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 wenzelm@12668  1 \chapter{The Basics}  nipkow@8743  2 nipkow@8743  3 \section{Introduction}  nipkow@8743  4 paulson@11405  5 This book is a tutorial on how to use the theorem prover Isabelle/HOL as a  paulson@11405  6 specification and verification system. Isabelle is a generic system for  paulson@11405  7 implementing logical formalisms, and Isabelle/HOL is the specialization  paulson@11405  8 of Isabelle for HOL, which abbreviates Higher-Order Logic. We introduce  paulson@11405  9 HOL step by step following the equation  nipkow@8743  10 $\mbox{HOL} = \mbox{Functional Programming} + \mbox{Logic}.$  paulson@11456  11 We do not assume that you are familiar with mathematical logic.  paulson@11456  12 However, we do assume that  paulson@11456  13 you are used to logical and set theoretic notation, as covered  paulson@11456  14 in a good discrete mathematics course~\cite{Rosen-DMA}, and  paulson@11450  15 that you are familiar with the basic concepts of functional  nipkow@11209  16 programming~\cite{Bird-Haskell,Hudak-Haskell,paulson-ml2,Thompson-Haskell}.  nipkow@11209  17 Although this tutorial initially concentrates on functional programming, do  nipkow@11209  18 not be misled: HOL can express most mathematical concepts, and functional  nipkow@11209  19 programming is just one particularly simple and ubiquitous instance.  nipkow@8743  20 nipkow@11205  21 Isabelle~\cite{paulson-isa-book} is implemented in ML~\cite{SML}. This has  nipkow@11205  22 influenced some of Isabelle/HOL's concrete syntax but is otherwise irrelevant  paulson@11450  23 for us: this tutorial is based on  nipkow@11213  24 Isabelle/Isar~\cite{isabelle-isar-ref}, an extension of Isabelle which hides  nipkow@11213  25 the implementation language almost completely. Thus the full name of the  nipkow@11213  26 system should be Isabelle/Isar/HOL, but that is a bit of a mouthful.  nipkow@11213  27 nipkow@11213  28 There are other implementations of HOL, in particular the one by Mike Gordon  paulson@11450  29 \index{Gordon, Mike}%  nipkow@11213  30 \emph{et al.}, which is usually referred to as the HOL system''  nipkow@11213  31 \cite{mgordon-hol}. For us, HOL refers to the logical system, and sometimes  paulson@11450  32 its incarnation Isabelle/HOL\@.  nipkow@8743  33 nipkow@8743  34 A tutorial is by definition incomplete. Currently the tutorial only  nipkow@8743  35 introduces the rudiments of Isar's proof language. To fully exploit the power  nipkow@11213  36 of Isar, in particular the ability to write readable and structured proofs,  nipkow@11213  37 you need to consult the Isabelle/Isar Reference  nipkow@12327  38 Manual~\cite{isabelle-isar-ref} and Wenzel's PhD thesis~\cite{Wenzel-PhD}  nipkow@12327  39 which discusses many proof patterns. If you want to use Isabelle's ML level  nipkow@8743  40 directly (for example for writing your own proof procedures) see the Isabelle  nipkow@8743  41 Reference Manual~\cite{isabelle-ref}; for details relating to HOL see the  nipkow@8743  42 Isabelle/HOL manual~\cite{isabelle-HOL}. All manuals have a comprehensive  nipkow@8743  43 index.  nipkow@8743  44 nipkow@8743  45 \section{Theories}  nipkow@8743  46 \label{sec:Basic:Theories}  nipkow@8743  47 paulson@11428  48 \index{theories|(}%  nipkow@8743  49 Working with Isabelle means creating theories. Roughly speaking, a  paulson@11428  50 \textbf{theory} is a named collection of types, functions, and theorems,  nipkow@8743  51 much like a module in a programming language or a specification in a  nipkow@8743  52 specification language. In fact, theories in HOL can be either. The general  nipkow@8743  53 format of a theory \texttt{T} is  nipkow@8743  54 \begin{ttbox}  nipkow@15136  55 theory T  nipkow@15141  56 imports B$$@1$$ $$\ldots$$ B$$@n$$  nipkow@15136  57 begin  paulson@11450  58 {\rmfamily\textit{declarations, definitions, and proofs}}  nipkow@8743  59 end  nipkow@8743  60 \end{ttbox}  nipkow@15136  61 where \texttt{B}$@1$ \dots\ \texttt{B}$@n$ are the names of existing  paulson@11450  62 theories that \texttt{T} is based on and \textit{declarations,  paulson@11450  63  definitions, and proofs} represents the newly introduced concepts  nipkow@8771  64 (types, functions etc.) and proofs about them. The \texttt{B}$@i$ are the  paulson@11450  65 direct \textbf{parent theories}\indexbold{parent theories} of~\texttt{T}\@.  