# HG changeset patch # User wenzelm # Date 1304358924 -7200 # Node ID 07fc82c444d25b9d95d30ad5a7701e2e31912a78 # Parent 613b9b589ca0bd42dfe95cc71537395ad6b691aa discontinued old version of old HOL manual; diff -r 613b9b589ca0 -r 07fc82c444d2 doc-src/Logics/Old_HOL.tex --- a/doc-src/Logics/Old_HOL.tex Mon May 02 17:43:42 2011 +0200 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,1884 +0,0 @@ -%% $Id$ -\chapter{Higher-Order Logic} -\index{higher-order logic|(} -\index{HOL system@{\sc hol} system} - -The theory~\thydx{HOL} implements higher-order logic. It is based on -Gordon's~{\sc hol} system~\cite{mgordon-hol}, which itself is based on -Church's original paper~\cite{church40}. Andrews's book~\cite{andrews86} is a -full description of higher-order logic. Experience with the {\sc hol} system -has demonstrated that higher-order logic is useful for hardware verification; -beyond this, it is widely applicable in many areas of mathematics. It is -weaker than ZF set theory but for most applications this does not matter. If -you prefer {\ML} to Lisp, you will probably prefer HOL to~ZF. - -Previous releases of Isabelle included a different version of~HOL, with -explicit type inference rules~\cite{paulson-COLOG}. This version no longer -exists, but \thydx{ZF} supports a similar style of reasoning. - -HOL has a distinct feel, compared with ZF and CTT. It identifies object-level -types with meta-level types, taking advantage of Isabelle's built-in type -checker. It identifies object-level functions with meta-level functions, so -it uses Isabelle's operations for abstraction and application. There is no -`apply' operator: function applications are written as simply~$f(a)$ rather -than $f{\tt`}a$. - -These identifications allow Isabelle to support HOL particularly nicely, but -they also mean that HOL requires more sophistication from the user --- in -particular, an understanding of Isabelle's type system. Beginners should work -with {\tt show_types} set to {\tt true}. Gain experience by working in -first-order logic before attempting to use higher-order logic. This chapter -assumes familiarity with~FOL. - - -\begin{figure} -\begin{center} -\begin{tabular}{rrr} - \it name &\it meta-type & \it description \\ - \cdx{Trueprop}& $bool\To prop$ & coercion to $prop$\\ - \cdx{not} & $bool\To bool$ & negation ($\neg$) \\ - \cdx{True} & $bool$ & tautology ($\top$) \\ - \cdx{False} & $bool$ & absurdity ($\bot$) \\ - \cdx{if} & $[bool,\alpha,\alpha]\To\alpha::term$ & conditional \\ - \cdx{Inv} & $(\alpha\To\beta)\To(\beta\To\alpha)$ & function inversion\\ - \cdx{Let} & $[\alpha,\alpha\To\beta]\To\beta$ & let binder -\end{tabular} -\end{center} -\subcaption{Constants} - -\begin{center} -\index{"@@{\tt\at} symbol} -\index{*"! symbol}\index{*"? symbol} -\index{*"?"! symbol}\index{*"E"X"! symbol} -\begin{tabular}{llrrr} - \it symbol &\it name &\it meta-type & \it description \\ - \tt\at & \cdx{Eps} & $(\alpha\To bool)\To\alpha::term$ & - Hilbert description ($\epsilon$) \\ - {\tt!~} or \sdx{ALL} & \cdx{All} & $(\alpha::term\To bool)\To bool$ & - universal quantifier ($\forall$) \\ - {\tt?~} or \sdx{EX} & \cdx{Ex} & $(\alpha::term\To bool)\To bool$ & - existential quantifier ($\exists$) \\ - {\tt?!} or {\tt EX!} & \cdx{Ex1} & $(\alpha::term\To bool)\To bool$ & - unique existence ($\exists!$) -\end{tabular} -\end{center} -\subcaption{Binders} - -\begin{center} -\index{*"= symbol} -\index{&@{\tt\&} symbol} -\index{*"| symbol} -\index{*"-"-"> symbol} -\begin{tabular}{rrrr} - \it symbol & \it meta-type & \it priority & \it description \\ - \sdx{o} & $[\beta\To\gamma,\alpha\To\beta]\To (\alpha\To\gamma)$ & - Right 50 & composition ($\circ$) \\ - \tt = & $[\alpha::term,\alpha]\To bool$ & Left 50 & equality ($=$) \\ - \tt < & $[\alpha::ord,\alpha]\To bool$ & Left 50 & less than ($<$) \\ - \tt <= & $[\alpha::ord,\alpha]\To bool$ & Left 50 & - less than or equals ($\leq$)\\ - \tt \& & $[bool,bool]\To bool$ & Right 35 & conjunction ($\conj$) \\ - \tt | & $[bool,bool]\To bool$ & Right 30 & disjunction ($\disj$) \\ - \tt --> & $[bool,bool]\To bool$ & Right 25 & implication ($\imp$) -\end{tabular} -\end{center} -\subcaption{Infixes} -\caption{Syntax of {\tt HOL}} \label{hol-constants} -\end{figure} - - -\begin{figure} -\index{*let symbol} -\index{*in symbol} -\dquotes -\[\begin{array}{rclcl} - term & = & \hbox{expression of class~$term$} \\ - & | & "\at~" id~id^* " . " formula \\ - & | & - \multicolumn{3}{l}{"let"~id~"="~term";"\dots";"~id~"="~term~"in"~term} - \\[2ex] - formula & = & \hbox{expression of type~$bool$} \\ - & | & term " = " term \\ - & | & term " \ttilde= " term \\ - & | & term " < " term \\ - & | & term " <= " term \\ - & | & "\ttilde\ " formula \\ - & | & formula " \& " formula \\ - & | & formula " | " formula \\ - & | & formula " --> " formula \\ - & | & "!~~~" id~id^* " . " formula - & | & "ALL~" id~id^* " . " formula \\ - & | & "?~~~" id~id^* " . " formula - & | & "EX~~" id~id^* " . " formula \\ - & | & "?!~~" id~id^* " . " formula - & | & "EX!~" id~id^* " . " formula - \end{array} -\] -\caption{Full grammar for HOL} \label{hol-grammar} -\end{figure} - - -\section{Syntax} -The type class of higher-order terms is called~\cldx{term}. Type variables -range over this class by default. The equality symbol and quantifiers are -polymorphic over class {\tt term}. - -Class \cldx{ord} consists of all ordered types; the relations $<$ and -$\leq$ are polymorphic over this class, as are the functions -\cdx{mono}, \cdx{min} and \cdx{max}. Three other -type classes --- \cldx{plus}, \cldx{minus} and \cldx{times} --- permit -overloading of the operators {\tt+}, {\tt-} and {\tt*}. In particular, -{\tt-} is overloaded for set difference and subtraction. -\index{*"+ symbol} -\index{*"- symbol} -\index{*"* symbol} - -Figure~\ref{hol-constants} lists the constants (including infixes and -binders), while Fig.\ts\ref{hol-grammar} presents the grammar of -higher-order logic. Note that $a$\verb|~=|$b$ is translated to -$\neg(a=b)$. - -\begin{warn} - HOL has no if-and-only-if connective; logical equivalence is expressed - using equality. But equality has a high priority, as befitting a - relation, while if-and-only-if typically has the lowest priority. Thus, - $\neg\neg P=P$ abbreviates $\neg\neg (P=P)$ and not $(\neg\neg P)=P$. - When using $=$ to mean logical equivalence, enclose both operands in - parentheses. -\end{warn} - -\subsection{Types}\label{HOL-types} -The type of formulae, \tydx{bool}, belongs to class \cldx{term}; thus, -formulae are terms. The built-in type~\tydx{fun}, which constructs function -types, is overloaded with arity {\tt(term,term)term}. Thus, $\sigma\To\tau$ -belongs to class~{\tt term} if $\sigma$ and~$\tau$ do, allowing quantification -over functions. - -Types in HOL must be non-empty; otherwise the quantifier rules would be -unsound. I have commented on this elsewhere~\cite[\S7]{paulson-COLOG}. - -\index{type definitions} -Gordon's {\sc hol} system supports {\bf type definitions}. A type is -defined by exhibiting an existing type~$\sigma$, a predicate~$P::\sigma\To -bool$, and a theorem of the form $\exists x::\sigma.P(x)$. Thus~$P$ -specifies a non-empty subset of~$\sigma$, and the new type denotes this -subset. New function constants are generated to establish an isomorphism -between the new type and the subset. If type~$\sigma$ involves type -variables $\alpha@1$, \ldots, $\alpha@n$, then the type definition creates -a type constructor $(\alpha@1,\ldots,\alpha@n)ty$ rather than a particular -type. Melham~\cite{melham89} discusses type definitions at length, with -examples. - -Isabelle does not support type definitions at present. Instead, they are -mimicked by explicit definitions of isomorphism functions. The definitions -should be supported by theorems of the form $\exists x::\sigma.P(x)$, but -Isabelle cannot enforce this. - - -\subsection{Binders} -Hilbert's {\bf description} operator~$\epsilon x.P[x]$ stands for some~$a$ -satisfying~$P[a]$, if such exists. Since all terms in HOL denote something, a -description is always meaningful, but we do not know its value unless $P[x]$ -defines it uniquely. We may write descriptions as \cdx{Eps}($P$) or use the -syntax \hbox{\tt \at $x$.$P[x]$}. - -Existential quantification is defined by -\[ \exists x.P(x) \;\equiv\; P(\epsilon x.P(x)). \] -The unique existence quantifier, $\exists!x.P[x]$, is defined in terms -of~$\exists$ and~$\forall$. An Isabelle binder, it admits nested -quantifications. For instance, $\exists!x y.P(x,y)$ abbreviates -$\exists!x. \exists!y.P(x,y)$; note that this does not mean that there -exists a unique pair $(x,y)$ satisfying~$P(x,y)$. - -\index{*"! symbol}\index{*"? symbol}\index{HOL system@{\sc hol} system} -Quantifiers have two notations. As in Gordon's {\sc hol} system, HOL -uses~{\tt!