\isasymInter}x\isasymin{}A. B(x))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   595
\tdx{INT_E}:  [| b\isasymin{}({\isasymInter}x\isasymin{}A. B(x));  a\isasymin{}A |] ==> b\isasymin{}B(a)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   596
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   601
%%% upair.thy
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   602
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   603
\begin{figure}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   604
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   605
\tdx{pairing}:   a\isasymin{}Upair(b,c) <-> (a=b | a=c)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   606
\tdx{UpairI1}:   a\isasymin{}Upair(a,b)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   607
\tdx{UpairI2}:   b\isasymin{}Upair(a,b)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   608
\tdx{UpairE}:    [| a\isasymin{}Upair(b,c);  a=b ==> P;  a=c ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   609
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   615
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   616
\tdx{UnI1}:      c\isasymin{}A ==> c\isasymin{}A \isasymunion B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   617
\tdx{UnI2}:      c\isasymin{}B ==> c\isasymin{}A \isasymunion B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   618
\tdx{UnCI}:      (c \isasymnotin B ==> c\isasymin{}A) ==> c\isasymin{}A \isasymunion B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   619
\tdx{UnE}:       [| c\isasymin{}A \isasymunion B;  c\isasymin{}A ==> P;  c\isasymin{}B ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   620
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   621
\tdx{IntI}:      [| c\isasymin{}A;  c\isasymin{}B |] ==> c\isasymin{}A \isasyminter B
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   622
\tdx{IntD1}:     c\isasymin{}A \isasyminter B ==> c\isasymin{}A
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   623
\tdx{IntD2}:     c\isasymin{}A \isasyminter B ==> c\isasymin{}B
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   624
\tdx{IntE}:      [| c\isasymin{}A \isasyminter B;  [| c\isasymin{}A; c\isasymin{}B |] ==> P |] ==> P
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   625
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   626
\tdx{DiffI}:     [| c\isasymin{}A;  c \isasymnotin B |] ==> c\isasymin{}A - B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   627
\tdx{DiffD1}:    c\isasymin{}A - B ==> c\isasymin{}A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   628
\tdx{DiffD2}:    c\isasymin{}A - B ==> c  \isasymnotin  B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   629
\tdx{DiffE}:     [| c\isasymin{}A - B;  [| c\isasymin{}A; c \isasymnotin B |] ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   630
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   636
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   637
\tdx{consI1}:    a\isasymin{}cons(a,B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   638
\tdx{consI2}:    a\isasymin{}B ==> a\isasymin{}cons(b,B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   639
\tdx{consCI}:    (a \isasymnotin B ==> a=b) ==> a\isasymin{}cons(b,B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   640
\tdx{consE}:     [| a\isasymin{}cons(b,A);  a=b ==> P;  a\isasymin{}A ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   641
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   642
\tdx{singletonI}:  a\isasymin{}{\ttlbrace}a{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   643
\tdx{singletonE}:  [| a\isasymin{}{\ttlbrace}b{\ttrbrace}; a=b ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   644
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   650
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   651
\tdx{succI1}:    i\isasymin{}succ(i)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   652
\tdx{succI2}:    i\isasymin{}j ==> i\isasymin{}succ(j)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   653
\tdx{succCI}:    (i \isasymnotin j ==> i=j) ==> i\isasymin{}succ(j)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   654
\tdx{succE}:     [| i\isasymin{}succ(j);  i=j ==> P;  i\isasymin{}j ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   655
\tdx{succ_neq_0}:  [| succ(n)=0 |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   656
\tdx{succ_inject}: succ(m) = succ(n) ==> m=n
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   657
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   663
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   664
\tdx{the_equality}: [| P(a); !!x. P(x) ==> x=a |] ==> (THE x. P(x))=a
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   665
\tdx{theI}:         \isasymexists! x. P(x) ==> P(THE x. P(x))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   666
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   667
\tdx{if_P}:          P ==> (if P then a else b) = a
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   668
\tdx{if_not_P}:     ~P ==> (if P then a else b) = b
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   669
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   670
\tdx{mem_asym}:     [| a\isasymin{}b;  b\isasymin{}a |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   671
\tdx{mem_irrefl}:   a\isasymin{}a ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   672
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   690
function, which is defined in terms of~\isa{cons}.  The proof that 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   691
\isa{succ} is injective appears to require the Axiom of Foundation.
6121
5fe77b9b5185 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   704
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   708
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   709
\tdx{Union_upper}:    B\isasymin{}A ==> B \isasymsubseteq Union(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   710
\tdx{Union_least}:    [| !!x. x\isasymin{}A ==> x \isasymsubseteq C |] ==> Union(A) \isasymsubseteq C
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   711
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   712
\tdx{Inter_lower}:    B\isasymin{}A ==> Inter(A) \isasymsubseteq B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   713
\tdx{Inter_greatest}: [| a\isasymin{}A; !!x. x\isasymin{}A ==> C \isasymsubseteq x |] ==> C\isasymsubseteq{}Inter(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   714
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   715
\tdx{Un_upper1}:      A \isasymsubseteq A \isasymunion B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   716
\tdx{Un_upper2}:      B \isasymsubseteq A \isasymunion B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   717
\tdx{Un_least}:       [| A \isasymsubseteq C;  B \isasymsubseteq C |] ==> A \isasymunion B \isasymsubseteq C
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   718
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   719
\tdx{Int_lower1}:     A \isasyminter B \isasymsubseteq A
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   720
\tdx{Int_lower2}:     A \isasyminter B \isasymsubseteq B
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   721
\tdx{Int_greatest}:   [| C \isasymsubseteq A;  C \isasymsubseteq B |] ==> C \isasymsubseteq A \isasyminter B
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   722
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   723
\tdx{Diff_subset}:    A-B \isasymsubseteq A
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   724
\tdx{Diff_contains}:  [| C \isasymsubseteq A;  C \isasyminter B = 0 |] ==> C \isasymsubseteq A-B
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   725
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   726
\tdx{Collect_subset}: Collect(A,P) \isasymsubseteq A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   727
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   739
below presents an example of this, proving the equation 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   740
${\isa{Pow}(A)\cap \isa{Pow}(B)}= \isa{Pow}(A\cap B)$.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   741
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   745
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   746
\tdx{Pair_inject1}: <a,b> = <c,d> ==> a=c
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   747
\tdx{Pair_inject2}: <a,b> = <c,d> ==> b=d
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   748
\tdx{Pair_inject}:  [| <a,b> = <c,d>;  [| a=c; b=d |] ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   749
\tdx{Pair_neq_0}:   <a,b>=0 ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   750
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   751
\tdx{fst_conv}:     fst(<a,b>) = a
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   752
\tdx{snd_conv}:     snd(<a,b>) = b
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   753
\tdx{split}:        split(\%x y. c(x,y), <a,b>) = c(a,b)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   754
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   755
\tdx{SigmaI}:     [| a\isasymin{}A;  b\isasymin{}B(a) |] ==> <a,b>\isasymin{}Sigma(A,B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   756
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   757
\tdx{SigmaE}:     [| c\isasymin{}Sigma(A,B);  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   758
                !!x y.[| x\isasymin{}A; y\isasymin{}B(x); c=<x,y> |] ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   759
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   760
\tdx{SigmaE2}:    [| <a,b>\isasymin{}Sigma(A,B);    
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   761
                [| a\isasymin{}A;  b\isasymin{}B(a) |] ==> P   |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   770
projections and general sums --- in particular, that
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   771
$\{\{a\},\{a,b\}\}$ functions as an ordered pair.  This property is
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   805
\item[Let:] \isa{let {\it pattern} = $t$ in $u$}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   806
\item[Choice:] \isa{THE~{\it pattern}~.~$P$}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   807
\item[Set operations:] \isa{\isasymUnion~{\it pattern}:$A$.~$B$}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   815
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   816
\tdx{domainI}:     <a,b>\isasymin{}r ==> a\isasymin{}domain(r)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   817
\tdx{domainE}:     [| a\isasymin{}domain(r); !!y. <a,y>\isasymin{}r ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   818
\tdx{domain_subset}: domain(Sigma(A,B)) \isasymsubseteq A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   819
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   820
\tdx{rangeI}:      <a,b>\isasymin{}r ==> b\isasymin{}range(r)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   821
\tdx{rangeE}:      [| b\isasymin{}range(r); !!x. <x,b>\isasymin{}r ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   822
\tdx{range_subset}: range(A*B) \isasymsubseteq B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   823
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   824
\tdx{fieldI1}:     <a,b>\isasymin{}r ==> a\isasymin{}field(r)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   825
\tdx{fieldI2}:     <a,b>\isasymin{}r ==> b\isasymin{}field(r)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   826
\tdx{fieldCI}:     (<c,a> \isasymnotin r ==> <a,b>\isasymin{}r) ==> a\isasymin{}field(r)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   827
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   828
\tdx{fieldE}:      [| a\isasymin{}field(r); 
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   829
                !!x. <a,x>\isasymin{}r ==> P; 
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   830
                !!x. <x,a>\isasymin{}r ==> P      
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   831
             |] ==> P
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
   832
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   833
\tdx{field_subset}:  field(A*A) \isasymsubseteq A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   839
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   840
\tdx{imageI}:      [| <a,b>\isasymin{}r; a\isasymin{}A |] ==> b\isasymin{}r``A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   841
\tdx{imageE}:      [| b\isasymin{}r``A; !!x.[| <x,b>\isasymin{}r; x\isasymin{}A |] ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   842
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   843
\tdx{vimageI}:     [| <a,b>\isasymin{}r; b\isasymin{}B |] ==> a\isasymin{}r-``B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   844
\tdx{vimageE}:     [| a\isasymin{}r-``B; !!x.[| <a,x>\isasymin{}r;  x\isasymin{}B |] ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   862
respectively. 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   863
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   864
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   868
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   869
\tdx{fun_is_rel}:     f\isasymin{}Pi(A,B) ==> f \isasymsubseteq Sigma(A,B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   870
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   871
\tdx{apply_equality}:  [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> f`a = b
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   872
\tdx{apply_equality2}: [| <a,b>\isasymin{}f; <a,c>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b=c
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   873
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   874
\tdx{apply_type}:     [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> f`a\isasymin{}B(a)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   875
\tdx{apply_Pair}:     [| f\isasymin{}Pi(A,B); a\isasymin{}A |] ==> <a,f`a>\isasymin{}f
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   876
\tdx{apply_iff}:      f\isasymin{}Pi(A,B) ==> <a,b>\isasymin{}f <-> a\isasymin{}A & f`a = b
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   877
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   878
\tdx{fun_extension}:  [| f\isasymin{}Pi(A,B); g\isasymin{}Pi(A,D);
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   879
                   !!x. x\isasymin{}A ==> f`x = g`x     |] ==> f=g
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   880
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   881
\tdx{domain_type}:    [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> a\isasymin{}A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   882
\tdx{range_type}:     [| <a,b>\isasymin{}f; f\isasymin{}Pi(A,B) |] ==> b\isasymin{}B(a)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   883
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   884