paulson@11450  66 Everything defined in the parent theories (and their parents, recursively) is  nipkow@8743  67 automatically visible. To avoid name clashes, identifiers can be  paulson@11450  68 \textbf{qualified}\indexbold{identifiers!qualified}  paulson@11450  69 by theory names as in \texttt{T.f} and~\texttt{B.f}.  paulson@11450  70 Each theory \texttt{T} must  paulson@11428  71 reside in a \textbf{theory file}\index{theory files} named \texttt{T.thy}.  nipkow@8743  72 nipkow@8743  73 This tutorial is concerned with introducing you to the different linguistic  paulson@11450  74 constructs that can fill the \textit{declarations, definitions, and  paulson@11450  75  proofs} above. A complete grammar of the basic  nipkow@12327  76 constructs is found in the Isabelle/Isar Reference  nipkow@12327  77 Manual~\cite{isabelle-isar-ref}.  nipkow@8743  78 paulson@10885  79 HOL's theory collection is available online at  nipkow@8743  80 \begin{center}\small  nipkow@10978  81  \url{http://isabelle.in.tum.de/library/HOL/}  nipkow@8743  82 \end{center}  paulson@10885  83 and is recommended browsing. Note that most of the theories  nipkow@9541  84 are based on classical Isabelle without the Isar extension. This means that  nipkow@9541  85 they look slightly different than the theories in this tutorial, and that all  nipkow@9541  86 proofs are in separate ML files.  nipkow@9541  87 nipkow@8743  88 \begin{warn}  paulson@11428  89  HOL contains a theory \thydx{Main}, the union of all the basic  paulson@10885  90  predefined theories like arithmetic, lists, sets, etc.  paulson@10885  91  Unless you know what you are doing, always include \isa{Main}  nipkow@10971  92  as a direct or indirect parent of all your theories.  nipkow@12332  93 \end{warn}  nipkow@12473  94 There is also a growing Library~\cite{HOL-Library}\index{Library}  nipkow@13814  95 of useful theories that are not part of \isa{Main} but can be included  nipkow@12473  96 among the parents of a theory and will then be loaded automatically.%  paulson@11428  97 \index{theories|)}  nipkow@8743  98 nipkow@8743  99 paulson@10885  100 \section{Types, Terms and Formulae}  nipkow@8743  101 \label{sec:TypesTermsForms}  nipkow@8743  102 paulson@10795  103 Embedded in a theory are the types, terms and formulae of HOL\@. HOL is a typed  nipkow@8771  104 logic whose type system resembles that of functional programming languages  paulson@11450  105 like ML or Haskell. Thus there are  paulson@11450  106 \index{types|(}  nipkow@8743  107 \begin{description}  paulson@11450  108 \item[base types,]  paulson@11450  109 in particular \tydx{bool}, the type of truth values,  paulson@11428  110 and \tydx{nat}, the type of natural numbers.  paulson@11450  111 \item[type constructors,]\index{type constructors}  paulson@11450  112  in particular \tydx{list}, the type of  paulson@11428  113 lists, and \tydx{set}, the type of sets. Type constructors are written  nipkow@8771  114 postfix, e.g.\ \isa{(nat)list} is the type of lists whose elements are  nipkow@8743  115 natural numbers. Parentheses around single arguments can be dropped (as in  nipkow@8771  116 \isa{nat list}), multiple arguments are separated by commas (as in  nipkow@8771  117 \isa{(bool,nat)ty}).  paulson@11450  118 \item[function types,]\index{function types}  paulson@11450  119 denoted by \isasymFun\indexbold{$IsaFun@\isasymFun}.  