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doc-src/TutorialI/basics.tex

author | wenzelm |

Wed Jul 25 12:38:54 2012 +0200 (2012-07-25) | |

changeset 48497 | ba61aceaa18a |

parent 38432 | 439f50a241c1 |

permissions | -rw-r--r-- |

some updates on "Building a repository version of Isabelle";

1 \chapter{The Basics}

3 \section{Introduction}

5 This book is a tutorial on how to use the theorem prover Isabelle/HOL as a

6 specification and verification system. Isabelle is a generic system for

7 implementing logical formalisms, and Isabelle/HOL is the specialization

8 of Isabelle for HOL, which abbreviates Higher-Order Logic. We introduce

9 HOL step by step following the equation

10 \[ \mbox{HOL} = \mbox{Functional Programming} + \mbox{Logic}. \]

11 We do not assume that you are familiar with mathematical logic.

12 However, we do assume that

13 you are used to logical and set theoretic notation, as covered

14 in a good discrete mathematics course~\cite{Rosen-DMA}, and

15 that you are familiar with the basic concepts of functional

16 programming~\cite{Bird-Haskell,Hudak-Haskell,paulson-ml2,Thompson-Haskell}.

17 Although this tutorial initially concentrates on functional programming, do

18 not be misled: HOL can express most mathematical concepts, and functional

19 programming is just one particularly simple and ubiquitous instance.

21 Isabelle~\cite{paulson-isa-book} is implemented in ML~\cite{SML}. This has

22 influenced some of Isabelle/HOL's concrete syntax but is otherwise irrelevant

23 for us: this tutorial is based on

24 Isabelle/Isar~\cite{isabelle-isar-ref}, an extension of Isabelle which hides

25 the implementation language almost completely. Thus the full name of the

26 system should be Isabelle/Isar/HOL, but that is a bit of a mouthful.

28 There are other implementations of HOL, in particular the one by Mike Gordon

29 \index{Gordon, Mike}%

30 \emph{et al.}, which is usually referred to as ``the HOL system''

31 \cite{mgordon-hol}. For us, HOL refers to the logical system, and sometimes

32 its incarnation Isabelle/HOL\@.

34 A tutorial is by definition incomplete. Currently the tutorial only

35 introduces the rudiments of Isar's proof language. To fully exploit the power

36 of Isar, in particular the ability to write readable and structured proofs,

37 you should start with Nipkow's overview~\cite{Nipkow-TYPES02} and consult

38 the Isabelle/Isar Reference Manual~\cite{isabelle-isar-ref} and Wenzel's

39 PhD thesis~\cite{Wenzel-PhD} (which discusses many proof patterns)

40 for further details. If you want to use Isabelle's ML level

41 directly (for example for writing your own proof procedures) see the Isabelle

42 Reference Manual~\cite{isabelle-ref}; for details relating to HOL see the

43 Isabelle/HOL manual~\cite{isabelle-HOL}. All manuals have a comprehensive

44 index.

46 \section{Theories}

47 \label{sec:Basic:Theories}

49 \index{theories|(}%

50 Working with Isabelle means creating theories. Roughly speaking, a

51 \textbf{theory} is a named collection of types, functions, and theorems,

52 much like a module in a programming language or a specification in a

53 specification language. In fact, theories in HOL can be either. The general

54 format of a theory \texttt{T} is

55 \begin{ttbox}

56 theory T

57 imports B\(@1\) \(\ldots\) B\(@n\)

58 begin

59 {\rmfamily\textit{declarations, definitions, and proofs}}

60 end

61 \end{ttbox}\cmmdx{theory}\cmmdx{imports}

62 where \texttt{B}$@1$ \dots\ \texttt{B}$@n$ are the names of existing

63 theories that \texttt{T} is based on and \textit{declarations,

64 definitions, and proofs} represents the newly introduced concepts

65 (types, functions etc.) and proofs about them. The \texttt{B}$@i$ are the

66 direct \textbf{parent theories}\indexbold{parent theories} of~\texttt{T}\@.

67 Everything defined in the parent theories (and their parents, recursively) is

68 automatically visible. To avoid name clashes, identifiers can be

69 \textbf{qualified}\indexbold{identifiers!qualified}

70 by theory names as in \texttt{T.f} and~\texttt{B.f}.

71 Each theory \texttt{T} must

72 reside in a \textbf{theory file}\index{theory files} named \texttt{T.thy}.

74 This tutorial is concerned with introducing you to the different linguistic

75 constructs that can fill the \textit{declarations, definitions, and

76 proofs} above. A complete grammar of the basic

77 constructs is found in the Isabelle/Isar Reference

78 Manual~\cite{isabelle-isar-ref}.

80 \begin{warn}

81 HOL contains a theory \thydx{Main}, the union of all the basic

82 predefined theories like arithmetic, lists, sets, etc.