\tdx{Pi_type}:        [| f\isasymin{}A->C; !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==> f\isasymin{}Pi(A,B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   885
\tdx{domain_of_fun}:  f\isasymin{}Pi(A,B) ==> domain(f)=A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   886
\tdx{range_of_fun}:   f\isasymin{}Pi(A,B) ==> f\isasymin{}A->range(f)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   887
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   888
\tdx{restrict}:       a\isasymin{}A ==> restrict(f,A) ` a = f`a
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   889
\tdx{restrict_type}:  [| !!x. x\isasymin{}A ==> f`x\isasymin{}B(x) |] ==> 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   890
                restrict(f,A)\isasymin{}Pi(A,B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   897
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   898
\tdx{lamI}:     a\isasymin{}A ==> <a,b(a)>\isasymin{}(lam x\isasymin{}A. b(x))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   899
\tdx{lamE}:     [| p\isasymin{}(lam x\isasymin{}A. b(x)); !!x.[| x\isasymin{}A; p=<x,b(x)> |] ==> P 
8249
3fc32155372c fixed some overfull lines
paulson
parents: 6745
diff changeset
   900
          |] ==>  P
3fc32155372c fixed some overfull lines
paulson
parents: 6745
diff changeset
   901
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   902
\tdx{lam_type}: [| !!x. x\isasymin{}A ==> b(x)\isasymin{}B(x) |] ==> (lam x\isasymin{}A. b(x))\isasymin{}Pi(A,B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   903
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   904
\tdx{beta}:     a\isasymin{}A ==> (lam x\isasymin{}A. b(x)) ` a = b(a)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   905
\tdx{eta}:      f\isasymin{}Pi(A,B) ==> (lam x\isasymin{}A. f`x) = f
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   912
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   913
\tdx{fun_empty}:           0\isasymin{}0->0
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   914
\tdx{fun_single}:          {\ttlbrace}<a,b>{\ttrbrace}\isasymin{}{\ttlbrace}a{\ttrbrace} -> {\ttlbrace}b{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   915
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   916
\tdx{fun_disjoint_Un}:     [| f\isasymin{}A->B; g\isasymin{}C->D; A \isasyminter C = 0  |] ==>  
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   917
                     (f \isasymunion g)\isasymin{}(A \isasymunion C) -> (B \isasymunion D)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   918
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   919
\tdx{fun_disjoint_apply1}: [| a\isasymin{}A; f\isasymin{}A->B; g\isasymin{}C->D;  A\isasyminter{}C = 0 |] ==>  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   920
                     (f \isasymunion g)`a = f`a
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   921
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   922
\tdx{fun_disjoint_apply2}: [| c\isasymin{}C; f\isasymin{}A->B; g\isasymin{}C->D;  A\isasyminter{}C = 0 |] ==>  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   923
                     (f \isasymunion g)`c = g`c
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   958
\begin{alltt*}\isastyleminor
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   959
\tdx{Int_absorb}:        A \isasyminter A = A
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   960
\tdx{Int_commute}:       A \isasyminter B = B \isasyminter A
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   961
\tdx{Int_assoc}:         (A \isasyminter B) \isasyminter C  =  A \isasyminter (B \isasyminter C)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   962
\tdx{Int_Un_distrib}:    (A \isasymunion B) \isasyminter C  =  (A \isasyminter C) \isasymunion (B \isasyminter C)
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   963
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   964
\tdx{Un_absorb}:         A \isasymunion A = A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   965
\tdx{Un_commute}:        A \isasymunion B = B \isasymunion A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   966
\tdx{Un_assoc}:          (A \isasymunion B) \isasymunion C  =  A \isasymunion (B \isasymunion C)
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   967
\tdx{Un_Int_distrib}:    (A \isasyminter B) \isasymunion C  =  (A \isasymunion C) \isasyminter (B \isasymunion C)
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   968
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   969
\tdx{Diff_cancel}:       A-A = 0
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   970
\tdx{Diff_disjoint}:     A \isasyminter (B-A) = 0
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   971
\tdx{Diff_partition}:    A \isasymsubseteq B ==> A \isasymunion (B-A) = B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   972
\tdx{double_complement}: [| A \isasymsubseteq B; B \isasymsubseteq C |] ==> (B - (C-A)) = A
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   973
\tdx{Diff_Un}:           A - (B \isasymunion C) = (A-B) \isasyminter (A-C)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   974
\tdx{Diff_Int}:          A - (B \isasyminter C) = (A-B) \isasymunion (A-C)
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   975
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   976
\tdx{Union_Un_distrib}:  Union(A \isasymunion B) = Union(A) \isasymunion Union(B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   977
\tdx{Inter_Un_distrib}:  [| a \isasymin A;  b \isasymin B |] ==> 
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   978
                   Inter(A \isasymunion B) = Inter(A) \isasyminter Inter(B)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   979
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   980
\tdx{Int_Union_RepFun}:  A \isasyminter Union(B) = ({\isasymUnion}C \isasymin B. A \isasyminter C)
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   981
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   982
\tdx{Un_Inter_RepFun}:   b \isasymin B ==> 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   983
                   A \isasymunion Inter(B) = ({\isasymInter}C \isasymin B. A \isasymunion C)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   984
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   985
\tdx{SUM_Un_distrib1}:   (SUM x \isasymin A \isasymunion B. C(x)) = 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   986
                   (SUM x \isasymin A. C(x)) \isasymunion (SUM x \isasymin B. C(x))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   987
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   988
\tdx{SUM_Un_distrib2}:   (SUM x \isasymin C. A(x) \isasymunion B(x)) =
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   989
                   (SUM x \isasymin C. A(x)) \isasymunion (SUM x \isasymin C. B(x))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   990
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   991
\tdx{SUM_Int_distrib1}:  (SUM x \isasymin A \isasyminter B. C(x)) =
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   992
                   (SUM x \isasymin A. C(x)) \isasyminter (SUM x \isasymin B. C(x))
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   993
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   994
\tdx{SUM_Int_distrib2}:  (SUM x \isasymin C. A(x) \isasyminter B(x)) =
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
   995
                   (SUM x \isasymin C. A(x)) \isasyminter (SUM x \isasymin C. B(x))
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
   996
\end{alltt*}
6121
5fe77b9b5185 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1005
%  \cdx{cond}   & $[i,i,i]\To i$ &       & conditional for \isa{bool}    \\
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1006
%  \cdx{not}    & $i\To i$       &       & negation for \isa{bool}       \\
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1007
%  \sdx{and}    & $[i,i]\To i$   & Left 70 & conjunction for \isa{bool}  \\
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1008
%  \sdx{or}     & $[i,i]\To i$   & Left 65 & disjunction for \isa{bool}  \\
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1012
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1013
\tdx{bool_def}:      bool == {\ttlbrace}0,1{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1014
\tdx{cond_def}:      cond(b,c,d) == if b=1 then c else d
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1015
\tdx{not_def}:       not(b)  == cond(b,0,1)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1016
\tdx{and_def}:       a and b == cond(a,b,0)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1017
\tdx{or_def}:        a or b  == cond(a,1,b)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1018
\tdx{xor_def}:       a xor b == cond(a,not(b),b)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1019
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1020
\tdx{bool_1I}:       1 \isasymin bool
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1021
\tdx{bool_0I}:       0 \isasymin bool
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1022
\tdx{boolE}:         [| c \isasymin bool;  c=1 ==> P;  c=0 ==> P |] ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1023
\tdx{cond_1}:        cond(1,c,d) = c
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1024
\tdx{cond_0}:        cond(0,c,d) = d
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1025
\end{alltt*}
6121
5fe77b9b5185 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1040
\isa{bool}-valued functions, for example.  The constant~\isa{1} is
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1051
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1052
\tdx{sum_def}:   A+B == {\ttlbrace}0{\ttrbrace}*A \isasymunion {\ttlbrace}1{\ttrbrace}*B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1053
\tdx{Inl_def}:   Inl(a) == <0,a>
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1054
\tdx{Inr_def}:   Inr(b) == <1,b>
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1055
\tdx{case_def}:  case(c,d,u) == split(\%y z. cond(y, d(z), c(z)), u)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1056
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1057
\tdx{InlI}:      a \isasymin A ==> Inl(a) \isasymin A+B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1058
\tdx{InrI}:      b \isasymin B ==> Inr(b) \isasymin A+B
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1059
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1060
\tdx{Inl_inject}:  Inl(a)=Inl(b) ==> a=b
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1061
\tdx{Inr_inject}:  Inr(a)=Inr(b) ==> a=b
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1062
\tdx{Inl_neq_Inr}: Inl(a)=Inr(b) ==> P
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1063
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1064
\tdx{sum_iff}:  u \isasymin A+B <-> ({\isasymexists}x\isasymin{}A. u=Inl(x)) | ({\isasymexists}y\isasymin{}B. u=Inr(y))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1065
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1066
\tdx{case_Inl}:  case(c,d,Inl(a)) = c(a)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1067
\tdx{case_Inr}:  case(c,d,Inr(b)) = d(b)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1081
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1082
\tdx{QPair_def}:      <a;b> == a+b
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1083
\tdx{qsplit_def}:     qsplit(c,p)  == THE y. {\isasymexists}a b. p=<a;b> & y=c(a,b)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1084
\tdx{qfsplit_def}:    qfsplit(R,z) == {\isasymexists}x y. z=<x;y> & R(x,y)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1085
\tdx{qconverse_def}:  qconverse(r) == {\ttlbrace}z. w \isasymin r, {\isasymexists}x y. w=<x;y> & z=<y;x>{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1086
\tdx{QSigma_def}:     QSigma(A,B)  == {\isasymUnion}x \isasymin A. {\isasymUnion}y \isasymin B(x). {\ttlbrace}<x;y>{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1087
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1088
\tdx{qsum_def}:       A <+> B      == ({\ttlbrace}0{\ttrbrace} <*> A) \isasymunion ({\ttlbrace}1{\ttrbrace} <*> B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1089
\tdx{QInl_def}:       QInl(a)      == <0;a>
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1090
\tdx{QInr_def}:       QInr(b)      == <1;b>
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1091
\tdx{qcase_def}:      qcase(c,d)   == qsplit(\%y z. cond(y, d(z), c(z)))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
c120262044b6 Now uses manual.bib; some references updated
paulson
parents: 6173
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1111
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1112
\tdx{bnd_mono_def}:  bnd_mono(D,h) == 
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1113
               h(D)\isasymsubseteq{}D & ({\isasymforall}W X. W\isasymsubseteq{}X --> X\isasymsubseteq{}D --> h(W)\isasymsubseteq{}h(X))
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1114
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1115
\tdx{lfp_def}:       lfp(D,h) == Inter({\ttlbrace}X \isasymin Pow(D). h(X) \isasymsubseteq X{\ttrbrace})
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1116
\tdx{gfp_def}:       gfp(D,h) == Union({\ttlbrace}X \isasymin Pow(D). X \isasymsubseteq h(X){\ttrbrace})
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1117
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1118
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1119
\tdx{lfp_lowerbound}: [| h(A) \isasymsubseteq A;  A \isasymsubseteq D |] ==> lfp(D,h) \isasymsubseteq A
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1120
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1121
\tdx{lfp_subset}:    lfp(D,h) \isasymsubseteq D
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1122
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1123
\tdx{lfp_greatest}:  [| bnd_mono(D,h);  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1124
                  !!X. [| h(X) \isasymsubseteq X;  X \isasymsubseteq D |] ==> A \isasymsubseteq X 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1125
               |] ==> A \isasymsubseteq lfp(D,h)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1126
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1127
\tdx{lfp_Tarski}:    bnd_mono(D,h) ==> lfp(D,h) = h(lfp(D,h))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1128
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1129
\tdx{induct}:        [| a \isasymin lfp(D,h);  bnd_mono(D,h);
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1133
\tdx{lfp_mono}:      [| bnd_mono(D,h);  bnd_mono(E,i);
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1134
                  !!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X)  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1135
               |] ==> lfp(D,h) \isasymsubseteq lfp(E,i)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1136
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1137