}\ and~{\tt?}\ to stand for $\forall$ and $\exists$. The -existential quantifier must be followed by a space; thus {\tt?x} is an -unknown, while \verb'? x.f(x)=y' is a quantification. Isabelle's usual -notation for quantifiers, \sdx{ALL} and \sdx{EX}, is also available. Both -notations are accepted for input. The {\ML} reference -\ttindexbold{HOL_quantifiers} governs the output notation. If set to {\tt - true}, then~{\tt!}\ and~{\tt?}\ are displayed; this is the default. If set -to {\tt false}, then~{\tt ALL} and~{\tt EX} are displayed. - -All these binders have priority 10. - - -\subsection{The \sdx{let} and \sdx{case} constructions} -Local abbreviations can be introduced by a {\tt let} construct whose -syntax appears in Fig.\ts\ref{hol-grammar}. Internally it is translated into -the constant~\cdx{Let}. It can be expanded by rewriting with its -definition, \tdx{Let_def}. - -HOL also defines the basic syntax -\[\dquotes"case"~e~"of"~c@1~"=>"~e@1~"|" \dots "|"~c@n~"=>"~e@n\] -as a uniform means of expressing {\tt case} constructs. Therefore {\tt - case} and \sdx{of} are reserved words. However, so far this is mere -syntax and has no logical meaning. By declaring translations, you can -cause instances of the {\tt case} construct to denote applications of -particular case operators. The patterns supplied for $c@1$,~\ldots,~$c@n$ -distinguish among the different case operators. For an example, see the -case construct for lists on page~\pageref{hol-list} below. - - -\begin{figure} -\begin{ttbox}\makeatother -\tdx{refl} t = (t::'a) -\tdx{subst} [| s=t; P(s) |] ==> P(t::'a) -\tdx{ext} (!!x::'a. (f(x)::'b) = g(x)) ==> (\%x.f(x)) = (\%x.g(x)) -\tdx{impI} (P ==> Q) ==> P-->Q -\tdx{mp} [| P-->Q; P |] ==> Q -\tdx{iff} (P-->Q) --> (Q-->P) --> (P=Q) -\tdx{selectI} P(x::'a) ==> P(@x.P(x)) -\tdx{True_or_False} (P=True) | (P=False) -\end{ttbox} -\caption{The {\tt HOL} rules} \label{hol-rules} -\end{figure} - - -\begin{figure}\hfuzz=4pt%suppress "Overfull \hbox" message -\begin{ttbox}\makeatother -\tdx{True_def} True == ((\%x::bool.x)=(\%x.x)) -\tdx{All_def} All == (\%P. P = (\%x.True)) -\tdx{Ex_def} Ex == (\%P. P(@x.P(x))) -\tdx{False_def} False == (!P.P) -\tdx{not_def} not == (\%P. P-->False) -\tdx{and_def} op & == (\%P Q. !R. (P-->Q-->R) --> R) -\tdx{or_def} op | == (\%P Q. !R. (P-->R) --> (Q-->R) --> R) -\tdx{Ex1_def} Ex1 == (\%P. ? x. P(x) & (! y. P(y) --> y=x)) - -\tdx{Inv_def} Inv == (\%(f::'a=>'b) y. @x. f(x)=y) -\tdx{o_def} op o == (\%(f::'b=>'c) g (x::'a). f(g(x))) -\tdx{if_def} if == (\%P x y.@z::'a.(P=True --> z=x) & (P=False --> z=y)) -\tdx{Let_def} Let(s,f) == f(s) -\end{ttbox} -\caption{The {\tt HOL} definitions} \label{hol-defs} -\end{figure} - - -\section{Rules of inference} -Figure~\ref{hol-rules} shows the inference rules of~HOL, with their~{\ML} -names. Some of the rules deserve additional comments: -\begin{ttdescription} -\item[\tdx{ext}] expresses extensionality of functions. -\item[\tdx{iff}] asserts that logically equivalent formulae are - equal. -\item[\tdx{selectI}] gives the defining property of the Hilbert - $\epsilon$-operator. It is a form of the Axiom of Choice. The derived rule - \tdx{select_equality} (see below) is often easier to use. -\item[\tdx{True_or_False}] makes the logic classical.\footnote{In - fact, the $\epsilon$-operator already makes the logic classical, as - shown by Diaconescu; see Paulson~\cite{paulson-COLOG} for details.} -\end{ttdescription} - -HOL follows standard practice in higher-order logic: only a few connectives -are taken as primitive, with the remainder defined obscurely -(Fig.\ts\ref{hol-defs}). Gordon's {\sc hol} system expresses the -corresponding definitions \cite[page~270]{mgordon-hol} using -object-equality~({\tt=}), which is possible because equality in higher-order -logic may equate formulae and even functions over formulae. But theory~HOL, -like all other Isabelle theories, uses meta-equality~({\tt==}) for -definitions. - -Some of the rules mention type variables; for -example, {\tt refl} mentions the type variable~{\tt'a}. This allows you to -instantiate type variables explicitly by calling {\tt res_inst_tac}. By -default, explicit type variables have class \cldx{term}. - -Include type constraints whenever you state a polymorphic goal. Type -inference may otherwise make the goal more polymorphic than you intended, -with confusing results. - -\begin{warn} - If resolution fails for no obvious reason, try setting - \ttindex{show_types} to {\tt true}, causing Isabelle to display types of - terms. Possibly set \ttindex{show_sorts} to {\tt true} as well, causing - Isabelle to display sorts. - - \index{unification!incompleteness of} - Where function types are involved, Isabelle's unification code does not - guarantee to find instantiations for type variables automatically. Be - prepared to use \ttindex{res_inst_tac} instead of {\tt resolve_tac}, - possibly instantiating type variables. Setting - \ttindex{Unify.trace_types} to {\tt true} causes Isabelle to report - omitted search paths during unification.\index{tracing!of unification} -\end{warn} - - -\begin{figure} -\begin{ttbox} -\tdx{sym} s=t ==> t=s -\tdx{trans} [| r=s; s=t |] ==> r=t -\tdx{ssubst} [| t=s; P(s) |] ==> P(t::'a) -\tdx{box_equals} [| a=b; a=c; b=d |] ==> c=d -\tdx{arg_cong} x=y ==> f(x)=f(y) -\tdx{fun_cong} f=g ==> f(x)=g(x) -\subcaption{Equality} - -\tdx{TrueI} True -\tdx{FalseE} False ==> P - -\tdx{conjI} [| P; Q |] ==> P&Q -\tdx{conjunct1} [| P&Q |] ==> P -\tdx{conjunct2} [| P&Q |] ==> Q -\tdx{conjE} [| P&Q; [| P; Q |] ==> R |] ==> R - -\tdx{disjI1} P ==> P|Q -\tdx{disjI2} Q ==> P|Q -\tdx{disjE} [| P | Q; P ==> R; Q ==> R |] ==> R - -\tdx{notI} (P ==> False) ==> ~ P -\tdx{notE} [| ~ P; P |] ==> R -\tdx{impE} [| P-->Q; P; Q ==> R |] ==> R -\subcaption{Propositional logic} - -\tdx{iffI} [| P ==> Q; Q ==> P |] ==> P=Q -\tdx{iffD1} [| P=Q; P |] ==> Q -\tdx{iffD2} [| P=Q; Q |] ==> P -\tdx{iffE} [| P=Q; [| P --> Q; Q --> P |] ==> R |] ==> R - -\tdx{eqTrueI} P ==> P=True -\tdx{eqTrueE} P=True ==> P -\subcaption{Logical equivalence} - -\end{ttbox} -\caption{Derived rules for HOL} \label{hol-lemmas1} -\end{figure} - - -\begin{figure} -\begin{ttbox}\makeatother -\tdx{allI} (!!x::'a. P(x)) ==> !x. P(x) -\tdx{spec} !x::'a.P(x) ==> P(x) -\tdx{allE} [| !x.P(x); P(x) ==> R |] ==> R -\tdx{all_dupE} [| !x.P(x); [| P(x); !x.P(x) |] ==> R |] ==> R - -\tdx{exI} P(x) ==> ? x::'a.P(x) -\tdx{exE} [| ? x::'a.P(x); !!x. P(x) ==> Q |] ==> Q - -\tdx{ex1I} [| P(a); !!x. P(x) ==> x=a |] ==> ?! x. P(x) -\tdx{ex1E} [| ?! x.P(x); !!x. [| P(x); ! y. P(y) --> y=x |] ==> R - |] ==> R - -\tdx{select_equality} [| P(a); !!x. P(x) ==> x=a |] ==> (@x.P(x)) = a -\subcaption{Quantifiers and descriptions} - -\tdx{ccontr} (~P ==> False) ==> P -\tdx{classical} (~P ==> P) ==> P -\tdx{excluded_middle} ~P | P - -\tdx{disjCI} (~Q ==> P) ==> P|Q -\tdx{exCI} (! x. ~ P(x) ==> P(a)) ==> ? x.P(x) -\tdx{impCE} [| P-->Q; ~ P ==> R; Q ==> R |] ==> R -\tdx{iffCE} [| P=Q; [| P;Q |] ==> R; [| ~P; ~Q |] ==> R |] ==> R -\tdx{notnotD} ~~P ==> P -\tdx{swap} ~P ==> (~Q ==> P) ==> Q -\subcaption{Classical logic} - -\tdx{if_True} if(True,x,y) = x -\tdx{if_False} if(False,x,y) = y -\tdx{if_P} P ==> if(P,x,y) = x -\tdx{if_not_P} ~ P ==> if(P,x,y) = y -\tdx{expand_if} P(if(Q,x,y)) = ((Q --> P(x)) & (~Q --> P(y))) -\subcaption{Conditionals} -\end{ttbox} -\caption{More derived rules} \label{hol-lemmas2} -\end{figure} - - -Some derived rules are shown in Figures~\ref{hol-lemmas1} -and~\ref{hol-lemmas2}, with their {\ML} names. These include natural rules -for the logical connectives, as well as sequent-style elimination rules for -conjunctions, implications, and universal quantifiers. - -Note the equality rules: \tdx{ssubst} performs substitution in -backward proofs, while \tdx{box_equals} supports reasoning by -simplifying both sides of an equation. - - -\begin{figure} -\begin{center} -\begin{tabular}{rrr} - \it name &\it meta-type & \it description \\ -\index{{}@\verb'{}' symbol} - \verb|{}| & $\alpha\,set$ & the empty set \\ - \cdx{insert} & $[\alpha,\alpha\,set]\To \alpha\,set$ - & insertion of element \\ - \cdx{Collect} & $(\alpha\To bool)\To\alpha\,set$ - & comprehension \\ - \cdx{Compl} & $(\alpha\,set)\To\alpha\,set$ - & complement \\ - \cdx{INTER} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$ - & intersection over a set\\ - \cdx{UNION} & $[\alpha\,set,\alpha\To\beta\,set]\To\beta\,set$ - & union over a set\\ - \cdx{Inter} & $(\alpha\,set)set\To\alpha\,set$ - &set of sets intersection \\ - \cdx{Union} & $(\alpha\,set)set\To\alpha\,set$ - &set of sets union \\ - \cdx{Pow} & $\alpha\,set \To (\alpha\,set)set$ - & powerset \\[1ex] - \cdx{range} & $(\alpha\To\beta )\To\beta\,set$ - & range of a function \\[1ex] - \cdx{Ball}~~\cdx{Bex} & $[\alpha\,set,\alpha\To bool]\To bool$ - & bounded quantifiers \\ - \cdx{mono} & $(\alpha\,set\To\beta\,set)\To bool$ - & monotonicity \\ - \cdx{inj}~~\cdx{surj}& $(\alpha\To\beta )\To bool$ - & injective/surjective \\ - \cdx{inj_onto} & $[\alpha\To\beta ,\alpha\,set]\To bool$ - & injective over subset -\end{tabular} -\end{center} -\subcaption{Constants} - -\begin{center} -\begin{tabular}{llrrr} - \it symbol &\it name &\it meta-type & \it priority & \it description \\ - \sdx{INT} & \cdx{INTER1} & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 & - intersection over a type\\ - \sdx{UN} & \cdx{UNION1} & $(\alpha\To\beta\,set)\To\beta\,set$ & 10 & - union over a type -\end{tabular} -\end{center} -\subcaption{Binders} - -\begin{center} -\index{*"`"` symbol} -\index{*": symbol} -\index{*"<"= symbol} -\begin{tabular}{rrrr} - \it symbol & \it meta-type & \it priority & \it description \\ - \tt `` & $[\alpha\To\beta ,\alpha\,set]\To (\beta\,set)$ - & Left 90 & image \\ - \sdx{Int} & $[\alpha\,set,\alpha\,set]\To\alpha\,set$ - & Left 70 & intersection ($\inter$) \\ - \sdx{Un} & $[\alpha\,set,\alpha\,set]\To\alpha\,set$ - & Left 65 & union ($\union$) \\ - \tt: & $[\alpha ,\alpha\,set]\To bool$ - & Left 50 & membership ($\in$) \\ - \tt <= & $[\alpha\,set,\alpha\,set]\To bool$ - & Left 50 & subset ($\subseteq$) -\end{tabular} -\end{center} -\subcaption{Infixes} -\caption{Syntax of the theory {\tt Set}} \label{hol-set-syntax} -\end{figure} - - -\begin{figure} -\begin{center} \tt\frenchspacing -\index{*"! symbol} -\begin{tabular}{rrr} - \it external & \it internal & \it description \\ - $a$ \ttilde: $b$ & \ttilde($a$ : $b$) & \rm non-membership\\ - \{$a@1$, $\ldots$\} & insert($a@1$, $\ldots$\{\}) & \rm finite set \\ - \{$x$.$P[x]$\} & Collect($\lambda x.P[x]$) & - \rm comprehension \\ - \sdx{INT} $x$:$A$.$B[x]$ & INTER($A$,$\lambda x.B[x]$) & - \rm intersection \\ - \sdx{UN}{\tt\ } $x$:$A$.$B[x]$ & UNION($A$,$\lambda x.B[x]$) & - \rm union \\ - \tt ! $x$:$A$.$P[x]$ or \sdx{ALL} $x$:$A$.$P[x]$ & - Ball($A$,$\lambda x.P[x]$) & - \rm bounded $\forall$ \\ - \sdx{?} $x$:$A$.$P[x]$ or \sdx{EX}{\tt\ } $x$:$A$.$P[x]$ & - Bex($A$,$\lambda x.P[x]$) & \rm bounded $\exists$ -\end{tabular} -\end{center} -\subcaption{Translations} - -\dquotes -\[\begin{array}{rclcl} - term & = & \hbox{other terms\ldots} \\ - & | & "\{\}" \\ - & | & "\{ " term\; ("," term)^* " \}" \\ - & | & "\{ " id " . " formula " \}" \\ - & | & term " `` " term \\ - & | & term " Int " term \\ - & | & term " Un " term \\ - & | & "INT~~" id ":" term " . " term \\ - & | & "UN~~~" id ":" term " . " term \\ - & | & "INT~~" id~id^* " . " term \\ - & | & "UN~~~" id~id^* " . " term \\[2ex] - formula & = & \hbox{other formulae\ldots} \\ - & | & term " : " term \\ - & | & term " \ttilde: " term \\ - & | & term " <= " term \\ - & | & "!~" id ":" term " . " formula - & | & "ALL " id ":" term " . " formula \\ - & | & "?~" id ":" term " . " formula - & | & "EX~~" id ":" term " . " formula - \end{array} -\] -\subcaption{Full Grammar} -\caption{Syntax of the theory {\tt Set} (continued)} \label{hol-set-syntax2} -\end{figure} - - -\section{A formulation of set theory} -Historically, higher-order logic gives a foundation for Russell and -Whitehead's theory of classes. Let us use modern terminology and call them -{\bf sets}, but note that these sets are distinct from those of ZF set theory, -and behave more like ZF classes. -\begin{itemize} -\item -Sets are given by predicates over some type~$\sigma$. Types serve to -define universes for sets, but type checking is still significant. -\item -There is a universal set (for each type). Thus, sets have complements, and -may be defined by absolute comprehension. -\item -Although sets may contain other sets as elements, the containing set must -have a more complex type. -\end{itemize} -Finite unions and intersections have the same behaviour in HOL as they do -in~ZF. In HOL the intersection of the empty set is well-defined, denoting the -universal set for the given type. - - -\subsection{Syntax of set theory}\index{*set type} -HOL's set theory is called \thydx{Set}. The type $\alpha\,set$ is essentially -the same as $\alpha\To bool$. The new type is defined for clarity and to -avoid complications involving function types in unification. Since Isabelle -does not support type definitions (as mentioned in \S\ref{HOL-types}), the -isomorphisms between the two types are declared explicitly. Here they are -natural: {\tt Collect} maps $\alpha\To bool$ to $\alpha\,set$, while \hbox{\tt - op :} maps in the other direction (ignoring argument order). - -Figure~\ref{hol-set-syntax} lists the constants, infixes, and syntax -translations. Figure~\ref{hol-set-syntax2} presents the grammar of the new -constructs. Infix operators include union and intersection ($A\union B$ -and $A\inter B$), the subset and membership relations, and the image -operator~{\tt``}\@. Note that $a$\verb|~:|$b$ is translated to -$\neg(a\in b)$. - -The {\tt\{\ldots\}} notation abbreviates finite sets constructed in the -obvious manner using~{\tt insert} and~$\{\}$: -\begin{eqnarray*} - \{a@1, \ldots, a@n\} & \equiv & - {\tt insert}(a@1,\ldots,{\tt insert}(a@n,\{\})) -\end{eqnarray*} - -The set \hbox{\tt\{$x$.$P[x]$\}} consists of all $x$ (of suitable type) that -satisfy~$P[x]$, where $P[x]$ is a formula that may contain free occurrences -of~$x$. This syntax expands to \cdx{Collect}$(\lambda x.P[x])$. It defines -sets by absolute comprehension, which is impossible in~ZF; the type of~$x$ -implicitly restricts the comprehension. - -The set theory defines two {\bf bounded quantifiers}: -\begin{eqnarray*} - \forall x\in A.P[x] &\hbox{abbreviates}& \forall x. x\in A\imp P[x] \\ - \exists x\in A.P[x] &\hbox{abbreviates}& \exists x. x\in A\conj P[x] -\end{eqnarray*} -The constants~\cdx{Ball} and~\cdx{Bex} are defined -accordingly. Instead of {\tt Ball($A$,$P$)} and {\tt Bex($A$,$P$)} we may -write\index{*"! symbol}\index{*"? symbol} -\index{*ALL symbol}\index{*EX symbol} -% -\hbox{\tt !~$x$:$A$.$P[x]$} and \hbox{\tt ?~$x$:$A$.$P[x]$}. Isabelle's -usual quantifier symbols, \sdx{ALL} and \sdx{EX}, are also accepted -for input. As with the primitive quantifiers, the {\ML} reference -\ttindex{HOL_quantifiers} specifies which notation to use for output. - -Unions and intersections over sets, namely $\bigcup@{x\in A}B[x]$ and -$\bigcap@{x\in A}B[x]$, are written -\sdx{UN}~\hbox{\tt$x$:$A$.$B[x]$} and -\sdx{INT}~\hbox{\tt$x$:$A$.$B[x]$}. - -Unions and intersections over types, namely $\bigcup@x B[x]$ and $\bigcap@x -B[x]$, are written \sdx{UN}~\hbox{\tt$x$.$B[x]$} and -\sdx{INT}~\hbox{\tt$x$.$B[x]$}. They are equivalent to the previous -union and intersection operators when $A$ is the universal set. - -The operators $\bigcup A$ and $\bigcap A$ act upon sets of sets. They are -not binders, but are equal to $\bigcup@{x\in A}x$ and $\bigcap@{x\in A}x$, -respectively. - - -\begin{figure} \underscoreon -\begin{ttbox} -\tdx{mem_Collect_eq} (a : \{x.P(x)\}) = P(a) -\tdx{Collect_mem_eq} \{x.x:A\} = A - -\tdx{empty_def} \{\} == \{x.False\} -\tdx{insert_def} insert(a,B) == \{x.x=a\} Un B -\tdx{Ball_def} Ball(A,P) == ! x. x:A --> P(x) -\tdx{Bex_def} Bex(A,P) == ? x. x:A & P(x) -\tdx{subset_def} A <= B == ! x:A. x:B -\tdx{Un_def} A Un B == \{x.x:A | x:B\} -\tdx{Int_def} A Int B == \{x.x:A & x:B\} -\tdx{set_diff_def} A - B == \{x.x:A & x~:B\} -\tdx{Compl_def} Compl(A) == \{x. ~ x:A\} -\tdx{INTER_def} INTER(A,B) == \{y. ! x:A. y: B(x)\} -\tdx{UNION_def} UNION(A,B) == \{y. ? x:A. y: B(x)\} -\tdx{INTER1_def} INTER1(B) == INTER(\{x.True\}, B) -\tdx{UNION1_def} UNION1(B) == UNION(\{x.True\}, B) -\tdx{Inter_def} Inter(S) == (INT x:S. x) -\tdx{Union_def} Union(S) == (UN x:S. x) -\tdx{Pow_def} Pow(A) == \{B. B <= A\} -\tdx{image_def} f``A == \{y. ? x:A. y=f(x)\} -\tdx{range_def} range(f) == \{y. ? x. y=f(x)\} -\tdx{mono_def} mono(f) == !A B. A <= B --> f(A) <= f(B) -\tdx{inj_def} inj(f) == ! x y. f(x)=f(y) --> x=y -\tdx{surj_def} surj(f) == ! y. ? x. y=f(x) -\tdx{inj_onto_def} inj_onto(f,A) == !x:A. !y:A. f(x)=f(y) --> x=y -\end{ttbox} -\caption{Rules of the theory {\tt Set}} \label{hol-set-rules} -\end{figure} - - -\begin{figure} \underscoreon -\begin{ttbox} -\tdx{CollectI} [| P(a) |] ==> a : \{x.P(x)\} -\tdx{CollectD} [| a : \{x.P(x)\} |] ==> P(a) -\tdx{CollectE} [| a : \{x.P(x)\}; P(a) ==> W |] ==> W - -\tdx{ballI} [| !!x. x:A ==> P(x) |] ==> ! x:A. P(x) -\tdx{bspec} [| ! x:A. P(x); x:A |] ==> P(x) -\tdx{ballE} [| ! x:A. P(x); P(x) ==> Q; ~ x:A ==> Q |] ==> Q - -\tdx{bexI} [| P(x); x:A |] ==> ? x:A. P(x) -\tdx{bexCI} [| ! x:A. ~ P(x) ==> P(a); a:A |] ==> ? x:A.P(x) -\tdx{bexE} [| ? x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q -\subcaption{Comprehension and Bounded quantifiers} - -\tdx{subsetI} (!!x.x:A ==> x:B) ==> A <= B -\tdx{subsetD} [| A <= B; c:A |] ==> c:B -\tdx{subsetCE} [| A <= B; ~ (c:A) ==> P; c:B ==> P |] ==> P - -\tdx{subset_refl} A <= A -\tdx{subset_trans} [| A<=B; B<=C |] ==> A<=C - -\tdx{equalityI} [| A <= B; B <= A |] ==> A = B -\tdx{equalityD1} A = B ==> A<=B -\tdx{equalityD2} A = B ==> B<=A -\tdx{equalityE} [| A = B; [| A<=B; B<=A |] ==> P |] ==> P - -\tdx{equalityCE} [| A = B; [| c:A; c:B |] ==> P; - [| ~ c:A; ~ c:B |] ==> P - |] ==> P -\subcaption{The subset and equality relations} -\end{ttbox} -\caption{Derived rules for set theory} \label{hol-set1} -\end{figure} - - -\begin{figure} \underscoreon -\begin{ttbox} -\tdx{emptyE} a : \{\} ==> P - -\tdx{insertI1} a : insert(a,B) -\tdx{insertI2} a : B ==> a : insert(b,B) -\tdx{insertE} [| a : insert(b,A); a=b ==> P; a:A ==> P |] ==> P - -\tdx{ComplI} [| c:A ==> False |] ==> c : Compl(A) -\tdx{ComplD} [| c : Compl(A) |] ==> ~ c:A - -\tdx{UnI1} c:A ==> c : A Un B -\tdx{UnI2} c:B ==> c : A Un B -\tdx{UnCI} (~c:B ==> c:A) ==> c : A Un B -\tdx{UnE} [| c : A Un B; c:A ==> P; c:B ==> P |] ==> P - -\tdx{IntI} [| c:A; c:B |] ==> c : A Int B -\tdx{IntD1} c : A Int B ==> c:A -\tdx{IntD2} c : A Int B ==> c:B -\tdx{IntE} [| c : A Int B; [| c:A; c:B |] ==> P |] ==> P - -\tdx{UN_I} [| a:A; b: B(a) |] ==> b: (UN x:A. B(x)) -\tdx{UN_E} [| b: (UN x:A. B(x)); !!x.[| x:A; b:B(x) |] ==> R |] ==> R - -\tdx{INT_I} (!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x)) -\tdx{INT_D} [| b: (INT x:A. B(x)); a:A |] ==> b: B(a) -\tdx{INT_E} [| b: (INT x:A. B(x)); b: B(a) ==> R; ~ a:A ==> R |] ==> R - -\tdx{UnionI} [| X:C; A:X |] ==> A : Union(C) -\tdx{UnionE} [| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R - -\tdx{InterI} [| !!X. X:C ==> A:X |] ==> A : Inter(C) -\tdx{InterD} [| A : Inter(C); X:C |] ==> A:X -\tdx{InterE} [| A : Inter(C); A:X ==> R; ~ X:C ==> R |] ==> R - -\tdx{PowI} A<=B ==> A: Pow(B) -\tdx{PowD} A: Pow(B) ==> A<=B -\end{ttbox} -\caption{Further derived rules for set theory} \label{hol-set2} -\end{figure} - - -\subsection{Axioms and rules of set theory} -Figure~\ref{hol-set-rules} presents the rules of theory \thydx{Set}. The -axioms \tdx{mem_Collect_eq} and \tdx{Collect_mem_eq} assert -that the functions {\tt Collect} and \hbox{\tt op :} are isomorphisms. Of -course, \hbox{\tt op :} also serves as the membership relation. - -All the other axioms are definitions. They include the empty set, bounded -quantifiers, unions, intersections, complements and the subset relation. -They also include straightforward properties of functions: image~({\tt``}) and -{\tt range}, and predicates concerning monotonicity, injectiveness and -surjectiveness. - -The predicate \cdx{inj_onto} is used for simulating type definitions. -The statement ${\tt inj_onto}(f,A)$ asserts that $f$ is injective on the -set~$A$, which specifies a subset of its domain type. In a type -definition, $f$ is the abstraction function and $A$ is the set of valid -representations; we should not expect $f$ to be injective outside of~$A$. - -\begin{figure} \underscoreon -\begin{ttbox} -\tdx{Inv_f_f} inj(f) ==> Inv(f,f(x)) = x -\tdx{f_Inv_f} y : range(f) ==> f(Inv(f,y)) = y - -%\tdx{Inv_injective} -% [| Inv(f,x)=Inv(f,y); x: range(f); y: range(f) |] ==> x=y -% -\tdx{imageI} [| x:A |] ==> f(x) : f``A -\tdx{imageE} [| b : f``A; !!x.[| b=f(x); x:A |] ==> P |] ==> P - -\tdx{rangeI} f(x) : range(f) -\tdx{rangeE} [| b : range(f); !!x.[| b=f(x) |] ==> P |] ==> P - -\tdx{monoI} [| !!A B. A <= B ==> f(A) <= f(B) |] ==> mono(f) -\tdx{monoD} [| mono(f); A <= B |] ==> f(A) <= f(B) - -\tdx{injI} [| !! x y. f(x) = f(y) ==> x=y |] ==> inj(f) -\tdx{inj_inverseI} (!!x. g(f(x)) = x) ==> inj(f) -\tdx{injD} [| inj(f); f(x) = f(y) |] ==> x=y - -\tdx{inj_ontoI} (!!x y. [| f(x)=f(y); x:A; y:A |] ==> x=y) ==> inj_onto(f,A) -\tdx{inj_ontoD} [| inj_onto(f,A); f(x)=f(y); x:A; y:A |] ==> x=y - -\tdx{inj_onto_inverseI} - (!!x. x:A ==> g(f(x)) = x) ==> inj_onto(f,A) -\tdx{inj_onto_contraD} - [| inj_onto(f,A); x~=y; x:A; y:A |] ==> ~ f(x)=f(y) -\end{ttbox} -\caption{Derived rules involving functions} \label{hol-fun} -\end{figure} - - -\begin{figure} \underscoreon -\begin{ttbox} -\tdx{Union_upper} B:A ==> B <= Union(A) -\tdx{Union_least} [| !!X. X:A ==> X<=C |] ==> Union(A) <= C - -\tdx{Inter_lower} B:A ==> Inter(A) <= B -\tdx{Inter_greatest} [| !!X. X:A ==> C<=X |] ==> C <= Inter(A) - -\tdx{Un_upper1} A <= A Un B -\tdx{Un_upper2} B <= A Un B -\tdx{Un_least} [| A<=C; B<=C |] ==> A Un B <= C - -\tdx{Int_lower1} A Int B <= A -\tdx{Int_lower2} A Int B <= B -\tdx{Int_greatest} [| C<=A; C<=B |] ==> C <= A Int B -\end{ttbox} -\caption{Derived rules involving subsets} \label{hol-subset} -\end{figure} - - -\begin{figure} \underscoreon \hfuzz=4pt%suppress "Overfull \hbox" message -\begin{ttbox} -\tdx{Int_absorb} A Int A = A -\tdx{Int_commute} A Int B = B Int A -\tdx{Int_assoc} (A Int B) Int C = A Int (B Int C) -\tdx{Int_Un_distrib} (A Un B) Int C = (A Int C) Un (B Int C) - -\tdx{Un_absorb} A Un A = A -\tdx{Un_commute} A Un B = B Un A -\tdx{Un_assoc} (A Un B) Un C = A Un (B Un C) -\tdx{Un_Int_distrib} (A Int B) Un C = (A Un C) Int (B Un C) - -\tdx{Compl_disjoint} A Int Compl(A) = \{x.False\} -\tdx{Compl_partition} A Un Compl(A) = \{x.True\} -\tdx{double_complement} Compl(Compl(A)) = A -\tdx{Compl_Un} Compl(A Un B) = Compl(A) Int Compl(B) -\tdx{Compl_Int} Compl(A Int B) = Compl(A) Un Compl(B) - -\tdx{Union_Un_distrib} Union(A Un B) = Union(A) Un Union(B) -\tdx{Int_Union} A Int Union(B) = (UN C:B. A Int C) -\tdx{Un_Union_image} (UN x:C. A(x) Un B(x)) = Union(A``C) Un Union(B``C) - -\tdx{Inter_Un_distrib} Inter(A Un B) = Inter(A) Int Inter(B) -\tdx{Un_Inter} A Un Inter(B) = (INT C:B. A Un C) -\tdx{Int_Inter_image} (INT x:C. A(x) Int B(x)) = Inter(A``C) Int Inter(B``C) -\end{ttbox} -\caption{Set equalities} \label{hol-equalities} -\end{figure} - - -Figures~\ref{hol-set1} and~\ref{hol-set2} present derived rules. Most are -obvious and resemble rules of Isabelle's ZF set theory. Certain rules, such -as \tdx{subsetCE}, \tdx{bexCI} and \tdx{UnCI}, are designed for classical -reasoning; the rules \tdx{subsetD}, \tdx{bexI}, \tdx{Un1} and~\tdx{Un2} are -not strictly necessary but yield more natural proofs. Similarly, -\tdx{equalityCE} supports classical reasoning about extensionality, after the -fashion of \tdx{iffCE}. See the file {\tt HOL/Set.ML} for proofs pertaining -to set theory. - -Figure~\ref{hol-fun} presents derived inference rules involving functions. -They also include rules for \cdx{Inv}, which is defined in theory~{\tt - HOL}; note that ${\tt Inv}(f)$ applies the Axiom of Choice to yield an -inverse of~$f$. They also include natural deduction rules for the image -and range operators, and for the predicates {\tt inj} and {\tt inj_onto}. -Reasoning about function composition (the operator~\sdx{o}) and the -predicate~\cdx{surj} is done simply by expanding the definitions. See -the file {\tt HOL/fun.ML} for a complete listing of the derived rules. - -Figure~\ref{hol-subset} presents lattice properties of the subset relation. -Unions form least upper bounds; non-empty intersections form greatest lower -bounds. Reasoning directly about subsets often yields clearer proofs than -reasoning about the membership relation. See the file {\tt HOL/subset.ML}. - -Figure~\ref{hol-equalities} presents many common set equalities. They -include commutative, associative and distributive laws involving unions, -intersections and complements. The proofs are mostly trivial, using the -classical reasoner; see file {\tt HOL/equalities.ML}. - - -\begin{figure} -\begin{constants} - \it symbol & \it meta-type & & \it description \\ - \cdx{Pair} & $[\alpha,\beta]\To \alpha\times\beta$ - & & ordered pairs $\langle a,b\rangle$ \\ - \cdx{fst} & $\alpha\times\beta \To \alpha$ & & first projection\\ - \cdx{snd} & $\alpha\times\beta \To \beta$ & & second projection\\ - \cdx{split} & $[[\alpha,\beta]\To\gamma, \alpha\times\beta] \To \gamma$ - & & generalized projection\\ - \cdx{Sigma} & - $[\alpha\,set, \alpha\To\beta\,set]\To(\alpha\times\beta)set$ & - & general sum of sets -\end{constants} -\begin{ttbox}\makeatletter -\tdx{fst_def} fst(p) == @a. ? b. p = -\tdx{snd_def} snd(p) == @b. ? a. p = -\tdx{split_def} split(c,p) == c(fst(p),snd(p)) -\tdx{Sigma_def} Sigma(A,B) == UN x:A. UN y:B(x). \{\} - - -\tdx{Pair_inject} [| = ; [| a=a'; b=b' |] ==> R |] ==> R -\tdx{fst_conv} fst() = a -\tdx{snd_conv} snd() = b -\tdx{split} split(c, ) = c(a,b) - -\tdx{surjective_pairing} p = - -\tdx{SigmaI} [| a:A; b:B(a) |] ==> : Sigma(A,B) - -\tdx{SigmaE} [| c: Sigma(A,B); - !!x y.