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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@15429 37 you should start with Nipkow's overview~\cite{Nipkow-TYPES02} and consult nipkow@15429 38 the Isabelle/Isar Reference Manual~\cite{isabelle-isar-ref} and Wenzel's nipkow@15429 39 PhD thesis~\cite{Wenzel-PhD} (which discusses many proof patterns) nipkow@15429 40 for further details. If you want to use Isabelle's ML level nipkow@8743 41 directly (for example for writing your own proof procedures) see the Isabelle nipkow@8743 42 Reference Manual~\cite{isabelle-ref}; for details relating to HOL see the nipkow@8743 43 Isabelle/HOL manual~\cite{isabelle-HOL}. All manuals have a comprehensive nipkow@8743 44 index. nipkow@8743 45 nipkow@8743 46 \section{Theories} nipkow@8743 47 \label{sec:Basic:Theories} nipkow@8743 48 paulson@11428 49 \index{theories|(}% nipkow@8743 50 Working with Isabelle means creating theories. Roughly speaking, a paulson@11428 51 \textbf{theory} is a named collection of types, functions, and theorems, nipkow@8743 52 much like a module in a programming language or a specification in a nipkow@8743 53 specification language. In fact, theories in HOL can be either. The general nipkow@8743 54 format of a theory \texttt{T} is nipkow@8743 55 \begin{ttbox} nipkow@15136 56 theory T nipkow@15141 57 imports B$$@1$$ $$\ldots$$ B$$@n$$ nipkow@15136 58 begin paulson@11450 59 {\rmfamily\textit{declarations, definitions, and proofs}} nipkow@8743 60 end nipkow@15358 61 \end{ttbox}\cmmdx{theory}\cmmdx{imports} nipkow@15136 62 where \texttt{B}$@1$ \dots\ \texttt{B}$@n$ are the names of existing paulson@11450 63 theories that \texttt{T} is based on and \textit{declarations, paulson@11450 64 definitions, and proofs} represents the newly introduced concepts nipkow@8771 65 (types, functions etc.) and proofs about them. The \texttt{B}$@i$ are the paulson@11450 66 direct \textbf{parent theories}\indexbold{parent theories} of~\texttt{T}\@. paulson@11450 67 Everything defined in the parent theories (and their parents, recursively) is nipkow@8743 68 automatically visible. To avoid name clashes, identifiers can be paulson@11450 69 \textbf{qualified}\indexbold{identifiers!qualified} paulson@11450 70 by theory names as in \texttt{T.f} and~\texttt{B.f}. paulson@11450 71 Each theory \texttt{T} must paulson@11428 72 reside in a \textbf{theory file}\index{theory files} named \texttt{T.thy}. nipkow@8743 73 nipkow@8743 74 This tutorial is concerned with introducing you to the different linguistic paulson@11450 75 constructs that can fill the \textit{declarations, definitions, and paulson@11450 76 proofs} above. A complete grammar of the basic nipkow@12327 77 constructs is found in the Isabelle/Isar Reference nipkow@12327 78 Manual~\cite{isabelle-isar-ref}. nipkow@8743 79 nipkow@8743 80 \begin{warn} paulson@11428 81 HOL contains a theory \thydx{Main}, the union of all the basic paulson@10885 82 predefined theories like arithmetic, lists, sets, etc. paulson@10885 83 Unless you know what you are doing, always include \isa{Main} nipkow@10971 84 as a direct or indirect parent of all your theories. nipkow@12332 85 \end{warn} nipkow@16306 86 HOL's theory collection is available online at nipkow@16306 87 \begin{center}\small nipkow@16306 88 \url{http://isabelle.in.tum.de/library/HOL/} nipkow@16306 89 \end{center} nipkow@16359 90 and is recommended browsing. In subdirectory \texttt{Library} you find nipkow@16359 91 a growing library of useful theories that are not part of \isa{Main} nipkow@16359 92 but can be included among the parents of a theory and will then be nipkow@16359 93 loaded automatically. nipkow@16306 94 nipkow@16306 95 For the more adventurous, there is the \emph{Archive of Formal Proofs}, nipkow@16306 96 a journal-like collection of more advanced Isabelle theories: nipkow@16306 97 \begin{center}\small nipkow@16306 98 \url{http://afp.sourceforge.net/} nipkow@16306 99 \end{center} nipkow@16306 100 We hope that you will contribute to it yourself one day.