\tdx{gfp_upperbound}: [| A \isasymsubseteq h(A);  A \isasymsubseteq D |] ==> A \isasymsubseteq gfp(D,h)
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1138
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1139
\tdx{gfp_subset}:    gfp(D,h) \isasymsubseteq D
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1140
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1141
\tdx{gfp_least}:     [| bnd_mono(D,h);  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1142
                  !!X. [| X \isasymsubseteq h(X);  X \isasymsubseteq D |] ==> X \isasymsubseteq A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1143
               |] ==> gfp(D,h) \isasymsubseteq A
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1144
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1145
\tdx{gfp_Tarski}:    bnd_mono(D,h) ==> gfp(D,h) = h(gfp(D,h))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1146
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1147
\tdx{coinduct}:      [| bnd_mono(D,h); a \isasymin X; X \isasymsubseteq h(X \isasymunion gfp(D,h)); X \isasymsubseteq D 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1148
               |] ==> a \isasymin gfp(D,h)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1149
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1150
\tdx{gfp_mono}:      [| bnd_mono(D,h);  D \isasymsubseteq E;
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1151
                  !!X. X \isasymsubseteq D ==> h(X) \isasymsubseteq i(X)  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1152
               |] ==> gfp(D,h) \isasymsubseteq gfp(E,i)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
74e8f703f5f2 tuned manual.bib;
wenzelm
parents: 6592
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1188
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1189
\tdx{Fin.emptyI}      0 \isasymin Fin(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1193
    [| b \isasymin Fin(A);
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1194
       P(0);
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1195
       !!x y. [| x\isasymin{}A; y\isasymin{}Fin(A); x\isasymnotin{}y; P(y) |] ==> P(cons(x,y))
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1196
    |] ==> P(b)
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1197
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1198
\tdx{Fin_mono}:       A \isasymsubseteq B ==> Fin(A) \isasymsubseteq Fin(B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1199
\tdx{Fin_UnI}:        [| b \isasymin Fin(A);  c \isasymin Fin(A) |] ==> b \isasymunion c \isasymin Fin(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1200
\tdx{Fin_UnionI}:     C \isasymin Fin(Fin(A)) ==> Union(C) \isasymin Fin(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1201
\tdx{Fin_subset}:     [| c \isasymsubseteq b;  b \isasymin Fin(A) |] ==> c \isasymin Fin(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1202
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1203
\caption{The finite set operator} \label{zf-fin}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1204
\end{figure}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1205
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1206
\begin{figure}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1207
\begin{constants}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1208
  \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
  1209
  \cdx{list}    & $i\To i$      && lists over some set\\
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1212
  \cdx{length}  & $i\To i$              &       & length of a list\\
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1213
  \cdx{rev}     & $i\To i$              &       & reverse of a list\\
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
paulson
parents:
diff changeset
  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
paulson
parents:
diff changeset
  1216
\end{constants}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1217
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1218
\underscoreon %%because @ is used here
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1219
\begin{alltt*}\isastyleminor
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1220
\tdx{NilI}:       Nil \isasymin list(A)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1221
\tdx{ConsI}:      [| a \isasymin A;  l \isasymin list(A) |] ==> Cons(a,l) \isasymin list(A)
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1222
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1223
\tdx{List.induct}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1224
    [| l \isasymin list(A);
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1225
       P(Nil);
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1226
       !!x y. [| x \isasymin A;  y \isasymin list(A);  P(y) |] ==> P(Cons(x,y))
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1227
    |] ==> P(l)
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1228
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1229
\tdx{Cons_iff}:       Cons(a,l)=Cons(a',l') <-> a=a' & l=l'
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1230
\tdx{Nil_Cons_iff}:    Nil \isasymnoteq Cons(a,l)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1231
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1232
\tdx{list_mono}:      A \isasymsubseteq B ==> list(A) \isasymsubseteq list(B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1233
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1234
\tdx{map_ident}:      l\isasymin{}list(A) ==> map(\%u. u, l) = l
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1235
\tdx{map_compose}:    l\isasymin{}list(A) ==> map(h, map(j,l)) = map(\%u. h(j(u)), l)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  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
paulson
parents:
diff changeset
  1237
\tdx{map_type}
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1238
    [| l\isasymin{}list(A); !!x. x\isasymin{}A ==> h(x)\isasymin{}B |] ==> map(h,l)\isasymin{}list(B)
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1241
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1242
\caption{Lists} \label{zf-list}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1243
\end{figure}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1244
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1245
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1246
Figure~\ref{zf-list} presents the set of lists over~$A$, $\isa{list}(A)$.  The
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
paulson
parents:
diff changeset
  1250
list functions by primitive recursion.  See theory \texttt{List}.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1251
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1252
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1253
\subsection{Miscellaneous}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1254
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1255
\begin{figure}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1256
\begin{constants} 
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1257
  \it symbol  & \it meta-type & \it priority & \it description \\ 
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1258
  \sdx{O}       & $[i,i]\To i$  &  Right 60     & composition ($\circ$) \\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1259
  \cdx{id}      & $i\To i$      &       & identity function \\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1260
  \cdx{inj}     & $[i,i]\To i$  &       & injective function space\\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1261
  \cdx{surj}    & $[i,i]\To i$  &       & surjective function space\\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1262
  \cdx{bij}     & $[i,i]\To i$  &       & bijective function space
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1263
\end{constants}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1264
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1265
\begin{alltt*}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1266
\tdx{comp_def}: r O s     == {\ttlbrace}xz \isasymin domain(s)*range(r) . 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1267
                        {\isasymexists}x y z. xz=<x,z> & <x,y> \isasymin s & <y,z> \isasymin r{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1268
\tdx{id_def}:   id(A)     == (lam x \isasymin A. x)
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  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}
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1270
\tdx{surj_def}: surj(A,B) == {\ttlbrace} f\isasymin{}A->B . {\isasymforall}y\isasymin{}B. {\isasymexists}x\isasymin{}A. f`x=y {\ttrbrace}
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1271
\tdx{bij_def}:  bij(A,B)  == inj(A,B) \isasyminter surj(A,B)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1272
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1273
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1274
\tdx{left_inverse}:    [| f\isasymin{}inj(A,B);  a\isasymin{}A |] ==> converse(f)`(f`a) = a
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1275
\tdx{right_inverse}:   [| f\isasymin{}inj(A,B);  b\isasymin{}range(f) |] ==> 
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1276
                 f`(converse(f)`b) = b
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1277
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1278
\tdx{inj_converse_inj}: f\isasymin{}inj(A,B) ==> converse(f) \isasymin inj(range(f),A)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1279
\tdx{bij_converse_bij}: f\isasymin{}bij(A,B) ==> converse(f) \isasymin bij(B,A)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1280
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1281
\tdx{comp_type}:     [| s \isasymsubseteq A*B;  r \isasymsubseteq B*C |] ==> (r O s) \isasymsubseteq A*C
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1282
\tdx{comp_assoc}:    (r O s) O t = r O (s O t)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1283
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1284
\tdx{left_comp_id}:  r \isasymsubseteq A*B ==> id(B) O r = r
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1285
\tdx{right_comp_id}: r \isasymsubseteq A*B ==> r O id(A) = r
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1286
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1287
\tdx{comp_func}:     [| g\isasymin{}A->B; f\isasymin{}B->C |] ==> (f O g) \isasymin A->C
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1288
\tdx{comp_func_apply}: [| g\isasymin{}A->B; f\isasymin{}B->C; a\isasymin{}A |] ==> (f O g)`a = f`(g`a)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1289
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1290
\tdx{comp_inj}:      [| g\isasymin{}inj(A,B);  f\isasymin{}inj(B,C)  |] ==> (f O g)\isasymin{}inj(A,C)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1291
\tdx{comp_surj}:     [| g\isasymin{}surj(A,B); f\isasymin{}surj(B,C) |] ==> (f O g)\isasymin{}surj(A,C)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1292
\tdx{comp_bij}:      [| g\isasymin{}bij(A,B); f\isasymin{}bij(B,C) |] ==> (f O g)\isasymin{}bij(A,C)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1293
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1294
\tdx{left_comp_inverse}:    f\isasymin{}inj(A,B) ==> converse(f) O f = id(A)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1295
\tdx{right_comp_inverse}:   f\isasymin{}surj(A,B) ==> f O converse(f) = id(B)
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1296
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1297
\tdx{bij_disjoint_Un}:  
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1298
    [| f\isasymin{}bij(A,B);  g\isasymin{}bij(C,D);  A \isasyminter C = 0;  B \isasyminter D = 0 |] ==> 
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1299
    (f \isasymunion g)\isasymin{}bij(A \isasymunion C, B \isasymunion D)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1300
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1301
\tdx{restrict_bij}: [| f\isasymin{}inj(A,B); C\isasymsubseteq{}A |] ==> restrict(f,C)\isasymin{}bij(C, f``C)
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1302
\end{alltt*}
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1303
\caption{Permutations} \label{zf-perm}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1304
\end{figure}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1305
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1306
The theory \thydx{Perm} is concerned with permutations (bijections) and
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1307
related concepts.  These include composition of relations, the identity
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1308
relation, and three specialized function spaces: injective, surjective and
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1309
bijective.  Figure~\ref{zf-perm} displays many of their properties that
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1310
have been proved.  These results are fundamental to a treatment of
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1311
equipollence and cardinality.
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1312
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1313
Theory \thydx{Univ} defines a `universe' $\isa{univ}(A)$, which is used by
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1314
the datatype package.  This set contains $A$ and the
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1315
natural numbers.  Vitally, it is closed under finite products: 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1316
$\isa{univ}(A)\times\isa{univ}(A)\subseteq\isa{univ}(A)$.  This theory also
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1317
defines the cumulative hierarchy of axiomatic set theory, which
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1318
traditionally is written $V@\alpha$ for an ordinal~$\alpha$.  The
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1319
`universe' is a simple generalization of~$V@\omega$.
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1320
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1321
Theory \thydx{QUniv} defines a `universe' $\isa{quniv}(A)$, which is used by
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1322
the datatype package to construct codatatypes such as streams.  It is
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1323
analogous to $\isa{univ}(A)$ (and is defined in terms of it) but is closed
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1324
under the non-standard product and sum.
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1325
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1326
6173
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1327
\section{Automatic Tools}
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1328
9695
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  1329
ZF provides the simplifier and the classical reasoner.  Moreover it supplies a
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  1330
specialized tool to infer `types' of terms.