[| x:A; y:B(x); c= |] ==> P |] ==> P -\end{ttbox} -\caption{Type $\alpha\times\beta$}\label{hol-prod} -\end{figure} - - -\begin{figure} -\begin{constants} - \it symbol & \it meta-type & & \it description \\ - \cdx{Inl} & $\alpha \To \alpha+\beta$ & & first injection\\ - \cdx{Inr} & $\beta \To \alpha+\beta$ & & second injection\\ - \cdx{sum_case} & $[\alpha\To\gamma, \beta\To\gamma, \alpha+\beta] \To\gamma$ - & & conditional -\end{constants} -\begin{ttbox}\makeatletter -\tdx{sum_case_def} sum_case == (\%f g p. @z. (!x. p=Inl(x) --> z=f(x)) & - (!y. p=Inr(y) --> z=g(y))) - -\tdx{Inl_not_Inr} ~ Inl(a)=Inr(b) - -\tdx{inj_Inl} inj(Inl) -\tdx{inj_Inr} inj(Inr) - -\tdx{sumE} [| !!x::'a. P(Inl(x)); !!y::'b. P(Inr(y)) |] ==> P(s) - -\tdx{sum_case_Inl} sum_case(f, g, Inl(x)) = f(x) -\tdx{sum_case_Inr} sum_case(f, g, Inr(x)) = g(x) - -\tdx{surjective_sum} sum_case(\%x::'a. f(Inl(x)), \%y::'b. f(Inr(y)), s) = f(s) -\end{ttbox} -\caption{Type $\alpha+\beta$}\label{hol-sum} -\end{figure} - - -\section{Generic packages and classical reasoning} -HOL instantiates most of Isabelle's generic packages; see {\tt HOL/ROOT.ML} -for details. -\begin{itemize} -\item Because it includes a general substitution rule, HOL instantiates the - tactic {\tt hyp_subst_tac}, which substitutes for an equality throughout a - subgoal and its hypotheses. -\item -It instantiates the simplifier, defining~\ttindexbold{HOL_ss} as the -simplification set for higher-order logic. Equality~($=$), which also -expresses logical equivalence, may be used for rewriting. See the file -{\tt HOL/simpdata.ML} for a complete listing of the simplification -rules. -\item -It instantiates the classical reasoner, as described below. -\end{itemize} -HOL derives classical introduction rules for $\disj$ and~$\exists$, as well as -classical elimination rules for~$\imp$ and~$\bimp$, and the swap rule; recall -Fig.\ts\ref{hol-lemmas2} above. - -The classical reasoner is set up as the structure {\tt Classical}. This -structure is open, so {\ML} identifiers such as {\tt step_tac}, {\tt - fast_tac}, {\tt best_tac}, etc., refer to it. HOL defines the following -classical rule sets: -\begin{ttbox} -prop_cs : claset -HOL_cs : claset -set_cs : claset -\end{ttbox} -\begin{ttdescription} -\item[\ttindexbold{prop_cs}] contains the propositional rules, namely -those for~$\top$, $\bot$, $\conj$, $\disj$, $\neg$, $\imp$ and~$\bimp$, -along with the rule~{\tt refl}. - -\item[\ttindexbold{HOL_cs}] extends {\tt prop_cs} with the safe rules - {\tt allI} and~{\tt exE} and the unsafe rules {\tt allE} - and~{\tt exI}, as well as rules for unique existence. Search using - this classical set is incomplete: quantified formulae are used at most - once. - -\item[\ttindexbold{set_cs}] extends {\tt HOL_cs} with rules for the bounded - quantifiers, subsets, comprehensions, unions and intersections, - complements, finite sets, images and ranges. -\end{ttdescription} -\noindent -See \iflabelundefined{chap:classical}{the {\em Reference Manual\/}}% - {Chap.\ts\ref{chap:classical}} -for more discussion of classical proof methods. - - -\section{Types} -The basic higher-order logic is augmented with a tremendous amount of -material, including support for recursive function and type definitions. A -detailed discussion appears elsewhere~\cite{paulson-coind}. The simpler -definitions are the same as those used the {\sc hol} system, but my -treatment of recursive types differs from Melham's~\cite{melham89}. The -present section describes product, sum, natural number and list types. - -\subsection{Product and sum types}\index{*"* type}\index{*"+ type} -Theory \thydx{Prod} defines the product type $\alpha\times\beta$, with -the ordered pair syntax {\tt<$a$,$b$>}. Theory \thydx{Sum} defines the -sum type $\alpha+\beta$. These use fairly standard constructions; see -Figs.\ts\ref{hol-prod} and~\ref{hol-sum}. Because Isabelle does not -support abstract type definitions, the isomorphisms between these types and -their representations are made explicitly. - -Most of the definitions are suppressed, but observe that the projections -and conditionals are defined as descriptions. Their properties are easily -proved using \tdx{select_equality}. - -\begin{figure} -\index{*"< symbol} -\index{*"* symbol} -\index{*div symbol} -\index{*mod symbol} -\index{*"+ symbol} -\index{*"- symbol} -\begin{constants} - \it symbol & \it meta-type & \it priority & \it description \\ - \cdx{0} & $nat$ & & zero \\ - \cdx{Suc} & $nat \To nat$ & & successor function\\ - \cdx{nat_case} & $[\alpha, nat\To\alpha, nat] \To\alpha$ - & & conditional\\ - \cdx{nat_rec} & $[nat, \alpha, [nat, \alpha]\To\alpha] \To \alpha$ - & & primitive recursor\\ - \cdx{pred_nat} & $(nat\times nat) set$ & & predecessor relation\\ - \tt * & $[nat,nat]\To nat$ & Left 70 & multiplication \\ - \tt div & $[nat,nat]\To nat$ & Left 70 & division\\ - \tt mod & $[nat,nat]\To nat$ & Left 70 & modulus\\ - \tt + & $[nat,nat]\To nat$ & Left 65 & addition\\ - \tt - & $[nat,nat]\To nat$ & Left 65 & subtraction -\end{constants} -\subcaption{Constants and infixes} - -\begin{ttbox}\makeatother -\tdx{nat_case_def} nat_case == (\%a f n. @z. (n=0 --> z=a) & - (!x. n=Suc(x) --> z=f(x))) -\tdx{pred_nat_def} pred_nat == \{p. ? n. p = \} -\tdx{less_def} m:pred_nat^+ -\tdx{nat_rec_def} nat_rec(n,c,d) == - wfrec(pred_nat, n, nat_case(\%g.c, \%m g. d(m,g(m)))) - -\tdx{add_def} m+n == nat_rec(m, n, \%u v.Suc(v)) -\tdx{diff_def} m-n == nat_rec(n, m, \%u v. nat_rec(v, 0, \%x y.x)) -\tdx{mult_def} m*n == nat_rec(m, 0, \%u v. n + v) -\tdx{mod_def} m mod n == wfrec(trancl(pred_nat), m, \%j f. if(j P(Suc(k)) |] ==> P(n) - -\tdx{Suc_not_Zero} Suc(m) ~= 0 -\tdx{inj_Suc} inj(Suc) -\tdx{n_not_Suc_n} n~=Suc(n) -\subcaption{Basic properties} - -\tdx{pred_natI} : pred_nat -\tdx{pred_natE} - [| p : pred_nat; !!x n. [| p = |] ==> R |] ==> R - -\tdx{nat_case_0} nat_case(a, f, 0) = a -\tdx{nat_case_Suc} nat_case(a, f, Suc(k)) = f(k) - -\tdx{wf_pred_nat} wf(pred_nat) -\tdx{nat_rec_0} nat_rec(0,c,h) = c -\tdx{nat_rec_Suc} nat_rec(Suc(n), c, h) = h(n, nat_rec(n,c,h)) -\subcaption{Case analysis and primitive recursion} - -\tdx{less_trans} [| i i ~ m P(m) |] ==> P(n) |] ==> P(n) - -\tdx{less_linear} m P(x#xs)) |] ==> P(l) - -\tdx{Cons_not_Nil} (x # xs) ~= [] -\tdx{Cons_Cons_eq} ((x # xs) = (y # ys)) = (x=y & xs=ys) -\subcaption{Induction and freeness} -\end{ttbox} -\caption{The theory \thydx{List}} \label{hol-list} -\end{figure} - -\begin{figure} -\begin{ttbox}\makeatother -\tdx{list_rec_Nil} list_rec([],c,h) = c -\tdx{list_rec_Cons} list_rec(a#l, c, h) = h(a, l, list_rec(l,c,h)) - -\tdx{list_case_Nil} list_case(c, h, []) = c -\tdx{list_case_Cons} list_case(c, h, x#xs) = h(x, xs) - -\tdx{map_Nil} map(f,[]) = [] -\tdx{map_Cons} map(f, x \# xs) = f(x) \# map(f,xs) - -\tdx{null_Nil} null([]) = True -\tdx{null_Cons} null(x#xs) = False - -\tdx{hd_Cons} hd(x#xs) = x -\tdx{tl_Cons} tl(x#xs) = xs - -\tdx{ttl_Nil} ttl([]) = [] -\tdx{ttl_Cons} ttl(x#xs) = xs - -\tdx{append_Nil} [] @ ys = ys -\tdx{append_Cons} (x#xs) \at ys = x # xs \at ys - -\tdx{mem_Nil} x mem [] = False -\tdx{mem_Cons} x mem (y#ys) = if(y=x, True, x mem ys) - -\tdx{filter_Nil} filter(P, []) = [] -\tdx{filter_Cons} filter(P,x#xs) = if(P(x), x#filter(P,xs), filter(P,xs)) - -\tdx{list_all_Nil} list_all(P,[]) = True -\tdx{list_all_Cons} list_all(P, x#xs) = (P(x) & list_all(P, xs)) -\end{ttbox} -\caption{Rewrite rules for lists} \label{hol-list-simps} -\end{figure} - - -\subsection{The type constructor for lists, {\tt list}} -\index{*list type} - -HOL's definition of lists is an example of an experimental method for handling -recursive data types. Figure~\ref{hol-list} presents the theory \thydx{List}: -the basic list operations with their types and properties. - -The \sdx{case} construct is defined by the following translation: -{\dquotes -\begin{eqnarray*} - \begin{array}{r@{\;}l@{}l} - "case " e " of" & "[]" & " => " a\\ - "|" & x"\#"xs & " => " b - \end{array} - & \equiv & - "list_case"(a, \lambda x\;xs.b, e) -\end{eqnarray*}}% -The theory includes \cdx{list_rec}, a primitive recursion operator -for lists. It is derived from well-founded recursion, a general principle -that can express arbitrary total recursive functions. - -The simpset \ttindex{list_ss} contains, along with additional useful lemmas, -the basic rewrite rules that appear in Fig.