% paulson@11428 101 \index{theories|)} nipkow@8743 102 nipkow@8743 103 paulson@10885 104 \section{Types, Terms and Formulae} nipkow@8743 105 \label{sec:TypesTermsForms} nipkow@8743 106 paulson@10795 107 Embedded in a theory are the types, terms and formulae of HOL\@. HOL is a typed nipkow@8771 108 logic whose type system resembles that of functional programming languages paulson@11450 109 like ML or Haskell. Thus there are paulson@11450 110 \index{types|(} nipkow@8743 111 \begin{description} paulson@11450 112 \item[base types,] paulson@11450 113 in particular \tydx{bool}, the type of truth values, paulson@11428 114 and \tydx{nat}, the type of natural numbers. paulson@11450 115 \item[type constructors,]\index{type constructors} paulson@11450 116 in particular \tydx{list}, the type of paulson@11428 117 lists, and \tydx{set}, the type of sets. Type constructors are written nipkow@8771 118 postfix, e.g.\ \isa{(nat)list} is the type of lists whose elements are nipkow@8743 119 natural numbers. Parentheses around single arguments can be dropped (as in nipkow@8771 120 \isa{nat list}), multiple arguments are separated by commas (as in nipkow@8771 121 \isa{(bool,nat)ty}). paulson@11450 122 \item[function types,]\index{function types} paulson@11450 123 denoted by \isasymFun\indexbold{$IsaFun@\isasymFun}. nipkow@8771 124 In HOL \isasymFun\ represents \emph{total} functions only. As is customary, nipkow@8771 125 \isa{$\tau@1$\isasymFun~$\tau@2$\isasymFun~$\tau@3$} means nipkow@8771 126 \isa{$\tau@1$\isasymFun~($\tau@2$\isasymFun~$\tau@3$)}. Isabelle also nipkow@8771 127 supports the notation \isa{[$\tau@1,\dots,\tau@n$] \isasymFun~$\tau$} nipkow@8771 128 which abbreviates \isa{$\tau@1$\isasymFun~$\cdots$\isasymFun~$\tau@n$nipkow@8743 129 \isasymFun~$\tau$}. paulson@11450 130 \item[type variables,]\index{type variables}\index{variables!type} paulson@10795 131 denoted by \ttindexboldpos{'a}{$Isatype}, \isa{'b} etc., just like in ML\@. They give rise nipkow@8771 132 to polymorphic types like \isa{'a \isasymFun~'a}, the type of the identity nipkow@8771 133 function. nipkow@8743 134 \end{description} nipkow@8743 135 \begin{warn} nipkow@8743 136 Types are extremely important because they prevent us from writing nipkow@16359 137 nonsense. Isabelle insists that all terms and formulae must be nipkow@16359 138 well-typed and will print an error message if a type mismatch is nipkow@16359 139 encountered. To reduce the amount of explicit type information that nipkow@16359 140 needs to be provided by the user, Isabelle infers the type of all nipkow@16359 141 variables automatically (this is called \bfindex{type inference}) nipkow@16359 142 and keeps quiet about it. Occasionally this may lead to nipkow@16359 143 misunderstandings between you and the system. If anything strange nipkow@16359 144 happens, we recommend that you ask Isabelle to display all type nipkow@16359 145 information via the Proof General menu item \textsf{Isabelle} $>$ nipkow@16359 146 \textsf{Settings} $>$ \textsf{Show Types} (see \S\ref{sec:interface} nipkow@16359 147 for details). paulson@11450 148 \end{warn}% paulson@11450 149 \index{types|)} nipkow@8743 150 nipkow@8743 151 paulson@11450 152 \index{terms|(} paulson@11450 153 \textbf{Terms} are formed as in functional programming by nipkow@8771 154 applying functions to arguments. If \isa{f} is a function of type nipkow@8771 155 \isa{$\tau@1$ \isasymFun~$\tau@2$} and \isa{t} is a term of type nipkow@8771 156 $\tau@1$ then \isa{f~t} is a term of type $\tau@2$. HOL also supports nipkow@8771 157 infix functions like \isa{+} and some basic constructs from functional paulson@11428 158 programming, such as conditional expressions: nipkow@8743 159 \begin{description} paulson@11450 160 \item[\isa{if $b$ then $t@1$ else $t@2$}]\index{*if expressions} paulson@11428 161 Here $b$ is of type \isa{bool} and $t@1$ and $t@2$ are of the same type. paulson@11450 162 \item[\isa{let $x$ = $t$ in $u$}]\index{*let expressions} nipkow@13814 163 is equivalent to $u$ where all free occurrences of $x$ have been replaced by nipkow@8743 164 $t$. For example, nipkow@8771 165 \isa{let x = 0 in x+x} is equivalent to \isa{0+0}. Multiple bindings are separated nipkow@13814 166 by semicolons: \isa{let $x@1$ = $t@1$;\dots; $x@n$ = $t@n$ in $u$}. nipkow@8771 167 \item[\isa{case $e$ of $c@1$ \isasymFun~$e@1$ |~\dots~| $c@n$ \isasymFun~$e@n$}] paulson@11450 168 \index{*case expressions} nipkow@8771 169 evaluates to $e@i$ if $e$ is of the form $c@i$. nipkow@8743 170 \end{description} nipkow@8743 171 nipkow@8743 172 Terms may also contain paulson@11450 173 \isasymlambda-abstractions.