6173
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1331
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1332
\subsection{Simplification and Classical Reasoning}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1333
9695
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  1334
ZF inherits simplification from FOL but adopts it for set theory.  The
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  1335
extraction of rewrite rules takes the ZF primitives into account.  It can
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1336
strip bounded universal quantifiers from a formula; for example, ${\forall
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1337
  x\in A. f(x)=g(x)}$ yields the conditional rewrite rule $x\in A \Imp
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1338
f(x)=g(x)$.  Given $a\in\{x\in A. P(x)\}$ it extracts rewrite rules from $a\in
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1339
A$ and~$P(a)$.  It can also break down $a\in A\int B$ and $a\in A-B$.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1340
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1341
The default simpset used by \isa{simp} contains congruence rules for all of ZF's
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1342
binding operators.  It contains all the conversion rules, such as
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1343
\isa{fst} and
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1344
\isa{snd}, as well as the rewrites shown in Fig.\ts\ref{zf-simpdata}.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1345
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1346
Classical reasoner methods such as \isa{blast} and \isa{auto} refer to
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1347
a rich collection of built-in axioms for all the set-theoretic
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1348
primitives.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1349
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1356
  a \in A-B             & \bimp &  a\in A \conj \lnot (a\in B)\\
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1367
\subsection{Type-Checking Tactics}
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1368
\index{type-checking tactics}
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1369
9695
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  1370
Isabelle/ZF provides simple tactics to help automate those proofs that are
6173
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1371
essentially type-checking.  Such proofs are built by applying rules such as
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1372
these:
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1373
\begin{ttbox}\isastyleminor
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1374
[| ?P ==> ?a \isasymin ?A; ~?P ==> ?b \isasymin ?A |] 
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1375
==> (if ?P then ?a else ?b) \isasymin ?A
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1376
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1377
[| ?m \isasymin nat; ?n \isasymin nat |] ==> ?m #+ ?n \isasymin nat
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1378
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1379
?a \isasymin ?A ==> Inl(?a) \isasymin ?A + ?B  
6173
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1380
\end{ttbox}
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1381
In typical applications, the goal has the form $t\in\Var{A}$: in other words,
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1382
we have a specific term~$t$ and need to infer its `type' by instantiating the
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1383
set variable~$\Var{A}$.  Neither the simplifier nor the classical reasoner
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1384
does this job well.  The if-then-else rule, and many similar ones, can make
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1385
the classical reasoner loop.  The simplifier refuses (on principle) to
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1386
instantiate variables during rewriting, so goals such as \isa{i\#+j \isasymin \ ?A}
6173
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1387
are left unsolved.
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1388
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1389
The simplifier calls the type-checker to solve rewritten subgoals: this stage
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1390
can indeed instantiate variables.  If you have defined new constants and
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1391
proved type-checking rules for them, then declare the rules using
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1392
the attribute \isa{TC} and the rest should be automatic.  In
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1393
particular, the simplifier will use type-checking to help satisfy
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1394
conditional rewrite rules. Call the method \ttindex{typecheck} to
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1395
break down all subgoals using type-checking rules. You can add new
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1396
type-checking rules temporarily like this:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1397
\begin{isabelle}
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1398
\isacommand{apply}\ (typecheck add:\ inj_is_fun)
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1399
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1400
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1401
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1402
%Though the easiest way to invoke the type-checker is via the simplifier,
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1403
%specialized applications may require more detailed knowledge of
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1404
%the type-checking primitives.  They are modelled on the simplifier's:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1405
%\begin{ttdescription}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1406
%\item[\ttindexbold{tcset}] is the type of tcsets: sets of type-checking rules.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1407
%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1408
%\item[\ttindexbold{addTCs}] is an infix operator to add type-checking rules to
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1409
%  a tcset.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1410
%  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1411
%\item[\ttindexbold{delTCs}] is an infix operator to remove type-checking rules
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1412
%  from a tcset.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1413
%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1414
%\item[\ttindexbold{typecheck_tac}] is a tactic for attempting to prove all
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1415
%  subgoals using the rules given in its argument, a tcset.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1416
%\end{ttdescription}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1417
%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1418
%Tcsets, like simpsets, are associated with theories and are merged when
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1419
%theories are merged.  There are further primitives that use the default tcset.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1420
%\begin{ttdescription}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1421
%\item[\ttindexbold{tcset}] is a function to return the default tcset; use the
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1422
%  expression \isa{tcset()}.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1423
%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1424
%\item[\ttindexbold{AddTCs}] adds type-checking rules to the default tcset.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1425
%  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1426
%\item[\ttindexbold{DelTCs}] removes type-checking rules from the default
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1427
%  tcset.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1428
%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1429
%\item[\ttindexbold{Typecheck_tac}] calls \isa{typecheck_tac} using the
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1430
%  default tcset.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1431
%\end{ttdescription}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1432
%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1433
%To supply some type-checking rules temporarily, using \isa{Addrules} and
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1434
%later \isa{Delrules} is the simplest way.  There is also a high-tech
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1435
%approach.  Call the simplifier with a new solver expressed using
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1436
%\ttindexbold{type_solver_tac} and your temporary type-checking rules.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1437
%\begin{ttbox}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1438
%by (asm_simp_tac 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1439
%     (simpset() setSolver type_solver_tac (tcset() addTCs prems)) 2);
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1440
%\end{ttbox}
6173
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1441
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1442
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1443
\section{Natural number and integer arithmetic}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1444
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1445
\index{arithmetic|(}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1446
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1447
\begin{figure}\small
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1448
\index{#*@{\tt\#*} symbol}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1449
\index{*div symbol}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1450
\index{*mod symbol}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1451
\index{#+@{\tt\#+} symbol}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1452
\index{#-@{\tt\#-} symbol}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1453
\begin{constants}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1454
  \it symbol  & \it meta-type & \it priority & \it description \\ 
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1455
  \cdx{nat}     & $i$                   &       & set of natural numbers \\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1456
  \cdx{nat_case}& $[i,i\To i,i]\To i$     &     & conditional for $nat$\\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1457
  \tt \#*       & $[i,i]\To i$  &  Left 70      & multiplication \\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1458
  \tt div       & $[i,i]\To i$  &  Left 70      & division\\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1459
  \tt mod       & $[i,i]\To i$  &  Left 70      & modulus\\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1460
  \tt \#+       & $[i,i]\To i$  &  Left 65      & addition\\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1461
  \tt \#-       & $[i,i]\To i$  &  Left 65      & subtraction
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1462
\end{constants}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1463
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1464
\begin{alltt*}\isastyleminor
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1465
\tdx{nat_def}: nat == lfp(lam r \isasymin Pow(Inf). {\ttlbrace}0{\ttrbrace} \isasymunion {\ttlbrace}succ(x). x \isasymin r{\ttrbrace}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1466
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1467
\tdx{nat_case_def}:  nat_case(a,b,k) == 
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1468
              THE y. k=0 & y=a | ({\isasymexists}x. k=succ(x) & y=b(x))
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1469
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1470
\tdx{nat_0I}:           0 \isasymin nat
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1471
\tdx{nat_succI}:        n \isasymin nat ==> succ(n) \isasymin nat
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1472
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1473
\tdx{nat_induct}:        
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1474
    [| n \isasymin nat;  P(0);  !!x. [| x \isasymin nat;  P(x) |] ==> P(succ(x)) 
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1475
    |] ==> P(n)
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1476
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1477
\tdx{nat_case_0}:       nat_case(a,b,0) = a
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1478
\tdx{nat_case_succ}:    nat_case(a,b,succ(m)) = b(m)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1479
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1480
\tdx{add_0_natify}:     0 #+ n = natify(n)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1481
\tdx{add_succ}:         succ(m) #+ n = succ(m #+ n)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1482
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1483
\tdx{mult_type}:        m #* n \isasymin nat
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1484
\tdx{mult_0}:           0 #* n = 0
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1485
\tdx{mult_succ}:        succ(m) #* n = n #+ (m #* n)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1486
\tdx{mult_commute}:     m #* n = n #* m
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1487
\tdx{add_mult_dist}:    (m #+ n) #* k = (m #* k) #+ (n #* k)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1488
\tdx{mult_assoc}:       (m #* n) #* k = m #* (n #* k)
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1489
\tdx{mod_div_equality}: m \isasymin nat ==> (m div n)#*n #+ m mod n = m
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1490
\end{alltt*}
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1491
\caption{The natural numbers} \label{zf-nat}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1492
\end{figure}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1493
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1494
\index{natural numbers}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1495
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1496
Theory \thydx{Nat} defines the natural numbers and mathematical
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1497
induction, along with a case analysis operator.  The set of natural
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1498
numbers, here called \isa{nat}, is known in set theory as the ordinal~$\omega$.
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1499
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1500
Theory \thydx{Arith} develops arithmetic on the natural numbers
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1501
(Fig.\ts\ref{zf-nat}).  Addition, multiplication and subtraction are defined
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1502
by primitive recursion.  Division and remainder are defined by repeated
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1503
subtraction, which requires well-founded recursion; the termination argument
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1504
relies on the divisor's being non-zero.  Many properties are proved:
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1505
commutative, associative and distributive laws, identity and cancellation
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1506
laws, etc.  The most interesting result is perhaps the theorem $a \bmod b +
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1507
(a/b)\times b = a$.
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1508
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1509
To minimize the need for tedious proofs of $t\in\isa{nat}$, the arithmetic
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1510
operators coerce their arguments to be natural numbers.  The function
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1511
\cdx{natify} is defined such that $\isa{natify}(n) = n$ if $n$ is a natural
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1512
number, $\isa{natify}(\isa{succ}(x)) =
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1513
\isa{succ}(\isa{natify}(x))$ for all $x$, and finally
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1514
$\isa{natify}(x)=0$ in all other cases.  The benefit is that the addition,
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1517
associative, distributive) are unconditional.  Occurrences of \isa{natify}
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1518
as operands of those operators are simplified away.  Any remaining occurrences
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1519
can either be tolerated or else eliminated by proving that the argument is a
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1520
natural number.
af21f4364c05 documented the integers and updated section on nat arithmetic
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1524
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1525
 1. i \#+ j \#+ k \#- j < k \#+ l\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1526
\isacommand{apply}\ simp\isanewline
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1527
 1. natify(i) < natify(l)
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1528
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
paulson
parents: 8249
diff changeset
  1530
\cdx{natify} would be simplified away.