nipkow@8771  120  In HOL \isasymFun\ represents \emph{total} functions only. As is customary,  nipkow@8771  121  \isa{$\tau@1$\isasymFun~$\tau@2$\isasymFun~$\tau@3$} means  nipkow@8771  122  \isa{$\tau@1$\isasymFun~($\tau@2$\isasymFun~$\tau@3$)}. Isabelle also  nipkow@8771  123  supports the notation \isa{[$\tau@1,\dots,\tau@n$] \isasymFun~$\tau$}  nipkow@8771  124  which abbreviates \isa{$\tau@1$\isasymFun~$\cdots$\isasymFun~$\tau@n$ nipkow@8743  125  \isasymFun~$\tau$}.  paulson@11450  126 \item[type variables,]\index{type variables}\index{variables!type}  paulson@10795  127  denoted by \ttindexboldpos{'a}{$Isatype}, \isa{'b} etc., just like in ML\@. They give rise  nipkow@8771  128  to polymorphic types like \isa{'a \isasymFun~'a}, the type of the identity  nipkow@8771  129  function.  nipkow@8743  130 \end{description}  nipkow@8743  131 \begin{warn}  nipkow@8743  132  Types are extremely important because they prevent us from writing  nipkow@8743  133  nonsense. Isabelle insists that all terms and formulae must be well-typed  nipkow@8743  134  and will print an error message if a type mismatch is encountered. To  nipkow@8743  135  reduce the amount of explicit type information that needs to be provided by  nipkow@8743  136  the user, Isabelle infers the type of all variables automatically (this is  nipkow@8743  137  called \bfindex{type inference}) and keeps quiet about it. Occasionally  nipkow@8743  138  this may lead to misunderstandings between you and the system. If anything  paulson@11428  139  strange happens, we recommend that you set the flag\index{flags}  paulson@11428  140  \isa{show_types}\index{*show_types (flag)}.  paulson@11428  141  Isabelle will then display type information  paulson@11450  142  that is usually suppressed. Simply type  nipkow@8743  143 \begin{ttbox}  nipkow@8743  144 ML "set show_types"  nipkow@8743  145 \end{ttbox}  nipkow@8743  146 nipkow@8743  147 \noindent  nipkow@10971  148 This can be reversed by \texttt{ML "reset show_types"}. Various other flags,  paulson@11428  149 which we introduce as we go along, can be set and reset in the same manner.%  paulson@11428  150 \index{flags!setting and resetting}  paulson@11450  151 \end{warn}%  paulson@11450  152 \index{types|)}  nipkow@8743  153 nipkow@8743  154 paulson@11450  155 \index{terms|(}  paulson@11450  156 \textbf{Terms} are formed as in functional programming by  nipkow@8771  157 applying functions to arguments. If \isa{f} is a function of type  nipkow@8771  158 \isa{$\tau@1$ \isasymFun~$\tau@2$} and \isa{t} is a term of type  nipkow@8771  159 $\tau@1$ then \isa{f~t} is a term of type $\tau@2$. HOL also supports  nipkow@8771  160 infix functions like \isa{+} and some basic constructs from functional  paulson@11428  161 programming, such as conditional expressions:  nipkow@8743  162 \begin{description}  paulson@11450  163 \item[\isa{if $b$ then $t@1$ else $t@2$}]\index{*if expressions}  paulson@11428  164 Here $b$ is of type \isa{bool} and $t@1$ and $t@2$ are of the same type.  paulson@11450  165 \item[\isa{let $x$ = $t$ in $u$}]\index{*let expressions}  nipkow@13814  166 is equivalent to $u$ where all free occurrences of $x$ have been replaced by  nipkow@8743  167 $t$. For example,  nipkow@8771  168 \isa{let x = 0 in x+x} is equivalent to \isa{0+0}. Multiple bindings are separated  nipkow@13814  169 by semicolons: \isa{let $x@1$ = $t@1$;\dots; $x@n$ = $t@n$ in $u$}.  nipkow@8771  170 \item[\isa{case $e$ of $c@1$ \isasymFun~$e@1$ |~\dots~| $c@n$ \isasymFun~$e@n$}]  paulson@11450  171 \index{*case expressions}  nipkow@8771  172 evaluates to $e@i$ if $e$ is of the form $c@i$.  nipkow@8743  173 \end{description}  nipkow@8743  174 nipkow@8743  175 Terms may also contain  paulson@11450  176 \isasymlambda-abstractions.