\index{lambda@$\lambda$ expressions} paulson@11450 174 For example, nipkow@8771 175 \isa{\isasymlambda{}x.~x+1} is the function that takes an argument \isa{x} and nipkow@8771 176 returns \isa{x+1}. Instead of nipkow@8771 177 \isa{\isasymlambda{}x.\isasymlambda{}y.\isasymlambda{}z.~$t$} we can write paulson@11450 178 \isa{\isasymlambda{}x~y~z.~$t$}.% paulson@11450 179 \index{terms|)} nipkow@8743 180 paulson@11450 181 \index{formulae|(}% paulson@11450 182 \textbf{Formulae} are terms of type \tydx{bool}. paulson@11428 183 There are the basic constants \cdx{True} and \cdx{False} and nipkow@8771 184 the usual logical connectives (in decreasing order of priority): paulson@11420 185 \indexboldpos{\protect\isasymnot}{$HOL0not}, \indexboldpos{\protect\isasymand}{$HOL0and}, paulson@11420 186 \indexboldpos{\protect\isasymor}{$HOL0or}, and \indexboldpos{\protect\isasymimp}{$HOL0imp}, nipkow@8743 187 all of which (except the unary \isasymnot) associate to the right. In nipkow@8771 188 particular \isa{A \isasymimp~B \isasymimp~C} means \isa{A \isasymimp~(B nipkow@8771 189 \isasymimp~C)} and is thus logically equivalent to \isa{A \isasymand~B nipkow@8771 190 \isasymimp~C} (which is \isa{(A \isasymand~B) \isasymimp~C}). nipkow@8743 191 paulson@11450 192 Equality\index{equality} is available in the form of the infix function paulson@11450 193 \isa{=} of type \isa{'a \isasymFun~'a nipkow@8771 194 \isasymFun~bool}. Thus \isa{$t@1$ = $t@2$} is a formula provided $t@1$ paulson@11450 195 and $t@2$ are terms of the same type. If $t@1$ and $t@2$ are of type paulson@11450 196 \isa{bool} then \isa{=} acts as \rmindex{if-and-only-if}. paulson@11450 197 The formula nipkow@8771 198 \isa{$t@1$~\isasymnoteq~$t@2$} is merely an abbreviation for nipkow@8771 199 \isa{\isasymnot($t@1$ = $t@2$)}. nipkow@8743 200 paulson@11450 201 Quantifiers\index{quantifiers} are written as paulson@11450 202 \isa{\isasymforall{}x.~$P$} and \isa{\isasymexists{}x.~$P$}. paulson@11420 203 There is even paulson@11450 204 \isa{\isasymuniqex{}x.~$P$}, which paulson@11420 205 means that there exists exactly one \isa{x} that satisfies \isa{$P$}. paulson@11420 206 Nested quantifications can be abbreviated: paulson@11420 207 \isa{\isasymforall{}x~y~z.~$P$} means paulson@11450 208 \isa{\isasymforall{}x.\isasymforall{}y.\isasymforall{}z.~$P$}.% paulson@11450 209 \index{formulae|)} nipkow@8743 210 nipkow@8743 211 Despite type inference, it is sometimes necessary to attach explicit paulson@11428 212 \bfindex{type constraints} to a term. The syntax is nipkow@8771 213 \isa{$t$::$\tau$} as in \isa{x < (y::nat)}. Note that nipkow@10538 214 \ttindexboldpos{::}{$Isatype} binds weakly and should therefore be enclosed paulson@11450 215 in parentheses. For instance, paulson@11450 216 \isa{x < y::nat} is ill-typed because it is interpreted as paulson@11450 217 \isa{(x < y)::nat}. Type constraints may be needed to disambiguate paulson@11450 218 expressions paulson@11450 219 involving overloaded functions such as~\isa{+}, paulson@11450 220 \isa{*} and~\isa{<}. Section~\ref{sec:overloading} paulson@11450 221 discusses overloading, while Table~\ref{tab:overloading} presents the most nipkow@10695 222 important overloaded function symbols. nipkow@8743 223 paulson@11450 224 In general, HOL's concrete \rmindex{syntax} tries to follow the conventions of paulson@11450 225 functional programming and mathematics. Here are the main rules that you paulson@11450 226 should be familiar with to avoid certain syntactic traps: nipkow@8743 227 \begin{itemize} nipkow@8743 228 \item nipkow@8771 229 Remember that \isa{f t