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1531
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1532
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1533
\begin{figure}\small
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1534
\index{$*@{\tt\$*} symbol}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1535
\index{$+@{\tt\$+} symbol}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1536
\index{$-@{\tt\$-} symbol}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1537
\begin{constants}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1538
  \it symbol  & \it meta-type & \it priority & \it description \\ 
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1539
  \cdx{int}     & $i$                   &       & set of integers \\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1540
  \tt \$*       & $[i,i]\To i$  &  Left 70      & multiplication \\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1541
  \tt \$+       & $[i,i]\To i$  &  Left 65      & addition\\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1542
  \tt \$-       & $[i,i]\To i$  &  Left 65      & subtraction\\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1543
  \tt \$<       & $[i,i]\To o$  &  Left 50      & $<$ on integers\\
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1544
  \tt \$<=      & $[i,i]\To o$  &  Left 50      & $\le$ on integers
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1545
\end{constants}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1546
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1547
\begin{alltt*}\isastyleminor
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1548
\tdx{zadd_0_intify}:    0 $+ n = intify(n)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1549
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1550
\tdx{zmult_type}:       m $* n \isasymin int
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1551
\tdx{zmult_0}:          0 $* n = 0
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1552
\tdx{zmult_commute}:    m $* n = n $* m
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1553
\tdx{zadd_zmult_dist}:   (m $+ n) $* k = (m $* k) $+ (n $* k)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1554
\tdx{zmult_assoc}:      (m $* n) $* k = m $* (n $* k)
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1555
\end{alltt*}
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
diff changeset
  1556
\caption{The integers} \label{zf-int}
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1579
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1580
 1. y \$+ z \$+ \#-3 \$* x \$+ y \$<=  x \$* \#2 \$+
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1581
z\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1582
\isacommand{apply}\ simp\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1583
 1. \#2 \$* y \$<= \#5 \$* x
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1584
\end{isabelle}
9584
af21f4364c05 documented the integers and updated section on nat arithmetic
paulson
parents: 8249
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
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  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
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  1595
The \ttindex{datatype} definition package of ZF constructs inductive datatypes
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1596
similar to \ML's.  It can also construct coinductive datatypes
9695
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  1597
(codatatypes), which are non-well-founded structures such as streams.  It
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  1598
defines the set using a fixed-point construction and proves induction rules,
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  1599
as well as theorems for recursion and case combinators.  It supplies
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  1600
mechanisms for reasoning about freeness.  The datatype package can handle both
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1641
\begin{alltt*}\isastyleminor
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1642
consts     list :: "i=>i"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1643
datatype  "list(A)" = Nil | Cons ("a \isasymin A", "l \isasymin list(A)")
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1646
However, the package declares the constructors.  Here, \isa{Nil} gets type
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1651
\begin{alltt*}\isastyleminor
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1652
consts   
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1653
  tree :: "i=>i"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1654
  forest :: "i=>i"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1655
  tree_forest :: "i=>i"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1656
datatype  "tree(A)"   = Tcons ("a{\isasymin}A",  "f{\isasymin}forest(A)")
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1657
and "forest(A)" = Fnil | Fcons ("t{\isasymin}tree(A)",  "f{\isasymin}forest(A)")
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1658
\end{alltt*}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1659
Here $\isa{tree}(A)$ is the set of trees over $A$, $\isa{forest}(A)$ is
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1663
The datatype \isa{term}, which is defined by
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1664
\begin{alltt*}\isastyleminor
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1665
consts     term :: "i=>i"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1666
datatype  "term(A)" = Apply ("a \isasymin A", "l \isasymin list(term(A))")
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1669
\end{alltt*}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1670
is an example of nested recursion.  (The theorem \isa{list_mono} is proved
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1671
in theory \isa{List}, and the \isa{term} example is developed in
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1672
theory
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1678
for example $\isa{Nil}\not=\isa{Cons}(a,l)$, and they are injective, for
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1679
example $\isa{Cons}(a,l)=\isa{Cons}(a',l') \bimp a=a' \conj l=l'$.
6121
5fe77b9b5185 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
1eb364a68c54 freeness reasoning: T.free_iffs
paulson
parents: 6141
diff changeset
  1681
datatype package does not prove them.  Instead, it ensures that simplification
1eb364a68c54 freeness reasoning: T.free_iffs
paulson
parents: 6141
diff changeset
  1682
will prove them dynamically: when the simplifier encounters a formula
1eb364a68c54 freeness reasoning: T.free_iffs
paulson
parents: 6141
diff changeset
  1683
asserting the equality of two datatype constructors, it performs freeness
1eb364a68c54 freeness reasoning: T.free_iffs
paulson
parents: 6141
diff changeset
  1684
reasoning.  
1eb364a68c54 freeness reasoning: T.free_iffs
paulson
parents: 6141
diff changeset
  1685
1eb364a68c54 freeness reasoning: T.free_iffs
paulson
parents: 6141
diff changeset
  1686
Freeness reasoning can also be done using the classical reasoner, but it is
1eb364a68c54 freeness reasoning: T.free_iffs
paulson
parents: 6141
diff changeset
  1687
more complicated.  You have to add some safe elimination rules rules to the
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1688
claset.  For the \isa{list} datatype, they are called
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1689
\isa{list.free_elims}.  Occasionally this exposes the underlying
6143
1eb364a68c54 freeness reasoning: T.free_iffs
paulson
parents: 6141
diff changeset
  1690
representation of some constructor, which can be rectified using the command
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1691
\isa{unfold list.con_defs [symmetric]}.
6143
1eb364a68c54 freeness reasoning: T.free_iffs
paulson
parents: 6141
diff changeset
  1692
6121
5fe77b9b5185 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1700
\isa{TF}.  The rule \isa{tree_forest.induct} performs induction over a
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1703
\begin{alltt*}\isastyleminor
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1704
[| x \isasymin tree_forest(A);
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1705
   !!a f. [| a \isasymin A; f \isasymin forest(A); P(f) |] ==> P(Tcons(a, f)); 
8249
3fc32155372c fixed some overfull lines
paulson
parents: 6745
diff changeset
  1706
   P(Fnil);
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1710
\end{alltt*}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1711
The rule \isa{tree_forest.mutual_induct} performs induction over two
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1712
distinct predicates, \isa{P_tree} and \isa{P_forest}.
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1717
   !!f t. [| t{\isasymin}tree(A); P_tree(t); f{\isasymin}forest(A); P_forest(f) |]
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1718
          ==> P_forest(Fcons(t, f)) 
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1719
|] ==> ({\isasymforall}za. za \isasymin tree(A) --> P_tree(za)) &
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1720
    ({\isasymforall}za. za \isasymin forest(A) --> P_forest(za))
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1721
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1722
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1723
For datatypes with nested recursion, such as the \isa{term} example from
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1724
above, things are a bit more complicated.  The rule \isa{term.induct}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1725
refers to the monotonic operator, \isa{list}:
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1726
\begin{alltt*}\isastyleminor
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1727
[| x \isasymin term(A);
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1728
   !!a l. [| a\isasymin{}A; l\isasymin{}list(Collect(term(A), P)) |] ==> P(Apply(a,l)) 
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1729
|] ==> P(x)
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1730
\end{alltt*}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1731
The theory \isa{Induct/Term.thy} derives two higher-level induction rules,
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1732
one of which is particularly useful for proving equations:
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1733
\begin{alltt*}\isastyleminor
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1734
[| t \isasymin term(A);
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1735
   !!x zs. [| x \isasymin A; zs \isasymin list(term(A)); map(f, zs) = map(g, zs) |]
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1736
           ==> f(Apply(x, zs)) = g(Apply(x, zs)) 
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1737
|] ==> f(t) = g(t)  
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1738
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1742
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1743
\subsubsection{The \isa{case} operator}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1744
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1745
The package defines an operator for performing case analysis over the
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1746
datatype.  For \isa{list}, it is called \isa{list_case} and satisfies
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1747
the equations
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1748
\begin{ttbox}\isastyleminor
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1749
list_case(f_Nil, f_Cons, []) = f_Nil
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1750
list_case(f_Nil, f_Cons, Cons(a, l)) = f_Cons(a, l)
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1751
\end{ttbox}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1752
Here \isa{f_Nil} is the value to return if the argument is \isa{Nil} and
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1753
\isa{f_Cons} is a function that computes the value to return if the
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1754
argument has the form $\isa{Cons}(a,l)$.  The function can be expressed as
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1755
an abstraction, over patterns if desired (\S\ref{sec:pairs}).
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1756
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1757
For mutually recursive datatypes, there is a single \isa{case} operator.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1758
In the tree/forest example, the constant \isa{tree_forest_case} handles all
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1759
of the constructors of the two datatypes.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1762
\subsection{Defining datatypes}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1763
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1764
The theory syntax for datatype definitions is shown in
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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 the separate FOL and ZF logics manual, with new material on datatypes and
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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1772
\begin{rail}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1773
datatype : ( 'datatype' | 'codatatype' ) datadecls;
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1778
         ;
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1779
consargs : '(' ('"' var ' : ' term '"' + ',') ')'
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1780
         ;
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1781
\end{rail}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1782
\caption{Syntax of datatype declarations}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1783
\label{datatype-grammar}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1784
\end{figure}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1785
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1786
Codatatypes are declared like datatypes and are identical to them in every
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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:
diff 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:
diff 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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1792
Most of the theorems about datatypes become part of the default simpset.  You
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1793
never need to see them again because the simplifier applies them
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1794
automatically.  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1795
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1796
\subsubsection{Specialized methods for datatypes}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1797
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1798
Induction and case-analysis can be invoked using these special-purpose
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1799
methods:
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1800
\begin{ttdescription}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1801
\item[\methdx{induct_tac} $x$] applies structural
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1802
  induction on variable $x$ to subgoal~1, provided the type of $x$ is a
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1803
  datatype.  The induction variable should not occur among other assumptions
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1804
  of the subgoal.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1805
\end{ttdescription}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1806
% 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1807
% we also have the ind_cases method, but what does it do?
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1808
In some situations, induction is overkill and a case distinction over all
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1809
constructors of the datatype suffices.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1810
\begin{ttdescription}
14202
643fc73e2910 case_tac tweak
paulson
parents: 14158
diff changeset
  1811
\item[\methdx{case_tac} $x$]
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1812
 performs a case analysis for the variable~$x$.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1813
\end{ttdescription}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1814
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1815
Both tactics can only be applied to a variable, whose typing must be given in
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1816
some assumption, for example the assumption \isa{x \isasymin \ list(A)}.  The tactics
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1817
also work for the natural numbers (\isa{nat}) and disjoint sums, although
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1818
these sets were not defined using the datatype package.  (Disjoint sums are
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1819
not recursive, so only \isa{case_tac} is available.)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1820
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1821
Structured Isar methods are also available. Below, $t$ 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1822
stands for the name of the datatype.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1823
\begin{ttdescription}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1824
\item[\methdx{induct} \isa{set:}\ $t$] is the Isar induction tactic.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1825
\item[\methdx{cases} \isa{set:}\ $t$] is the Isar case-analysis tactic.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1826
\end{ttdescription}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1827
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1828
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1829
\subsubsection{The theorems proved by a datatype declaration}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1830
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1831
Here are some more details for the technically minded.  Processing the
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1832
datatype declaration of a set~$t$ produces a name space~$t$ containing
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1833
the following theorems:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1834
\begin{ttbox}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1835
intros          \textrm{the introduction rules}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1836
cases           \textrm{the case analysis rule}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1837
induct          \textrm{the standard induction rule}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1838
mutual_induct   \textrm{the mutual induction rule, if needed}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1839
case_eqns       \textrm{equations for the case operator}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1840
recursor_eqns   \textrm{equations for the recursor}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1841
simps           \textrm{the union of} case_eqns \textrm{and} recursor_eqns
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1842
con_defs        \textrm{definitions of the case operator and constructors}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1843
free_iffs       \textrm{logical equivalences for proving freeness}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1844
free_elims      \textrm{elimination rules for proving freeness}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1845
defs            \textrm{datatype definition(s)}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1846
\end{ttbox}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1847
Furthermore there is the theorem $C$ for every constructor~$C$; for
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1848
example, the \isa{list} datatype's introduction rules are bound to the
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1849
identifiers \isa{Nil} and \isa{Cons}.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1850
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1851
For a codatatype, the component \isa{coinduct} is the coinduction rule,
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1852
replacing the \isa{induct} component.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1853
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1854
See the theories \isa{Induct/Ntree} and \isa{Induct/Brouwer} for examples of
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1855
infinitely branching datatypes.  See theory \isa{Induct/LList} for an example
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1856
of a codatatype.  Some of these theories illustrate the use of additional,
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1857
undocumented features of the datatype package.  Datatype definitions are
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1858
reduced to inductive definitions, and the advanced features should be
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1859
understood in that light.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1860
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1861
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1862
\subsection{Examples}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1863
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1864
\subsubsection{The datatype of binary trees}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1865
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1866
Let us define the set $\isa{bt}(A)$ of binary trees over~$A$.  The theory
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1867
must contain these lines:
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1868
\begin{alltt*}\isastyleminor
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1869
consts   bt :: "i=>i"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1870
datatype "bt(A)" = Lf | Br ("a\isasymin{}A", "t1\isasymin{}bt(A)", "t2\isasymin{}bt(A)")
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1871
\end{alltt*}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1872
After loading the theory, we can prove some theorem.  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1873
We begin by declaring the constructor's typechecking rules
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1874
as simplification rules:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1875
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1876
\isacommand{declare}\ bt.intros\ [simp]%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1877
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1878
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1879
Our first example is the theorem that no tree equals its
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1880
left branch.  To make the inductive hypothesis strong enough, 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1881
the proof requires a quantified induction formula, but 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1882
the \isa{rule\_format} attribute will remove the quantifiers 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1883
before the theorem is stored.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1884
\begin{isabelle}
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1885
\isacommand{lemma}\ Br\_neq\_left\ [rule\_format]:\ "l\isasymin bt(A)\ ==>\ \isasymforall x\ r.\ Br(x,l,r)\isasymnoteq{}l"\isanewline
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1886
\ 1.\ l\ \isasymin \ bt(A)\ \isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1887
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1888
This can be proved by the structural induction tactic:
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1889
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1890
\ \ \isacommand{apply}\ (induct\_tac\ l)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1891
\ 1.\ \isasymforall x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1892
\ 2.\ \isasymAnd a\ t1\ t2.\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1894
\isaindent{\ 2.\ \ \ \ \ \ \ }\isasymforall x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1895
\isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ \isasymforall x\ r.\ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1896
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1897
Both subgoals are proved using \isa{auto}, which performs the necessary
6143
1eb364a68c54 freeness reasoning: T.free_iffs
paulson
parents: 6141
diff changeset
  1898
freeness reasoning. 