\index{lambda@$\lambda$ expressions}  paulson@11450  177 For example,  nipkow@8771  178 \isa{\isasymlambda{}x.~x+1} is the function that takes an argument \isa{x} and  nipkow@8771  179 returns \isa{x+1}. Instead of  nipkow@8771  180 \isa{\isasymlambda{}x.\isasymlambda{}y.\isasymlambda{}z.~$t$} we can write  paulson@11450  181 \isa{\isasymlambda{}x~y~z.~$t$}.%  paulson@11450  182 \index{terms|)}  nipkow@8743  183 paulson@11450  184 \index{formulae|(}%  paulson@11450  185 \textbf{Formulae} are terms of type \tydx{bool}.  paulson@11428  186 There are the basic constants \cdx{True} and \cdx{False} and  nipkow@8771  187 the usual logical connectives (in decreasing order of priority):  paulson@11420  188 \indexboldpos{\protect\isasymnot}{$HOL0not}, \indexboldpos{\protect\isasymand}{$HOL0and},  paulson@11420  189 \indexboldpos{\protect\isasymor}{$HOL0or}, and \indexboldpos{\protect\isasymimp}{$HOL0imp},  nipkow@8743  190 all of which (except the unary \isasymnot) associate to the right. In  nipkow@8771  191 particular \isa{A \isasymimp~B \isasymimp~C} means \isa{A \isasymimp~(B  nipkow@8771  192  \isasymimp~C)} and is thus logically equivalent to \isa{A \isasymand~B  nipkow@8771  193  \isasymimp~C} (which is \isa{(A \isasymand~B) \isasymimp~C}).  nipkow@8743  194 paulson@11450  195 Equality\index{equality} is available in the form of the infix function  paulson@11450  196 \isa{=} of type \isa{'a \isasymFun~'a  nipkow@8771  197  \isasymFun~bool}. Thus \isa{$t@1$ = $t@2$} is a formula provided $t@1$  paulson@11450  198 and $t@2$ are terms of the same type. If $t@1$ and $t@2$ are of type  paulson@11450  199 \isa{bool} then \isa{=} acts as \rmindex{if-and-only-if}.  paulson@11450  200 The formula  nipkow@8771  201 \isa{$t@1$~\isasymnoteq~$t@2$} is merely an abbreviation for  nipkow@8771  202 \isa{\isasymnot($t@1$ = $t@2$)}.  nipkow@8743  203 paulson@11450  204 Quantifiers\index{quantifiers} are written as  paulson@11450  205 \isa{\isasymforall{}x.~$P$} and \isa{\isasymexists{}x.~$P$}.  paulson@11420  206 There is even  paulson@11450  207 \isa{\isasymuniqex{}x.~$P$}, which  paulson@11420  208 means that there exists exactly one \isa{x} that satisfies \isa{$P$}.  paulson@11420  209 Nested quantifications can be abbreviated:  paulson@11420  210 \isa{\isasymforall{}x~y~z.~$P$} means  paulson@11450  211 \isa{\isasymforall{}x.\isasymforall{}y.\isasymforall{}z.~$P$}.%  paulson@11450  212 \index{formulae|)}  nipkow@8743  213 nipkow@8743  214 Despite type inference, it is sometimes necessary to attach explicit  paulson@11428  215 \bfindex{type constraints} to a term. The syntax is  nipkow@8771  216 \isa{$t$::$\tau$} as in \isa{x < (y::nat)}. Note that  nipkow@10538  217 \ttindexboldpos{::}{$Isatype} binds weakly and should therefore be enclosed  paulson@11450  218 in parentheses. For instance,  paulson@11450  219 \isa{x < y::nat} is ill-typed because it is interpreted as  paulson@11450  220 \isa{(x < y)::nat}. Type constraints may be needed to disambiguate  paulson@11450  221 expressions  paulson@11450  222 involving overloaded functions such as~\isa{+},  paulson@11450  223 \isa{*} and~\isa{<}. Section~\ref{sec:overloading}  paulson@11450  224 discusses overloading, while Table~\ref{tab:overloading} presents the most  nipkow@10695  225 important overloaded function symbols.  nipkow@8743  226 paulson@11450  227 In general, HOL's concrete \rmindex{syntax} tries to follow the conventions of  paulson@11450  228 functional programming and mathematics. Here are the main rules that you  paulson@11450  229 should be familiar with to avoid certain syntactic traps:  nipkow@8743  230 \begin{itemize}  nipkow@8743  231 \item  nipkow@8771  232 Remember that \isa{f t u} means \isa{(f t) u} and not \isa{f(t u)}!  nipkow@8743  233 \item  nipkow@8771  234 Isabelle allows infix functions like \isa{+}. The prefix form of function  nipkow@8771  235 application binds more strongly than anything else and hence \isa{f~x + y}  nipkow@8771  236 means \isa{(f~x)~+~y} and not \isa{f(x+y)}.  