\ts\ref{hol-list-simps}. - -The tactic {\tt\ttindex{list_ind_tac} "$xs$" $i$} performs induction over the -variable~$xs$ in subgoal~$i$. - - -\section{Datatype declarations} -\index{*datatype|(} - -\underscoreon - -It is often necessary to extend a theory with \ML-like datatypes. This -extension consists of the new type, declarations of its constructors and -rules that describe the new type. The theory definition section {\tt - datatype} represents a compact way of doing this. - - -\subsection{Foundations} - -A datatype declaration has the following general structure: -\[ \mbox{\tt datatype}~ (\alpha_1,\dots,\alpha_n)t ~=~ - C_1(\tau_{11},\dots,\tau_{1k_1}) ~\mid~ \dots ~\mid~ - C_m(\tau_{m1},\dots,\tau_{mk_m}) -\] -where $\alpha_i$ are type variables, $C_i$ are distinct constructor names and -$\tau_{ij}$ are one of the following: -\begin{itemize} -\item type variables $\alpha_1,\dots,\alpha_n$, -\item types $(\beta_1,\dots,\beta_l)s$ where $s$ is a previously declared - type or type synonym and $\{\beta_1,\dots,\beta_l\} \subseteq - \{\alpha_1,\dots,\alpha_n\}$, -\item the newly defined type $(\alpha_1,\dots,\alpha_n)t$ \footnote{This - makes it a recursive type. To ensure that the new type is not empty at - least one constructor must consist of only non-recursive type - components.} -\end{itemize} -If you would like one of the $\tau_{ij}$ to be a complex type expression -$\tau$ you need to declare a new type synonym $syn = \tau$ first and use -$syn$ in place of $\tau$. Of course this does not work if $\tau$ mentions the -recursive type itself, thus ruling out problematic cases like \[ \mbox{\tt - datatype}~ t ~=~ C(t \To t) \] together with unproblematic ones like \[ -\mbox{\tt datatype}~ t ~=~ C(t~list). \] - -The constructors are automatically defined as functions of their respective -type: -\[ C_j : [\tau_{j1},\dots,\tau_{jk_j}] \To (\alpha_1,\dots,\alpha_n)t \] -These functions have certain {\em freeness} properties: -\begin{description} -\item[\tt distinct] They are distinct: -\[ C_i(x_1,\dots,x_{k_i}) \neq C_j(y_1,\dots,y_{k_j}) \qquad - \mbox{for all}~ i \neq j. -\] -\item[\tt inject] They are injective: -\[ (C_j(x_1,\dots,x_{k_j}) = C_j(y_1,\dots,y_{k_j})) = - (x_1 = y_1 \land \dots \land x_{k_j} = y_{k_j}) -\] -\end{description} -Because the number of inequalities is quadratic in the number of -constructors, a different method is used if their number exceeds -a certain value, currently 4. In that case every constructor is mapped to a -natural number -\[ -\begin{array}{lcl} -\mbox{\it t\_ord}(C_1(x_1,\dots,x_{k_1})) & = & 0 \\ -& \vdots & \\ -\mbox{\it t\_ord}(C_m(x_1,\dots,x_{k_m})) & = & m-1 -\end{array} -\] -and distinctness of constructors is expressed by: -\[ -\mbox{\it t\_ord}(x) \neq \mbox{\it t\_ord}(y) \Imp x \neq y. -\] -In addition a structural induction axiom {\tt induct} is provided: -\[ -\infer{P(x)} -{\begin{array}{lcl} -\Forall x_1\dots x_{k_1}. - \List{P(x_{r_{11}}); \dots; P(x_{r_{1l_1}})} & - \Imp & P(C_1(x_1,\dots,x_{k_1})) \\ - & \vdots & \\ -\Forall x_1\dots x_{k_m}. - \List{P(x_{r_{m1}}); \dots; P(x_{r_{ml_m}})} & - \Imp & P(C_m(x_1,\dots,x_{k_m})) -\end{array}} -\] -where $\{r_{j1},\dots,r_{jl_j}\} = \{i \in \{1,\dots k_j\} ~\mid~ \tau_{ji} -= (\alpha_1,\dots,\alpha_n)t \}$, i.e.\ the property $P$ can be assumed for -all arguments of the recursive type. - -The type also comes with an \ML-like \sdx{case}-construct: -\[ -\begin{array}{rrcl} -\mbox{\tt case}~e~\mbox{\tt of} & C_1(x_{11},\dots,x_{1k_1}) & \To & e_1 \\ - \vdots \\ - \mid & C_m(x_{m1},\dots,x_{mk_m}) & \To & e_m -\end{array} -\] -In contrast to \ML, {\em all} constructors must be present, their order is -fixed, and nested patterns are not supported. - - -\subsection{Defining datatypes} - -A datatype is defined in a theory definition file using the keyword {\tt - datatype}. The definition following {\tt datatype} must conform to the -syntax of {\em typedecl} specified in Fig.~\ref{datatype-grammar} and must -obey the rules in the previous section. As a result the theory is extended -with the new type, the constructors, and the theorems listed in the previous -section. - -\begin{figure} -\begin{rail} -typedecl : typevarlist id '=' (cons + '|') - ; -cons : (id | string) ( () | '(' (typ + ',') ')' ) ( () | mixfix ) - ; -typ : typevarlist id - | tid - ; -typevarlist : () | tid | '(' (tid + ',') ')' - ; -\end{rail} -\caption{Syntax of datatype declarations} -\label{datatype-grammar} -\end{figure} - -Reading the theory file produces a structure which, in addition to the usual -components, contains a structure named $t$ for each datatype $t$ defined in -the file.\footnote{Otherwise multiple datatypes in the same theory file would - lead to name clashes.} Each structure $t$ contains the following elements: -\begin{ttbox} -val distinct : thm list -val inject : thm list -val induct : thm -val cases : thm list -val simps : thm list -val induct_tac : string -> int -> tactic -\end{ttbox} -{\tt distinct}, {\tt inject} and {\tt induct} contain the theorems described -above. For convenience {\tt distinct} contains inequalities in both -directions. -\begin{warn} - If there are five or more constructors, the {\em t\_ord} scheme is used for - {\tt distinct}. In this case the theory {\tt Arith} must be contained - in the current theory, if necessary by including it explicitly. -\end{warn} -The reduction rules of the {\tt case}-construct are in {\tt cases}. All -theorems from {\tt distinct}, {\tt inject} and {\tt cases} are combined in -{\tt simps} for use with the simplifier. The tactic {\verb$induct_tac$~{\em - var i}\/} applies structural induction over variable {\em var} to -subgoal {\em i}. - - -\subsection{Examples} - -\subsubsection{The datatype $\alpha~list$} - -We want to define the type $\alpha~list$.\footnote{Of course there is a list - type in HOL already. This is only an example.} To do this we have to build -a new theory that contains the type definition. We start from {\tt HOL}. -\begin{ttbox} -MyList = HOL + - datatype 'a list = Nil | Cons ('a, 'a list) -end -\end{ttbox} -After loading the theory (\verb$use_thy "MyList"$), we can prove -$Cons(x,xs)\neq xs$. First we build a suitable simpset for the simplifier: -\begin{ttbox} -val mylist_ss = HOL_ss addsimps MyList.list.simps; -goal MyList.thy "!x. Cons(x,xs) ~= xs"; -{\out Level 0} -{\out ! x. Cons(x, xs) ~= xs} -{\out 1. ! x. Cons(x, xs) ~= xs} -\end{ttbox} -This can be proved by the structural induction tactic: -\begin{ttbox} -by (MyList.list.induct_tac "xs" 1); -{\out Level 1} -{\out ! x. Cons(x, xs) ~= xs} -{\out 1. ! x. Cons(x, Nil) ~= Nil} -{\out 2. !!a list.} -{\out ! x. Cons(x, list) ~= list ==>} -{\out ! x. Cons(x, Cons(a, list)) ~= Cons(a, list)} -\end{ttbox} -The first subgoal can be proved with the simplifier and the distinctness -axioms which are part of \verb$mylist_ss$. -\begin{ttbox} -by (simp_tac mylist_ss 1); -{\out Level 2} -{\out ! x. Cons(x, xs) ~= xs} -{\out 1. !!a list.} -{\out ! x. Cons(x, list) ~= list ==>} -{\out ! x. Cons(x, Cons(a, list)) ~= Cons(a, list)} -\end{ttbox} -Using the freeness axioms we can quickly prove the remaining goal. -\begin{ttbox} -by (asm_simp_tac mylist_ss 1); -{\out Level 3} -{\out ! x. Cons(x, xs) ~= xs} -{\out No subgoals!} -\end{ttbox} -Because both subgoals were proved by almost the same tactic we could have -done that in one step using -\begin{ttbox} -by (ALLGOALS (asm_simp_tac mylist_ss)); -\end{ttbox} - - -\subsubsection{The datatype $\alpha~list$ with mixfix syntax} - -In this example we define the type $\alpha~list$ again but this time we want -to write {\tt []} instead of {\tt Nil} and we want to use the infix operator -\verb|#| instead of {\tt Cons}. To do this we simply add mixfix annotations -after the constructor declarations as follows: -\begin{ttbox} -MyList = HOL + - datatype 'a list = "[]" ("[]") - | "#" ('a, 'a list) (infixr 70) -end -\end{ttbox} -Now the theorem in the previous example can be written \verb|x#xs ~= xs|. The -proof is the same. - - -\subsubsection{A datatype for weekdays} - -This example shows a datatype that consists of more than four constructors: -\begin{ttbox} -Days = Arith + - datatype days = Mo | Tu | We | Th | Fr | Sa | So -end -\end{ttbox} -Because there are more than four constructors, the theory must be based on -{\tt Arith}. Inequality is defined via a function \verb|days_ord|. Although -the expression \verb|Mo ~= Tu| is not directly contained in {\tt distinct}, -it can be proved by the simplifier if \verb$arith_ss$ is used: -\begin{ttbox} -val days_ss = arith_ss addsimps Days.days.simps; - -goal Days.thy "Mo ~= Tu"; -by (simp_tac days_ss 1); -\end{ttbox} -Note that usually it is not necessary to derive these inequalities explicitly -because the simplifier will dispose of them automatically. - -\subsection{Primitive recursive functions} -\index{primitive recursion|(} -\index{*primrec|(} - -Datatypes come with a uniform way of defining functions, {\bf primitive - recursion}. Although it is possible to define primitive recursive functions -by asserting their reduction rules as new axioms, e.g.