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1899
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1900
\ \ \isacommand{apply}\ auto\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1901
No\ subgoals!\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1902
\isacommand{done}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1903
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1904
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1905
An alternative proof uses Isar's fancy \isa{induct} method, which 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1906
automatically quantifies over all free variables:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1907
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1908
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1909
\isacommand{lemma}\ Br\_neq\_left':\ "l\ \isasymin \ bt(A)\ ==>\ (!!x\ r.\ Br(x,\ l,\ r)\ \isasymnoteq \ l)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1910
\ \ \isacommand{apply}\ (induct\ set:\ bt)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1911
\ 1.\ \isasymAnd x\ r.\ Br(x,\ Lf,\ r)\ \isasymnoteq \ Lf\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1912
\ 2.\ \isasymAnd a\ t1\ t2\ x\ r.\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1914
\isaindent{\ 2.\ \ \ \ \ \ \ }\isasymAnd x\ r.\ Br(x,\ t2,\ r)\ \isasymnoteq \ t2\isasymrbrakk \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1915
\isaindent{\ 2.\ \ \ \ }\isasymLongrightarrow \ Br(x,\ Br(a,\ t1,\ t2),\ r)\ \isasymnoteq \ Br(a,\ t1,\ t2)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1916
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1917
Compare the form of the induction hypotheses with the corresponding ones in
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1918
the previous proof. As before, to conclude requires only \isa{auto}.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1919
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1920
When there are only a few constructors, we might prefer to prove the freenness
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1921
theorems for each constructor.  This is simple:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1922
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1923
\isacommand{lemma}\ Br\_iff:\ "Br(a,l,r)\ =\ Br(a',l',r')\ <->\ a=a'\ \&\ l=l'\ \&\ r=r'"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1924
\ \ \isacommand{by}\ (blast\ elim!:\ bt.free\_elims)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1925
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1926
Here we see a demonstration of freeness reasoning using
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1927
\isa{bt.free\_elims}, but simpler still is just to apply \isa{auto}.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1928
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1929
An \ttindex{inductive\_cases} declaration generates instances of the
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1930
case analysis rule that have been simplified  using freeness
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1931
reasoning. 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1932
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1933
\isacommand{inductive\_cases}\ Br\_in\_bt:\ "Br(a,\ l,\ r)\ \isasymin \ bt(A)"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1934
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1935
The theorem just created is 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1936
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1938
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1939
It is an elimination rule that from $\isa{Br}(a,l,r)\in\isa{bt}(A)$
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1940
lets us infer $a\in A$, $l\in\isa{bt}(A)$ and
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1941
$r\in\isa{bt}(A)$.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1942
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1943
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1944
\subsubsection{Mixfix syntax in datatypes}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1945
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1946
Mixfix syntax is sometimes convenient.  The theory \isa{Induct/PropLog} makes a
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1947
deep embedding of propositional logic:
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1948
\begin{alltt*}\isastyleminor
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1949
consts     prop :: i
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1950
datatype  "prop" = Fls
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1951
                 | Var ("n \isasymin nat")                ("#_" [100] 100)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1952
                 | "=>" ("p \isasymin prop", "q \isasymin prop")   (infixr 90)
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1953
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1954
The second constructor has a special $\#n$ syntax, while the third constructor
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1955
is an infixed arrow.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1956
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1957
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1958
\subsubsection{A giant enumeration type}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1959
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1960
This example shows a datatype that consists of 60 constructors:
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1961
\begin{alltt*}\isastyleminor
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1962
consts  enum :: i
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1963
datatype
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1964
  "enum" = C00 | C01 | C02 | C03 | C04 | C05 | C06 | C07 | C08 | C09
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1965
         | C10 | C11 | C12 | C13 | C14 | C15 | C16 | C17 | C18 | C19
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1966
         | C20 | C21 | C22 | C23 | C24 | C25 | C26 | C27 | C28 | C29
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1967
         | C30 | C31 | C32 | C33 | C34 | C35 | C36 | C37 | C38 | C39
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1968
         | C40 | C41 | C42 | C43 | C44 | C45 | C46 | C47 | C48 | C49
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1969
         | C50 | C51 | C52 | C53 | C54 | C55 | C56 | C57 | C58 | C59
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1970
end
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  1971
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1972
The datatype package scales well.  Even though all properties are proved
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1973
rather than assumed, full processing of this definition takes around two seconds
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1974
(on a 1.8GHz machine).  The constructors have a balanced representation,
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1975
related to binary notation, so freeness properties can be proved fast.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1976
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1977
\isacommand{lemma}\ "C00 \isasymnoteq\ C01"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1978
\ \ \isacommand{by}\ simp
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1979
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1980
You need not derive such inequalities explicitly.  The simplifier will
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1981
dispose of them automatically.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1982
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1983
\index{*datatype|)}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1984
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1985
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1986
\subsection{Recursive function definitions}\label{sec:ZF:recursive}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1987
\index{recursive functions|see{recursion}}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1988
\index{*primrec|(}
6173
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  1989
\index{recursion!primitive|(}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
paulson
parents:
diff 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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1995
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  1996
In principle, one could introduce primitive recursive functions by asserting
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1997
their reduction rules as axioms.  Here is a dangerous way of defining a
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1998
recursive function over binary trees:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  1999
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2000
\isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2001
\isacommand{axioms}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2002
\ \ n\_nodes\_Lf:\ "n\_nodes(Lf)\ =\ 0"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2003
\ \ n\_nodes\_Br:\ "n\_nodes(Br(a,l,r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2004
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2005
Asserting axioms brings the danger of accidentally introducing
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2006
contradictions.  It should be avoided whenever possible.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2007
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2008
The \ttindex{primrec} declaration is a safe means of defining primitive
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2009
recursive functions on datatypes:
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2010
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2011
\isacommand{consts}\ \ n\_nodes\ ::\ "i\ =>\ i"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2012
\isacommand{primrec}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2013
\ \ "n\_nodes(Lf)\ =\ 0"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2014
\ \ "n\_nodes(Br(a,\ l,\ r))\ =\ succ(n\_nodes(l)\ \#+\ n\_nodes(r))"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2015
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2016
Isabelle will now derive the two equations from a low-level definition  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2017
based upon well-founded recursion.  If they do not define a legitimate
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2018
recursion, then Isabelle will reject the declaration.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2019
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2020
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2021
\subsubsection{Syntax of recursive definitions}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2022
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2023
The general form of a primitive recursive definition is
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2024
\begin{ttbox}\isastyleminor
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2025
primrec
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2026
    {\it reduction rules}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2027
\end{ttbox}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2028
where \textit{reduction rules} specify one or more equations of the form
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2029
\[ f \, x@1 \, \dots \, x@m \, (C \, y@1 \, \dots \, y@k) \, z@1 \,
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2030
\dots \, z@n = r \] such that $C$ is a constructor of the datatype, $r$
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
paulson
parents:
diff changeset
  2032
calls in $r$ are of the form $f \, \dots \, y@i \, \dots$ for some $i$.  
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2033
There must be at most one reduction rule for each constructor.  The order is
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2039
\isa{rules} section of a theory, will be visible in proof scripts.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2040
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2041
The reduction rules become part of the default simpset, which
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2042
leads to short proof scripts:
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2043
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2044
\isacommand{lemma}\ n\_nodes\_type\ [simp]:\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes(t)\ \isasymin \ nat"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2045
\ \ \isacommand{by}\ (induct\_tac\ t,\ auto)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2046
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2047
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2048
You can even use the \isa{primrec} form with non-recursive datatypes and
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2049
with codatatypes.  Recursion is not allowed, but it provides a convenient
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2050
syntax for defining functions by cases.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2051
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2052
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2053
\subsubsection{Example: varying arguments}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2054
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2055
All arguments, other than the recursive one, must be the same in each equation
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2056
and in each recursive call.  To get around this restriction, use explict
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2057
$\lambda$-abstraction and function application.  For example, let us
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2058
define the tail-recursive version of \isa{n\_nodes}, using an 
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2059
accumulating argument for the counter.  The second argument, $k$, varies in
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2060
recursive calls.
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2061
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2062
\isacommand{consts}\ \ n\_nodes\_aux\ ::\ "i\ =>\ i"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2063
\isacommand{primrec}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2064
\ \ "n\_nodes\_aux(Lf)\ =\ (\isasymlambda k\ \isasymin \ nat.\ k)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2065
\ \ "n\_nodes\_aux(Br(a,l,r))\ =\ \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2066
\ \ \ \ \ \ (\isasymlambda k\ \isasymin \ nat.\ n\_nodes\_aux(r)\ `\ \ (n\_nodes\_aux(l)\ `\ succ(k)))"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2067
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2068
Now \isa{n\_nodes\_aux(t)\ `\ k} is our function in two arguments. We
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2069
can prove a theorem relating it to \isa{n\_nodes}. Note the quantification
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2070
over \isa{k\ \isasymin \ nat}:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2071
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2072
\isacommand{lemma}\ n\_nodes\_aux\_eq\ [rule\_format]:\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2073
\ \ \ \ \ "t\ \isasymin \ bt(A)\ ==>\ \isasymforall k\ \isasymin \ nat.\ n\_nodes\_aux(t)`k\ =\ n\_nodes(t)\ \#+\ k"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2074
\ \ \isacommand{by}\ (induct\_tac\ t,\ simp\_all)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2075
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2076
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2077
Now, we can use \isa{n\_nodes\_aux} to define a tail-recursive version
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2078
of \isa{n\_nodes}:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2079
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2080
\isacommand{constdefs}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2081
\ \ n\_nodes\_tail\ ::\ "i\ =>\ i"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2082
\ \ \ "n\_nodes\_tail(t)\ ==\ n\_nodes\_aux(t)\ `\ 0"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2083
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2084
It is easy to
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2085
prove that \isa{n\_nodes\_tail} is equivalent to \isa{n\_nodes}:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2086
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2087
\isacommand{lemma}\ "t\ \isasymin \ bt(A)\ ==>\ n\_nodes\_tail(t)\ =\ n\_nodes(t)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2088
\ \isacommand{by}\ (simp\ add:\ n\_nodes\_tail\_def\ n\_nodes\_aux\_eq)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2089
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2090
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2091
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2117
proves some theorems.  It behaves identially to the analogous
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2118
inductive definition except that instead of an induction rule there is
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2119
a coinduction rule.  Its treatment of coinduction is described in
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2131
  domains     {\it domain declarations}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2132
  intros      {\it introduction rules}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2133
  monos       {\it monotonicity theorems}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2134
  con_defs    {\it constructor definitions}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2135
  type_intros {\it introduction rules for type-checking}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2139
\isa{co\-inductive}.  