nipkow@8743  237 \item Remember that in HOL if-and-only-if is expressed using equality. But  nipkow@8743  238  equality has a high priority, as befitting a relation, while if-and-only-if  nipkow@8771  239  typically has the lowest priority. Thus, \isa{\isasymnot~\isasymnot~P =  nipkow@8771  240  P} means \isa{\isasymnot\isasymnot(P = P)} and not  nipkow@8771  241  \isa{(\isasymnot\isasymnot P) = P}. When using \isa{=} to mean  nipkow@8771  242  logical equivalence, enclose both operands in parentheses, as in \isa{(A  nipkow@8743  243  \isasymand~B) = (B \isasymand~A)}.  nipkow@8743  244 \item  nipkow@8743  245 Constructs with an opening but without a closing delimiter bind very weakly  nipkow@8743  246 and should therefore be enclosed in parentheses if they appear in subterms, as  paulson@11450  247 in \isa{(\isasymlambda{}x.~x) = f}. This includes  paulson@11450  248 \isa{if},\index{*if expressions}  paulson@11450  249 \isa{let},\index{*let expressions}  paulson@11450  250 \isa{case},\index{*case expressions}  paulson@11450  251 \isa{\isasymlambda}, and quantifiers.  nipkow@8743  252 \item  nipkow@8771  253 Never write \isa{\isasymlambda{}x.x} or \isa{\isasymforall{}x.x=x}  nipkow@12327  254 because \isa{x.x} is always taken as a single qualified identifier. Write  nipkow@8771  255 \isa{\isasymlambda{}x.~x} and \isa{\isasymforall{}x.~x=x} instead.  paulson@11450  256 \item Identifiers\indexbold{identifiers} may contain the characters \isa{_}  nipkow@12327  257 and~\isa{'}, except at the beginning.  nipkow@8743  258 \end{itemize}  nipkow@8743  259 paulson@11450  260 For the sake of readability, we use the usual mathematical symbols throughout  nipkow@10983  261 the tutorial. Their \textsc{ascii}-equivalents are shown in table~\ref{tab:ascii} in  nipkow@8771  262 the appendix.  nipkow@8771  263 paulson@11450  264 \begin{warn}  paulson@11450  265 A particular  paulson@11450  266 problem for novices can be the priority of operators. If you are unsure, use  paulson@11450  267 additional parentheses. In those cases where Isabelle echoes your  paulson@11450  268 input, you can see which parentheses are dropped --- they were superfluous. If  paulson@11450  269 you are unsure how to interpret Isabelle's output because you don't know  paulson@11450  270 where the (dropped) parentheses go, set the flag\index{flags}  paulson@11450  271 \isa{show_brackets}\index{*show_brackets (flag)}:  paulson@11450  272 \begin{ttbox}  paulson@11450  273 ML "set show_brackets"; $$\dots$$; ML "reset show_brackets";  paulson@11450  274 \end{ttbox}  paulson@11450  275 \end{warn}  paulson@11450  276 nipkow@8743  277 nipkow@8743  278 \section{Variables}  nipkow@8743  279 \label{sec:variables}  paulson@11450  280 \index{variables|(}  nipkow@8743  281 paulson@11450  282 Isabelle distinguishes free and bound variables, as is customary. Bound  nipkow@8743  283 variables are automatically renamed to avoid clashes with free variables. In  paulson@11428  284 addition, Isabelle has a third kind of variable, called a \textbf{schematic  paulson@11428  285  variable}\index{variables!schematic} or \textbf{unknown}\index{unknowns},  nipkow@13439  286 which must have a~\isa{?} as its first character.  paulson@11428  287 Logically, an unknown is a free variable. But it may be  nipkow@8743  288 instantiated by another term during the proof process. For example, the  nipkow@8771  289 mathematical theorem$x = x$is represented in Isabelle as \isa{?x = ?x},  nipkow@8743  290 which means that Isabelle can instantiate it arbitrarily. This is in contrast  nipkow@8743  291 to ordinary variables, which remain fixed. The programming language Prolog  nipkow@8743  292 calls unknowns {\em logical\/} variables.  