u} means \isa{(f t) u} and not \isa{f(t u)}! nipkow@8743 230 \item nipkow@8771 231 Isabelle allows infix functions like \isa{+}. The prefix form of function nipkow@8771 232 application binds more strongly than anything else and hence \isa{f~x + y} nipkow@8771 233 means \isa{(f~x)~+~y} and not \isa{f(x+y)}. nipkow@8743 234 \item Remember that in HOL if-and-only-if is expressed using equality. But nipkow@8743 235 equality has a high priority, as befitting a relation, while if-and-only-if nipkow@8771 236 typically has the lowest priority. Thus, \isa{\isasymnot~\isasymnot~P = nipkow@8771 237 P} means \isa{\isasymnot\isasymnot(P = P)} and not nipkow@8771 238 \isa{(\isasymnot\isasymnot P) = P}. When using \isa{=} to mean nipkow@8771 239 logical equivalence, enclose both operands in parentheses, as in \isa{(A nipkow@8743 240 \isasymand~B) = (B \isasymand~A)}. nipkow@8743 241 \item nipkow@8743 242 Constructs with an opening but without a closing delimiter bind very weakly nipkow@8743 243 and should therefore be enclosed in parentheses if they appear in subterms, as paulson@11450 244 in \isa{(\isasymlambda{}x.~x) = f}. This includes paulson@11450 245 \isa{if},\index{*if expressions} paulson@11450 246 \isa{let},\index{*let expressions} paulson@11450 247 \isa{case},\index{*case expressions} paulson@11450 248 \isa{\isasymlambda}, and quantifiers. nipkow@8743 249 \item nipkow@8771 250 Never write \isa{\isasymlambda{}x.x} or \isa{\isasymforall{}x.x=x} nipkow@12327 251 because \isa{x.x} is always taken as a single qualified identifier. Write nipkow@8771 252 \isa{\isasymlambda{}x.~x} and \isa{\isasymforall{}x.~x=x} instead. paulson@11450 253 \item Identifiers\indexbold{identifiers} may contain the characters \isa{_} nipkow@12327 254 and~\isa{'}, except at the beginning. nipkow@8743 255 \end{itemize} nipkow@8743 256 paulson@11450 257 For the sake of readability, we use the usual mathematical symbols throughout nipkow@10983 258 the tutorial. Their \textsc{ascii}-equivalents are shown in table~\ref{tab:ascii} in nipkow@8771 259 the appendix. nipkow@8771 260 paulson@11450 261 \begin{warn} nipkow@16359 262 A particular problem for novices can be the priority of operators. If nipkow@16359 263 you are unsure, use additional parentheses. In those cases where nipkow@16359 264 Isabelle echoes your input, you can see which parentheses are dropped nipkow@16359 265 --- they were superfluous. If you are unsure how to interpret nipkow@16359 266 Isabelle's output because you don't know where the (dropped) nipkow@16359 267 parentheses go, set the Proof General flag \textsf{Isabelle}$>$nipkow@16359 268 \textsf{Settings}$>$\textsf{Show Brackets} (see \S\ref{sec:interface}). paulson@11450 269 \end{warn} paulson@11450 270 nipkow@8743 271 nipkow@8743 272 \section{Variables} nipkow@8743 273 \label{sec:variables} paulson@11450 274 \index{variables|(} nipkow@8743 275 paulson@11450 276 Isabelle distinguishes free and bound variables, as is customary. Bound nipkow@8743 277 variables are automatically renamed to avoid clashes with free variables. In paulson@11428 278 addition, Isabelle has a third kind of variable, called a \textbf{schematic paulson@11428 279 variable}\index{variables!schematic} or \textbf{unknown}\index{unknowns}, nipkow@13439 280 which must have a~\isa{?} as its first character. paulson@11428 281 Logically, an unknown is a free variable. But it may be nipkow@8743 282 instantiated by another term during the proof process. For example, the nipkow@8771 283 mathematical theorem$x = x$is represented in Isabelle as \isa{?x = ?x}, nipkow@8743 284 which means that Isabelle can instantiate it arbitrarily. This is in contrast nipkow@8743 285 to ordinary variables, which remain fixed. The programming language Prolog nipkow@8743 286 calls unknowns {\em logical\/} variables. nipkow@8743 287 nipkow@8743 288 Most of the time you can and should ignore unknowns and work with ordinary nipkow@8743 289 variables. Just don't be surprised that after you have finished the proof of paulson@11450 290 a theorem, Isabelle will turn your free variables into unknowns. It nipkow@8743 291 indicates that Isabelle will automatically instantiate those unknowns nipkow@8743 292 suitably when the theorem is used in some other proof. nipkow@9689 293 Note that for readability we often drop the \isa{?