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2140
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2141
The \isa{monos}, \isa{con\_defs}, \isa{type\_intros} and \isa{type\_elims}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2142
sections are optional.  If present, each is specified as a list of
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3fc32155372c fixed some overfull lines
paulson
parents: 6745
diff changeset
  2146
\item[\it domain declarations] are items of the form
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2164
  see theory \isa{Induct/Primrec}.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2165
  
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2193
Below, we shall see how Isabelle/ZF defines the finite powerset
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2194
operator.  The first step is to declare the constant~\isa{Fin}.  Then we
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2195
must declare it inductively, with two introduction rules:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2196
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2197
\isacommand{consts}\ \ Fin\ ::\ "i=>i"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2198
\isacommand{inductive}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2199
\ \ \isakeyword{domains}\ \ \ "Fin(A)"\ \isasymsubseteq\ "Pow(A)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2200
\ \ \isakeyword{intros}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2201
\ \ \ \ emptyI:\ \ "0\ \isasymin\ Fin(A)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2202
\ \ \ \ consI:\ \ \ "[|\ a\ \isasymin\ A;\ \ b\ \isasymin\ Fin(A)\ |]\ ==>\ cons(a,b)\ \isasymin\ Fin(A)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2203
\ \ \isakeyword{type\_intros}\ \ empty\_subsetI\ cons\_subsetI\ PowI\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2204
\ \ \isakeyword{type\_elims}\ \ \ PowD\ [THEN\ revcut\_rl]\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2205
The resulting theory contains a name space, called~\isa{Fin}.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2206
The \isa{Fin}$~A$ introduction rules can be referred to collectively as
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2207
\isa{Fin.intros}, and also individually as \isa{Fin.emptyI} and
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2217
In the example above, $\isa{Pow}(A)$ is given as the domain of
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  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
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  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
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  2232
The type-checking subgoal:
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2233
0 \isasymin Fin(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2237
0 \isasymin Fin(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2242
$\emptyset\in\isa{Pow}(A)$.  Restoring the \isa{type_intros} but not the
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2243
\isa{type_elims}, we again get an error message:
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  2244
\begin{alltt*}\isastyleminor
6173
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  2245
The type-checking subgoal:
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2246
0 \isasymin Fin(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2247
 1. 0 \isasymin Pow(A)
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2248
\ttbreak
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2249
The subgoal after monos, type_elims:
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2250
0 \isasymin Fin(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2251
 1. 0 \isasymin Pow(A)
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2252
\ttbreak
6173
2c0579e8e6fa documented typecheck_tac, etc
paulson
parents: 6143
diff changeset
  2253
The type-checking subgoal:
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2254
cons(a, b) \isasymin Fin(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2255
 1. [| a \isasymin A; b \isasymin Fin(A) |] ==> cons(a, b) \isasymin Pow(A)
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2256
\ttbreak
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2257
The subgoal after monos, type_elims:
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2258
cons(a, b) \isasymin Fin(A)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2259
 1. [| a \isasymin A; b \isasymin Pow(A) |] ==> cons(a, b) \isasymin Pow(A)
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2260
*** prove_goal: tactic failed
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  2261
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
paulson
parents:
diff changeset
  2263
simplest solution to such problems is to prove the failed subgoal separately
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2264
and to supply it under \isa{type_intros}.  The solution actually used is
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2265
to supply, under \isa{type_elims}, a rule that changes
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2266
$b\in\isa{Pow}(A)$ to $b\subseteq A$; together with \isa{cons_subsetI}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2267
and \isa{PowI}, it is enough to complete the type-checking.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2268
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2271
\subsection{Further examples}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2272
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2273
An inductive definition may involve arbitrary monotonic operators.  Here is a
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2274
standard example: the accessible part of a relation.  Note the use
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2275
of~\isa{Pow} in the introduction rule and the corresponding mention of the
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2276
rule \isa{Pow\_mono} in the \isa{monos} list.  If the desired rule has a
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2277
universally quantified premise, usually the effect can be obtained using
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2278
\isa{Pow}.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2279
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2280
\isacommand{consts}\ \ acc\ ::\ "i\ =>\ i"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2281
\isacommand{inductive}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2282
\ \ \isakeyword{domains}\ "acc(r)"\ \isasymsubseteq \ "field(r)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2283
\ \ \isakeyword{intros}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2284
\ \ \ \ vimage:\ \ "[|\ r-``\isacharbraceleft a\isacharbraceright\ \isasymin\ Pow(acc(r));\ a\ \isasymin \ field(r)\ |]
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2285
\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2286
\ \ \ \ \ \ \ \ \ \ \ \ \ \ ==>\ a\ \isasymin \ acc(r)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2287
\ \ \isakeyword{monos}\ \ Pow\_mono
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2288
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2289
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2290
Finally, here are some coinductive definitions.  We begin by defining
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2291
lazy (potentially infinite) lists as a codatatype:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2292
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2293
\isacommand{consts}\ \ llist\ \ ::\ "i=>i"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2294
\isacommand{codatatype}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2295
\ \ "llist(A)"\ =\ LNil\ |\ LCons\ ("a\ \isasymin \ A",\ "l\ \isasymin \ llist(A)")\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2296
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2297
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2298
The notion of equality on such lists is modelled as a bisimulation:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2299
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2300
\isacommand{consts}\ \ lleq\ ::\ "i=>i"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2301
\isacommand{coinductive}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2302
\ \ \isakeyword{domains}\ "lleq(A)"\ <=\ "llist(A)\ *\ llist(A)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2303
\ \ \isakeyword{intros}\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2304
\ \ \ \ LNil:\ \ "<LNil,\ LNil>\ \isasymin \ lleq(A)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2305
\ \ \ \ LCons:\ "[|\ a\ \isasymin \ A;\ <l,l'>\ \isasymin \ lleq(A)\ |]\ \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2306
\ \ \ \ \ \ \ \ \ \ \ \ ==>\ <LCons(a,l),\ LCons(a,l')>\ \isasymin \ lleq(A)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2307
\ \ \isakeyword{type\_intros}\ \ llist.intros
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2308
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2309
This use of \isa{type_intros} is typical: the relation concerns the
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2310
codatatype \isa{llist}, so naturally the introduction rules for that
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2313
The Isabelle distribution contains many other inductive definitions.  Simple
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2314
examples are collected on subdirectory \isa{ZF/Induct}.  The directory
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2315
\isa{Coind} and the theory \isa{ZF/Induct/LList} contain coinductive
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2316
definitions.  Larger examples may be found on other subdirectories of
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2317
\isa{ZF}, such as \isa{IMP}, and \isa{Resid}.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2318
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2319
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2320
\subsection{Theorems generated}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2321
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2322
Each (co)inductive set defined in a theory file generates a name space
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2323
containing the following elements:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2324
\begin{ttbox}\isastyleminor
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2325
intros        \textrm{the introduction rules}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2326
elim          \textrm{the elimination (case analysis) rule}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2327
induct        \textrm{the standard induction rule}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2328
mutual_induct \textrm{the mutual induction rule, if needed}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2329
defs          \textrm{definitions of inductive sets}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2330
bnd_mono      \textrm{monotonicity property}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2331
dom_subset    \textrm{inclusion in `bounding set'}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2332
\end{ttbox}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2333
Furthermore, each introduction rule is available under its declared
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2334
name. For a codatatype, the component \isa{coinduct} is the coinduction rule,
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2335
replacing the \isa{induct} component.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2336
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2337
Recall that the \ttindex{inductive\_cases} declaration generates
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2338
simplified instances of the case analysis rule.  It is as useful for
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2339
inductive definitions as it is for datatypes.  There are many examples
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2340
in the theory
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2341
\isa{Induct/Comb}, which is discussed at length
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2342
elsewhere~\cite{paulson-generic}.  The theory first defines the
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2343
datatype
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2344
\isa{comb} of combinators:
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  2345
\begin{alltt*}\isastyleminor
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2346
consts comb :: i
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2347
datatype  "comb" = K
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2348
                 | S
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2349
                 | "#" ("p \isasymin comb", "q \isasymin comb")   (infixl 90)
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  2350
\end{alltt*}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2351
The theory goes on to define contraction and parallel contraction
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2352
inductively.  Then the theory \isa{Induct/Comb.thy} defines special
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2353
cases of contraction, such as this one:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2354
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2355
\isacommand{inductive\_cases}\ K\_contractE [elim!]:\ "K -1-> r"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2356
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2357
The theorem just created is \isa{K -1-> r \ \isasymLongrightarrow \ Q},
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2358
which expresses that the combinator \isa{K} cannot reduce to
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2359
anything.  (From the assumption \isa{K-1->r}, we can conclude any desired
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2360
formula \isa{Q}\@.)  Similar elimination rules for \isa{S} and application are also
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2361
generated. The attribute \isa{elim!}\ shown above supplies the generated
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2362
theorem to the classical reasoner.  This mode of working allows
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2363
effective reasoniung about operational semantics.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2364
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2365
\index{*coinductive|)} \index{*inductive|)}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
paulson
parents:
diff changeset
  2369
\section{The outer reaches of set theory}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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 changeset
  2375
\begin{itemize}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2376
  \item Theory \isa{Trancl} defines the transitive closure of a relation
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2383
  \item Theory \isa{Ord} defines the notions of transitive set and ordinal
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2389
  \item Theory \isa{Epsilon} derives $\varepsilon$-induction and
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2390
    $\varepsilon$-recursion, which are generalisations of transfinite
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2391
    induction and recursion.  It also defines \cdx{rank}$(x)$, which is the
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2392
    least ordinal $\alpha$ such that $x$ is constructed at stage $\alpha$ of
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2393
    the cumulative hierarchy (thus $x\in V@{\alpha+1}$).
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2399
\item Theory \isa{Rel} defines the basic properties of relations, such as
14158
15bab630ae31 finished conversion to Isar format
paulson
parents: 14154
diff changeset
  2400
  reflexivity, symmetry and transitivity.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2401
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2402
\item Theory \isa{EquivClass} develops a theory of equivalence
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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:
diff changeset
  2404
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2405
\item Theory \isa{Order} defines partial orderings, total orderings and
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2406
  wellorderings.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2407
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2408
\item Theory \isa{OrderArith} defines orderings on sum and product sets.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2412
\item Theory \isa{OrderType} defines order types.  Every wellordering is
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2415
\item Theory \isa{Cardinal} defines equipollence and cardinal numbers.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2416
 
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2417
\item Theory \isa{CardinalArith} defines cardinal addition and
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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:
diff 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
paulson
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2427
\item Theory \isa{AC} asserts the Axiom of Choice and proves some simple
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2428
  equivalent forms.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2429
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2430
\item Theory \isa{Zorn} proves Hausdorff's Maximal Principle, Zorn's Lemma
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2433
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2434
\item Theory \isa{Cardinal\_AC} uses AC to prove simplified theorems about
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2440
\item Theory \isa{InfDatatype} proves theorems to justify infinitely
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2450
Directory \isa{HOL/IMP} contains a mechanised version of a semantic
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2456
The directory \isa{ZF/ex} contains further developments in ZF set theory.