\ -\begin{ttbox} -Append = MyList + -consts app :: "['a list,'a list] => 'a list" -rules - app_Nil "app([],ys) = ys" - app_Cons "app(x#xs, ys) = x#app(xs,ys)" -end -\end{ttbox} -this carries with it the danger of accidentally asserting an inconsistency, -as in \verb$app([],ys) = us$. Therefore primitive recursive functions on -datatypes can be defined with a special syntax: -\begin{ttbox} -Append = MyList + -consts app :: "['a list,'a list] => 'a list" -primrec app MyList.list - app_Nil "app([],ys) = ys" - app_Cons "app(x#xs, ys) = x#app(xs,ys)" -end -\end{ttbox} -The system will now check that the two rules \verb$app_Nil$ and -\verb$app_Cons$ do indeed form a primitive recursive definition, thus -ensuring that consistency is maintained. For example -\begin{ttbox} -primrec app MyList.list - app_Nil "app([],ys) = us" -\end{ttbox} -is rejected: -\begin{ttbox} -Extra variables on rhs -\end{ttbox} -\bigskip - -The general form of a primitive recursive definition is -\begin{ttbox} -primrec {\it function} {\it type} - {\it reduction rules} -\end{ttbox} -where -\begin{itemize} -\item {\it function} is the name of the function, either as an {\it id} or a - {\it string}. The function must already have been declared. -\item {\it type} is the name of the datatype, either as an {\it id} or in the - long form {\it Thy.t}, where {\it Thy} is the name of the parent theory the - datatype was declared in, and $t$ the name of the datatype. The long form - is required if the {\tt datatype} and the {\tt primrec} sections are in - different theories. -\item {\it reduction rules} specify one or more named equations of the form - {\it id\/}~{\it string}, where the identifier gives the name of the rule in - the result structure, and {\it string} is a reduction rule of the form \[ - f(x_1,\dots,x_m,C(y_1,\dots,y_k),z_1,\dots,z_n) = r \] such that $C$ is a - constructor of the datatype, $r$ contains only the free variables on the - left-hand side, and all recursive calls in $r$ are of the form - $f(\dots,y_i,\dots)$ for some $i$. There must be exactly one reduction - rule for each constructor. -\end{itemize} -A theory file may contain any number of {\tt primrec} sections which may be -intermixed with other declarations. - -For the consistency-sensitive user it may be reassuring to know that {\tt - primrec} does not assert the reduction rules as new axioms but derives them -as theorems from an explicit definition of the recursive function in terms of -a recursion operator on the datatype. - -The primitive recursive function can also use infix or mixfix syntax: -\begin{ttbox} -Append = MyList + -consts "@" :: "['a list,'a list] => 'a list" (infixr 60) -primrec "op @" MyList.list - app_Nil "[] @ ys = ys" - app_Cons "(x#xs) @ ys = x#(xs @ ys)" -end -\end{ttbox} - -The reduction rules become part of the ML structure \verb$Append$ and can -be used to prove theorems about the function: -\begin{ttbox} -val append_ss = HOL_ss addsimps [Append.app_Nil,Append.app_Cons]; - -goal Append.thy "(xs @ ys) @ zs = xs @ (ys @ zs)"; -by (MyList.list.induct_tac "xs" 1); -by (ALLGOALS(asm_simp_tac append_ss)); -\end{ttbox} - -%Note that underdefined primitive recursive functions are allowed: -%\begin{ttbox} -%Tl = MyList + -%consts tl :: "'a list => 'a list" -%primrec tl MyList.list -% tl_Cons "tl(x#xs) = xs" -%end -%\end{ttbox} -%Nevertheless {\tt tl} is total, although we do not know what the result of -%\verb$tl([])$ is. - -\index{primitive recursion|)} -\index{*primrec|)} -\index{*datatype|)} - - -\section{Inductive and coinductive definitions} -\index{*inductive|(} -\index{*coinductive|(} - -An {\bf inductive definition} specifies the least set closed under given -rules. For example, a structural operational semantics is an inductive -definition of an evaluation relation. Dually, a {\bf coinductive - definition} specifies the greatest set consistent with given rules. An -important example is using bisimulation relations to formalize equivalence -of processes and infinite data structures. - -A theory file may contain any number of inductive and coinductive -definitions. They may be intermixed with other declarations; in -particular, the (co)inductive sets {\bf must} be declared separately as -constants, and may have mixfix syntax or be subject to syntax translations. - -Each (co)inductive definition adds definitions to the theory and also -proves some theorems. Each definition creates an ML structure, which is a -substructure of the main theory structure. - -This package is derived from the ZF one, described in a -separate paper,\footnote{It appeared in CADE~\cite{paulson-CADE} and a - longer version is distributed with Isabelle.} which you should refer to -in case of difficulties. The package is simpler than ZF's, thanks to HOL's -automatic type-checking. The type of the (co)inductive determines the -domain of the fixedpoint definition, and the package does not use inference -rules for type-checking. - - -\subsection{The result structure} -Many of the result structure's components have been discussed in the paper; -others are self-explanatory. -\begin{description} -\item[\tt thy] is the new theory containing the recursive sets. - -\item[\tt defs] is the list of definitions of the recursive sets. - -\item[\tt mono] is a monotonicity theorem for the fixedpoint operator. - -\item[\tt unfold] is a fixedpoint equation for the recursive set (the union of -the recursive sets, in the case of mutual recursion). - -\item[\tt intrs] is the list of introduction rules, now proved as theorems, for -the recursive sets. The rules are also available individually, using the -names given them in the theory file. - -\item[\tt elim] is the elimination rule. - -\item[\tt mk\_cases] is a function to create simplified instances of {\tt -elim}, using freeness reasoning on some underlying datatype. -\end{description} - -For an inductive definition, the result structure contains two induction rules, -{\tt induct} and \verb|mutual_induct|. For a coinductive definition, it -contains the rule \verb|coinduct|. - -Figure~\ref{def-result-fig} summarizes the two result signatures, -specifying the types of all these components. - -\begin{figure} -\begin{ttbox} -sig -val thy : theory -val defs : thm list -val mono : thm -val unfold : thm -val intrs : thm list -val elim : thm -val mk_cases : thm list -> string -> thm -{\it(Inductive definitions only)} -val induct : thm -val mutual_induct: thm -{\it(Coinductive definitions only)} -val coinduct : thm -end -\end{ttbox} -\hrule -\caption{The result of a (co)inductive definition} \label{def-result-fig} -\end{figure} - -\subsection{The syntax of a (co)inductive definition} -An inductive definition has the form -\begin{ttbox} -inductive {\it inductive sets} - intrs {\it introduction rules} - monos {\it monotonicity theorems} - con_defs {\it constructor definitions} -\end{ttbox} -A coinductive definition is identical, except that it starts with the keyword -{\tt coinductive}. - -The {\tt monos} and {\tt con\_defs} sections are optional. If present, -each is specified as a string, which must be a valid ML expression of type -{\tt thm list}. It is simply inserted into the {\tt .thy.ML} file; if it -is ill-formed, it will trigger ML error messages. You can then inspect the -file on your directory. - -\begin{itemize} -\item The {\it inductive sets} are specified by one or more strings. - -\item The {\it introduction rules} specify one or more introduction rules in - the form {\it ident\/}~{\it string}, where the identifier gives the name of - the rule in the result structure. - -\item The {\it monotonicity theorems} are required for each operator - applied to a recursive set in the introduction rules. There {\bf must} - be a theorem of the form $A\subseteq B\Imp M(A)\subseteq M(B)$, for each - premise $t\in M(R_i)$ in an introduction rule! - -\item The {\it constructor definitions} contain definitions of constants - appearing in the introduction rules. In most cases it can be omitted. -\end{itemize} - -The package has a few notable restrictions: -\begin{itemize} -\item The theory must separately declare the recursive sets as - constants. - -\item The names of the recursive sets must be identifiers, not infix -operators. - -\item Side-conditions must not be conjunctions. However, an