}s when displaying a theorem. nipkow@8743 294 \begin{warn} paulson@11450 295 For historical reasons, Isabelle accepts \isa{?} as an ASCII representation paulson@11450 296 of the $$\exists$$ symbol. However, the \isa{?} character must then be followed paulson@11450 297 by a space, as in \isa{?~x. f(x) = 0}. Otherwise, \isa{?x} is paulson@11450 298 interpreted as a schematic variable. The preferred ASCII representation of paulson@11450 299 the $$\exists$$ symbol is \isa{EX}\@. paulson@11450 300 \end{warn}% paulson@11450 301 \index{variables|)} nipkow@8743 302 paulson@10885 303 \section{Interaction and Interfaces} nipkow@16306 304 \label{sec:interface} nipkow@8771 305 nipkow@16359 306 The recommended interface for Isabelle/Isar is the (X)Emacs-based nipkow@16359 307 \bfindex{Proof General}~\cite{proofgeneral,Aspinall:TACAS:2000}. nipkow@16359 308 Interaction with Isabelle at the shell level, although possible, nipkow@16359 309 should be avoided. Most of the tutorial is independent of the nipkow@16359 310 interface and is phrased in a neutral language. For example, the nipkow@16359 311 phrase to abandon a proof'' corresponds to the obvious nipkow@16359 312 action of clicking on the \textsf{Undo} symbol in Proof General. nipkow@16359 313 Proof General specific information is often displayed in paragraphs nipkow@16359 314 identified by a miniature Proof General icon. Here are two examples: nipkow@16359 315 \begin{pgnote} nipkow@16359 316 Proof General supports a special font with mathematical symbols known nipkow@16359 317 as x-symbols''. All symbols have \textsc{ascii}-equivalents: for nipkow@16359 318 example, you can enter either \verb!&! or \verb!\! to obtain nipkow@16359 319$\land$. For a list of the most frequent symbols see table~\ref{tab:ascii} nipkow@16359 320 in the appendix. nipkow@8771 321 nipkow@16359 322 Note that by default x-symbols are not enabled. You have to switch nipkow@16359 323 them on via the menu item \textsf{Proof-General}$>$\textsf{Options}$>$nipkow@16359 324 \textsf{X-Symbols} (and save the option via the top-level nipkow@16359 325 \textsf{Options} menu). nipkow@16306 326 \end{pgnote} nipkow@8771 327 nipkow@16306 328 \begin{pgnote} nipkow@16359 329 Proof General offers the \textsf{Isabelle} menu for displaying nipkow@16359 330 information and setting flags. A particularly useful flag is nipkow@16359 331 \textsf{Isabelle}$>$\textsf{Settings}$>$\textsf{Show Types} which nipkow@16359 332 causes Isabelle to output the type information that is usually nipkow@16306 333 suppressed. This is indispensible in case of errors of all kinds nipkow@16359 334 because often the types reveal the source of the problem. Once you nipkow@16359 335 have diagnosed the problem you may no longer want to see the types nipkow@16359 336 because they clutter all output. Simply reset the flag. nipkow@16306 337 \end{pgnote} nipkow@8771 338 paulson@10885 339 \section{Getting Started} nipkow@8743 340 nipkow@16359 341 Assuming you have installed Isabelle and Proof General, you start it by typing nipkow@16359 342 \texttt{Isabelle} in a shell window. This launches a Proof General window. nipkow@16359 343 By default, you are in HOL\footnote{This is controlled by the nipkow@16359 344 \texttt{ISABELLE_LOGIC} setting, see \emph{The Isabelle System Manual} nipkow@16359 345 for more details.}. nipkow@16359 346 nipkow@16359 347 \begin{pgnote} nipkow@16359 348 You can choose a different logic via the \textsf{Isabelle}$>\$ nipkow@16359 349 \textsf{Logics} menu. For example, you may want to work in the real nipkow@16359 350 numbers, an extension of HOL (see \S\ref{sec:real}). nipkow@16359 351 % This is just excess baggage: nipkow@16359 352 %(You have to restart Proof General if you only compile the new logic nipkow@16359 353 %after having launching Proof General already). nipkow@16359 354 \end{pgnote}