9695
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  2457
Here is an overview; see the files themselves for more details.  I describe
ec7d7f877712 proper setup of iman.sty/extra.sty/ttbox.sty;
wenzelm
parents: 9584
diff changeset
  2458
much of this material in other
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2459
publications~\cite{paulson-set-I,paulson-set-II,paulson-fixedpt-milner}.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2460
\begin{itemize}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2461
\item File \isa{misc.ML} contains miscellaneous examples such as
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2476
\item Theory \isa{Primes} defines the Greatest Common Divisor of two
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2479
\item Theory \isa{Bin} defines a datatype for two's complement binary
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2480
  integers, then proves rewrite rules to perform binary arithmetic.  For
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2481
  instance, $1359\times {-}2468 = {-}3354012$ takes 0.3 seconds.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2482
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2483
\item Theory \isa{BT} defines the recursive data structure $\isa{bt}(A)$, labelled binary trees.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2484
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2485
\item Theory \isa{Term} defines a recursive data structure for terms
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2498
\item Theory \isa{ListN} inductively defines the lists of $n$
6592
c120262044b6 Now uses manual.bib; some references updated
paulson
parents: 6173
diff changeset
  2499
  elements~\cite{paulin-tlca}.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2500
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2501
\item Theory \isa{Acc} inductively defines the accessible part of a
6592
c120262044b6 Now uses manual.bib; some references updated
paulson
parents: 6173
diff changeset
  2502
  relation~\cite{paulin-tlca}.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2503
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2521
from \isa{ZF/mono.ML}:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2522
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2523
\tdx{Pow_mono}:     A \isasymsubseteq B ==> Pow(A) \isasymsubseteq Pow(B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2524
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2527
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2528
\isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2529
\ 1.\ Pow(A\ \isasyminter \ B)\ =\ Pow(A)\ \isasyminter \ Pow(B)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2530
\isacommand{apply}\ (rule\ equalityI)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2531
\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\ \isasyminter \ Pow(B)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2532
\ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2533
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2534
Both inclusions could be tackled straightforwardly using \isa{subsetI}.
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2537
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2538
\isacommand{apply}\ (rule\ Int\_greatest)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2539
\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(A)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2540
\ 2.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2541
\ 3.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2542
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2543
Subgoal~1 follows by applying the monotonicity of \isa{Pow} to $A\int
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2544
B\subseteq A$; 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2546
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2547
\isacommand{apply}\ (rule\ Int\_lower1\ [THEN\ Pow\_mono])\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2548
\ 1.\ Pow(A\ \isasyminter \ B)\ \isasymsubseteq \ Pow(B)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2549
\ 2.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2550
\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2551
\isacommand{apply}\ (rule\ Int\_lower2\ [THEN\ Pow\_mono])\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2552
\ 1.\ Pow(A)\ \isasyminter \ Pow(B)\ \isasymsubseteq \ Pow(A\ \isasyminter \ B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2553
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2556
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2557
\isacommand{apply}\ (rule\ subsetI)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2558
\ 1.\ \isasymAnd x.\ x\ \isasymin \ Pow(A)\ \isasyminter \ Pow(B)\ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2559
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2563
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2564
\isacommand{apply}\ (erule\ IntE)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2565
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ Pow(A\ \isasyminter \ B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2566
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2567
The next step replaces the \isa{Pow} by the subset
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2568
relation~($\subseteq$).\index{*PowI theorem}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2569
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2570
\isacommand{apply}\ (rule\ PowI)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2571
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymin \ Pow(A);\ x\ \isasymin \ Pow(B)\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2572
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2573
We perform the same replacement in the assumptions.  This is a good
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2574
demonstration of the tactic \ttindex{drule}:\index{*PowD theorem}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2575
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2576
\isacommand{apply}\ (drule\ PowD)+\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2577
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\ \isasyminter \ B%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2578
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2581
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2582
\isacommand{apply}\ (rule\ Int\_greatest)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2583
\ 1.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ A\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2584
\ 2.\ \isasymAnd x.\ \isasymlbrakk x\ \isasymsubseteq \ A;\ x\ \isasymsubseteq \ B\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymsubseteq \ B%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2585
\end{isabelle}
6121
5fe77b9b5185 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2587
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2588
\isacommand{apply}\ (assumption+)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2589
\isacommand{done}%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2590
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2591
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2592
We could have performed this proof instantly by calling
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2593
\ttindex{blast}:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2594
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2595
\isacommand{lemma}\ "Pow(A\ Int\ B)\ =\ Pow(A)\ Int\ Pow(B)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2596
\isacommand{by}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2597
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2598
Past researchers regarded this as a difficult proof, as indeed it is if all
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2599
the symbols are replaced by their definitions.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2600
\goodbreak
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2601
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2602
\section{Monotonicity of the union operator}
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2603
For another example, we prove that general union is monotonic:
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2604
${C\subseteq D}$ implies $\bigcup(C)\subseteq \bigcup(D)$.  To begin, we
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2605
tackle the inclusion using \tdx{subsetI}:
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2606
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2607
\isacommand{lemma}\ "C\isasymsubseteq D\ ==>\ Union(C)\
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2608
\isasymsubseteq \ Union(D)"\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2609
\isacommand{apply}\ (rule\ subsetI)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2610
\ 1.\ \isasymAnd x.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ \isasymUnion C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2611
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2612
Big union is like an existential quantifier --- the occurrence in the
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2613
assumptions must be eliminated early, since it creates parameters.
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2614
\index{*UnionE theorem}
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2615
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2616
\isacommand{apply}\ (erule\ UnionE)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2617
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ \isasymUnion D%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2618
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2619
Now we may apply \tdx{UnionI}, which creates an unknown involving the
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2620
parameters.  To show \isa{x\ \isasymin \ \isasymUnion D} it suffices to show that~\isa{x} belongs
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2621
to some element, say~\isa{?B2(x,B)}, of~\isa{D}\@.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2622
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2623
\isacommand{apply}\ (rule\ UnionI)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2624
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ D\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2625
\ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2626
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2627
Combining the rule \tdx{subsetD} with the assumption \isa{C\ \isasymsubseteq \ D} yields 
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2628
$\Var{a}\in C \Imp \Var{a}\in D$, which reduces subgoal~1.  Note that
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2629
\isa{erule} removes the subset assumption.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2630
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2631
\isacommand{apply}\ (erule\ subsetD)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2632
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ ?B2(x,\ B)\ \isasymin \ C\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2633
\ 2.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ ?B2(x,\ B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2634
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2635
The rest is routine.  Observe how the first call to \isa{assumption}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2636
instantiates \isa{?B2(x,B)} to~\isa{B}\@.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2637
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2638
\isacommand{apply}\ assumption\ \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2639
\ 1.\ \isasymAnd x\ B.\ \isasymlbrakk C\ \isasymsubseteq \ D;\ x\ \isasymin \ B;\ B\ \isasymin \ C\isasymrbrakk \ \isasymLongrightarrow \ x\ \isasymin \ B%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2640
\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2641
\isacommand{apply}\ assumption\ \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2642
No\ subgoals!\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2643
\isacommand{done}%
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2644
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2645
Again, \isa{blast} can prove this theorem in one step.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2646
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2647
The theory \isa{ZF/equalities.thy} has many similar proofs.  Reasoning about
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2648
general intersection can be difficult because of its anomalous behaviour on
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2649
the empty set.  However, \isa{blast} copes well with these.  Here is
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
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
\[ a\in C \,\Imp\, \inter@{x\in C} \Bigl(A(x) \int B(x)\Bigr) =        
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2652
       \Bigl(\inter@{x\in C} A(x)\Bigr)  \int  
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2653
       \Bigl(\inter@{x\in C} B(x)\Bigr)  \]
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2656
The derived rules \isa{lamI}, \isa{lamE}, \isa{lam_type}, \isa{beta}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2657
and \isa{eta} support reasoning about functions in a
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2664
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2665
\isacommand{lemma}\ "[|\ a\ \isasymin \ A;\ \ f\ \isasymin \ A->B;\ \ g\ \isasymin \ C->D;\ \ A\ \isasyminter \ C\ =\ 0\ |]
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2666
\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2667
\ \ \ \ \ \ \ \ ==>\ (f\ \isasymunion \ g)`a\ =\ f`a"
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2668
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2669
Using \tdx{apply_equality}, we reduce the equality to reasoning about
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2670
ordered pairs.  The second subgoal is to verify that \isa{f\ \isasymunion \ g} is a function, since
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2671
\isa{Pi(?A,?B)} denotes a dependent function space.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2672
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2673
\isacommand{apply}\ (rule\ apply\_equality)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2674
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2675
\isaindent{\ 1.\ }\isasymLongrightarrow \ \isasymlangle a,\ f\ `\ a\isasymrangle \ \isasymin \ f\ \isasymunion \ g\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2676
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2677
\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2678
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2681
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2682
\isacommand{apply}\ (rule\ UnI1)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2684
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2685
\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2686
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2689
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2690
\isacommand{apply}\ (rule\ apply\_Pair)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2693
\ 3.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2694
\isaindent{\ 3.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2695
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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
paulson
parents:
diff changeset
  2697
from \tdx{apply_Pair}.  Recall that a $\Pi$-set is merely a generalized
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2698
function space, and observe that~{\tt?A2} gets instantiated to~\isa{A}.
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2699
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2700
\isacommand{apply}\ assumption\ \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2702
\ 2.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2703
\isaindent{\ 2.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2704
\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2705
\isacommand{apply}\ assumption\ \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2706
\ 1.\ \isasymlbrakk a\ \isasymin \ A;\ f\ \isasymin \ A\ \isasymrightarrow \ B;\ g\ \isasymin \ C\ \isasymrightarrow \ D;\ A\ \isasyminter \ C\ =\ 0\isasymrbrakk \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2707
\isaindent{\ 1.\ }\isasymLongrightarrow \ f\ \isasymunion \ g\ \isasymin \ Pi(?A,\ ?B)
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2708
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff 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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2711
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2712
\isacommand{apply}\ (rule\ fun\_disjoint\_Un)\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2716
\end{isabelle}
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  2717
The remaining subgoals are instances of the assumptions.  Again, observe how
14154
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2718
unknowns become instantiated:
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2719
\begin{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2720
\isacommand{apply}\ assumption\ \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2723
\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2724
\isacommand{apply}\ assumption\ \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  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
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2726
\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2727
\isacommand{apply}\ assumption\ \isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2728
No\ subgoals!\isanewline
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2729
\isacommand{done}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2730
\end{isabelle}
3bc0128e2c74 partial conversion to Isar format
paulson
parents: 9836
diff changeset
  2731
See the theories \isa{ZF/func.thy} and \isa{ZF/WF.thy} for more
6121
5fe77b9b5185 the separate FOL and ZF logics manual, with new material on datatypes and
paulson
parents:
diff changeset
  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|)}