nipkow@8743  293 nipkow@8743  294 Most of the time you can and should ignore unknowns and work with ordinary  nipkow@8743  295 variables. Just don't be surprised that after you have finished the proof of  paulson@11450  296 a theorem, Isabelle will turn your free variables into unknowns. It  nipkow@8743  297 indicates that Isabelle will automatically instantiate those unknowns  nipkow@8743  298 suitably when the theorem is used in some other proof.  nipkow@9689  299 Note that for readability we often drop the \isa{?}s when displaying a theorem.  nipkow@8743  300 \begin{warn}  paulson@11450  301  For historical reasons, Isabelle accepts \isa{?} as an ASCII representation  paulson@11450  302  of the $$\exists$$ symbol. However, the \isa{?} character must then be followed  paulson@11450  303  by a space, as in \isa{?~x. f(x) = 0}. Otherwise, \isa{?x} is  paulson@11450  304  interpreted as a schematic variable. The preferred ASCII representation of  paulson@11450  305  the $$\exists$$ symbol is \isa{EX}\@.  paulson@11450  306 \end{warn}%  paulson@11450  307 \index{variables|)}  nipkow@8743  308 paulson@10885  309 \section{Interaction and Interfaces}  nipkow@8771  310 nipkow@8771  311 Interaction with Isabelle can either occur at the shell level or through more  paulson@11301  312 advanced interfaces. To keep the tutorial independent of the interface, we  paulson@11301  313 have phrased the description of the interaction in a neutral language. For  nipkow@8771  314 example, the phrase to abandon a proof'' means to type \isacommand{oops} at the  nipkow@8771  315 shell level, which is explained the first time the phrase is used. Other  nipkow@8771  316 interfaces perform the same act by cursor movements and/or mouse clicks.  nipkow@8771  317 Although shell-based interaction is quite feasible for the kind of proof  nipkow@8771  318 scripts currently presented in this tutorial, the recommended interface for  nipkow@8771  319 Isabelle/Isar is the Emacs-based \bfindex{Proof  paulson@11450  320  General}~\cite{proofgeneral,Aspinall:TACAS:2000}.  nipkow@8771  321 nipkow@8771  322 Some interfaces (including the shell level) offer special fonts with  nipkow@10983  323 mathematical symbols. For those that do not, remember that \textsc{ascii}-equivalents  nipkow@10978  324 are shown in table~\ref{tab:ascii} in the appendix.  nipkow@8771  325 nipkow@9541  326 Finally, a word about semicolons.\indexbold{$Isar@\texttt{;}}  nipkow@9541  327 Commands may but need not be terminated by semicolons.  nipkow@9541  328 At the shell level it is advisable to use semicolons to enforce that a command  nipkow@8771  329 is executed immediately; otherwise Isabelle may wait for the next keyword  nipkow@9541  330 before it knows that the command is complete.  nipkow@8771  331 nipkow@8771  332 paulson@10885  333 \section{Getting Started}  nipkow@8743  334 nipkow@8743  335 Assuming you have installed Isabelle, you start it by typing \texttt{isabelle  nipkow@8743  336  -I HOL} in a shell window.\footnote{Simply executing \texttt{isabelle -I}  nipkow@8743  337  starts the default logic, which usually is already \texttt{HOL}. This is  nipkow@8743  338  controlled by the \texttt{ISABELLE_LOGIC} setting, see \emph{The Isabelle  nipkow@8743  339  System Manual} for more details.} This presents you with Isabelle's most  nipkow@10983  340 basic \textsc{ascii} interface. In addition you need to open an editor window to  paulson@11450  341 create theory files. While you are developing a theory, we recommend that you  nipkow@8743  342 type each command into the file first and then enter it into Isabelle by  nipkow@8743  343 copy-and-paste, thus ensuring that you have a complete record of your theory.  nipkow@8771  344 As mentioned above, Proof General offers a much superior interface.  paulson@10795  345 If you have installed Proof General, you can start it by typing \texttt{Isabelle}.