introduction rule -may contain any number of side-conditions. - -\item Side-conditions of the form $x=t$, where the variable~$x$ does not - occur in~$t$, will be substituted through the rule \verb|mutual_induct|. -\end{itemize} - - -\subsection{Example of an inductive definition} -Two declarations, included in a theory file, define the finite powerset -operator. First we declare the constant~{\tt Fin}. Then we declare it -inductively, with two introduction rules: -\begin{ttbox} -consts Fin :: "'a set => 'a set set" -inductive "Fin(A)" - intrs - emptyI "{} : Fin(A)" - insertI "[| a: A; b: Fin(A) |] ==> insert(a,b) : Fin(A)" -\end{ttbox} -The resulting theory structure contains a substructure, called~{\tt Fin}. -It contains the {\tt Fin}$(A)$ introduction rules as the list {\tt Fin.intrs}, -and also individually as {\tt Fin.emptyI} and {\tt Fin.consI}. The induction -rule is {\tt Fin.induct}. - -For another example, here is a theory file defining the accessible part of a -relation. The main thing to note is the use of~{\tt Pow} in the sole -introduction rule, and the corresponding mention of the rule -\verb|Pow_mono| in the {\tt monos} list. The paper discusses a ZF version -of this example in more detail. -\begin{ttbox} -Acc = WF + -consts pred :: "['b, ('a * 'b)set] => 'a set" (*Set of predecessors*) - acc :: "('a * 'a)set => 'a set" (*Accessible part*) -defs pred_def "pred(x,r) == {y. :r}" -inductive "acc(r)" - intrs - pred "pred(a,r): Pow(acc(r)) ==> a: acc(r)" - monos "[Pow_mono]" -end -\end{ttbox} -The HOL distribution contains many other inductive definitions, such as the -theory {\tt HOL/ex/PropLog.thy} and the directory {\tt HOL/IMP}. The -theory {\tt HOL/ex/LList.thy} contains coinductive definitions. - -\index{*coinductive|)} \index{*inductive|)} \underscoreoff - - -\section{The examples directories} -Directory {\tt HOL/Subst} contains Martin Coen's mechanisation of a theory of -substitutions and unifiers. It is based on Paulson's previous -mechanisation in {\LCF}~\cite{paulson85} of Manna and Waldinger's -theory~\cite{mw81}. - -Directory {\tt HOL/IMP} contains a mechanised version of a semantic -equivalence proof taken from Winskel~\cite{winskel93}. It formalises the -denotational and operational semantics of a simple while-language, then -proves the two equivalent. It contains several datatype and inductive -definitions, and demonstrates their use. - -Directory {\tt HOL/ex} contains other examples and experimental proofs in HOL. -Here is an overview of the more interesting files. -\begin{itemize} -\item File {\tt cla.ML} demonstrates the classical reasoner on over sixty - predicate calculus theorems, ranging from simple tautologies to - moderately difficult problems involving equality and quantifiers. - -\item File {\tt meson.ML} contains an experimental implementation of the {\sc - meson} proof procedure, inspired by Plaisted~\cite{plaisted90}. It is - much more powerful than Isabelle's classical reasoner. But it is less - useful in practice because it works only for pure logic; it does not - accept derived rules for the set theory primitives, for example. - -\item File {\tt mesontest.ML} contains test data for the {\sc meson} proof - procedure. These are mostly taken from Pelletier \cite{pelletier86}. - -\item File {\tt set.ML} proves Cantor's Theorem, which is presented in - \S\ref{sec:hol-cantor} below, and the Schr\"oder-Bernstein Theorem. - -\item Theories {\tt InSort} and {\tt Qsort} prove correctness properties of - insertion sort and quick sort. - -\item The definition of lazy lists demonstrates methods for handling - infinite data structures and coinduction in higher-order - logic~\cite{paulson-coind}. Theory \thydx{LList} defines an operator for - corecursion on lazy lists, which is used to define a few simple functions - such as map and append. Corecursion cannot easily define operations such - as filter, which can compute indefinitely before yielding the next - element (if any!) of the lazy list. A coinduction principle is defined - for proving equations on lazy lists. - -\item Theory {\tt PropLog} proves the soundness and completeness of classical - propositional logic, given a truth table semantics. The only connective is - $\imp$. A Hilbert-style axiom system is specified, and its set of theorems - defined inductively. A similar proof in ZF is described - elsewhere~\cite{paulson-set-II}. - -\item Theory {\tt Term} develops an experimental recursive type definition; - the recursion goes through the type constructor~\tydx{list}. - -\item Theory {\tt Simult} constructs mutually recursive sets of trees and - forests, including induction and recursion rules. - -\item Theory {\tt MT} contains Jacob Frost's formalization~\cite{frost93} of - Milner and Tofte's coinduction example~\cite{milner-coind}. This - substantial proof concerns the soundness of a type system for a simple - functional language. The semantics of recursion is given by a cyclic - environment, which makes a coinductive argument appropriate. -\end{itemize} - - -\goodbreak -\section{Example: Cantor's Theorem}\label{sec:hol-cantor} -Cantor's Theorem states that every set has more subsets than it has -elements. It has become a favourite example in higher-order logic since -it is so easily expressed: -\[ \forall f::[\alpha,\alpha]\To bool. \exists S::\alpha\To bool. - \forall x::\alpha. f(x) \not= S -\] -% -Viewing types as sets, $\alpha\To bool$ represents the powerset -of~$\alpha$. This version states that for every function from $\alpha$ to -its powerset, some subset is outside its range. - -The Isabelle proof uses HOL's set theory, with the type $\alpha\,set$ and the -operator \cdx{range}. The set~$S$ is given as an unknown instead of a -quantified variable so that we may inspect the subset found by the proof. -\begin{ttbox} -goal Set.thy "~ ?S : range(f :: 'a=>'a set)"; -{\out Level 0} -{\out ~ ?S : range(f)} -{\out 1. ~ ?S : range(f)} -\end{ttbox} -The first two steps are routine. The rule \tdx{rangeE} replaces -$\Var{S}\in {\tt range}(f)$ by $\Var{S}=f(x)$ for some~$x$. -\begin{ttbox} -by (resolve_tac [notI] 1); -{\out Level 1} -{\out ~ ?S : range(f)} -{\out 1. ?S : range(f) ==> False} -\ttbreak -by (eresolve_tac [rangeE] 1); -{\out Level 2} -{\out ~ ?S : range(f)} -{\out 1. !!x. ?S = f(x) ==> False} -\end{ttbox} -Next, we apply \tdx{equalityCE}, reasoning that since $\Var{S}=f(x)$, -we have $\Var{c}\in \Var{S}$ if and only if $\Var{c}\in f(x)$ for -any~$\Var{c}$. -\begin{ttbox} -by (eresolve_tac [equalityCE] 1); -{\out Level 3} -{\out ~ ?S : range(f)} -{\out 1. !!x. [| ?c3(x) : ?S; ?c3(x) : f(x) |] ==> False} -{\out 2. !!x. [| ~ ?c3(x) : ?S; ~ ?c3(x) : f(x) |] ==> False} -\end{ttbox} -Now we use a bit of creativity. Suppose that~$\Var{S}$ has the form of a -comprehension. Then $\Var{c}\in\{x.\Var{P}(x)\}$ implies -$\Var{P}(\Var{c})$. Destruct-resolution using \tdx{CollectD} -instantiates~$\Var{S}$ and creates the new assumption. -\begin{ttbox} -by (dresolve_tac [CollectD] 1); -{\out Level 4} -{\out ~ \{x. ?P7(x)\} : range(f)} -{\out 1. !!x. [| ?c3(x) : f(x); ?P7(?c3(x)) |] ==> False} -{\out 2. !!x. [| ~ ?c3(x) : \{x. ?P7(x)\}; ~ ?c3(x) : f(x) |] ==> False} -\end{ttbox} -Forcing a contradiction between the two assumptions of subgoal~1 completes -the instantiation of~$S$. It is now the set $\{x. x\not\in f(x)\}$, which -is the standard diagonal construction. -\begin{ttbox} -by (contr_tac 1); -{\out Level 5} -{\out ~ \{x. ~ x : f(x)\} : range(f)} -{\out 1. !!x. [| ~ x : \{x. ~ x : f(x)\}; ~ x : f(x) |] ==> False} -\end{ttbox} -The rest should be easy. To apply \tdx{CollectI} to the negated -assumption, we employ \ttindex{swap_res_tac}: -\begin{ttbox} -by (swap_res_tac [CollectI] 1); -{\out Level 6} -{\out ~ \{x. ~ x : f(x)\} : range(f)} -{\out 1. !!x. [| ~ x : f(x); ~ False |] ==> ~ x : f(x)} -\ttbreak -by (assume_tac 1); -{\out Level 7} -{\out ~ \{x. ~ x : f(x)\} : range(f)} -{\out No subgoals!} -\end{ttbox} -How much creativity is required? As it happens, Isabelle can prove this -theorem automatically. The classical set \ttindex{set_cs} contains rules for -most of the constructs of HOL's set theory. We must augment it with -\tdx{equalityCE} to break up set equalities, and then apply best-first search. -Depth-first search would diverge, but best-first search successfully navigates -through the large search space. \index{search!best-first} -\begin{ttbox} -choplev 0; -{\out Level 0} -{\out ~ ?S : range(f)} -{\out 1. ~ ?S : range(f)} -\ttbreak -by (best_tac (set_cs addSEs [equalityCE]) 1); -{\out Level 1} -{\out ~ \{x. ~ x : f(x)\} : range(f)} -{\out No subgoals!} -\end{ttbox} - -\index{higher-order logic|)}