83 Unless you know what you are doing, always include \isa{Main}

84 as a direct or indirect parent of all your theories.

85 \end{warn}

86 HOL's theory collection is available online at

87 \begin{center}\small

88 \url{http://isabelle.in.tum.de/library/HOL/}

89 \end{center}

90 and is recommended browsing. In subdirectory \texttt{Library} you find

91 a growing library of useful theories that are not part of \isa{Main}

92 but can be included among the parents of a theory and will then be

93 loaded automatically.

95 For the more adventurous, there is the \emph{Archive of Formal Proofs},

96 a journal-like collection of more advanced Isabelle theories:

97 \begin{center}\small

98 \url{http://afp.sourceforge.net/}

99 \end{center}

100 We hope that you will contribute to it yourself one day.%

101 \index{theories|)}

104 \section{Types, Terms and Formulae}

105 \label{sec:TypesTermsForms}

107 Embedded in a theory are the types, terms and formulae of HOL\@. HOL is a typed

108 logic whose type system resembles that of functional programming languages

109 like ML or Haskell. Thus there are

110 \index{types|(}

111 \begin{description}

112 \item[base types,]

113 in particular \tydx{bool}, the type of truth values,

114 and \tydx{nat}, the type of natural numbers.

115 \item[type constructors,]\index{type constructors}

116 in particular \tydx{list}, the type of

117 lists, and \tydx{set}, the type of sets. Type constructors are written

118 postfix, e.g.\ \isa{(nat)list} is the type of lists whose elements are

119 natural numbers. Parentheses around single arguments can be dropped (as in

120 \isa{nat list}), multiple arguments are separated by commas (as in

121 \isa{(bool,nat)ty}).

122 \item[function types,]\index{function types}

123 denoted by \isasymFun\indexbold{$IsaFun@\isasymFun}.

124 In HOL \isasymFun\ represents \emph{total} functions only. As is customary,

125 \isa{$\tau@1$ \isasymFun~$\tau@2$ \isasymFun~$\tau@3$} means

126 \isa{$\tau@1$ \isasymFun~($\tau@2$ \isasymFun~$\tau@3$)}. Isabelle also

127 supports the notation \isa{[$\tau@1,\dots,\tau@n$] \isasymFun~$\tau$}

128 which abbreviates \isa{$\tau@1$ \isasymFun~$\cdots$ \isasymFun~$\tau@n$

129 \isasymFun~$\tau$}.

130 \item[type variables,]\index{type variables}\index{variables!type}

131 denoted by \ttindexboldpos{'a}{$Isatype}, \isa{'b} etc., just like in ML\@. They give rise

132 to polymorphic types like \isa{'a \isasymFun~'a}, the type of the identity

133 function.

134 \end{description}

135 \begin{warn}

136 Types are extremely important because they prevent us from writing

137 nonsense. Isabelle insists that all terms and formulae must be

138 well-typed and will print an error message if a type mismatch is

139 encountered. To reduce the amount of explicit type information that

140 needs to be provided by the user, Isabelle infers the type of all

141 variables automatically (this is called \bfindex{type inference})

142 and keeps quiet about it. Occasionally this may lead to

143 misunderstandings between you and the system. If anything strange

144 happens, we recommend that you ask Isabelle to display all type

145 information via the Proof General menu item \pgmenu{Isabelle} $>$

146 \pgmenu{Settings} $>$ \pgmenu{Show Types} (see \S\ref{sec:interface}

147 for details).

148 \end{warn}%

149 \index{types|)}

152 \index{terms|(}

153 \textbf{Terms} are formed as in functional programming by

154 applying functions to arguments. If \isa{f} is a function of type

155 \isa{$\tau@1$ \isasymFun~$\tau@2$} and \isa{t} is a term of type

156 $\tau@1$ then \isa{f~t} is a term of type $\tau@2$. HOL also supports

157 infix functions like \isa{+} and some basic constructs from functional

158 programming, such as conditional expressions:

159 \begin{description}

160 \item[\isa{if $b$ then $t@1$ else $t@2$}]\index{*if expressions}

161 Here $b$ is of type \isa{bool} and $t@1$ and $t@2$ are of the same type.

162 \item[\isa{let $x$ = $t$ in $u$}]\index{*let expressions}

163 is equivalent to $u$ where all free occurrences of $x$ have been replaced by

164 $t$. For example,

165 \isa{let x = 0 in x+x} is equivalent to \isa{0+0}. Multiple bindings are separated

166 by semicolons: \isa{let $x@1$ = $t@1$;\dots; $x@n$ = $t@n$ in $u$}.

167 \item[\isa{case $e$ of $c@1$ \isasymFun~$e@1$ |~\dots~| $c@n$ \isasymFun~$e@n$}]

168 \index{*case expressions}

169 evaluates to $e@i$ if $e$ is of the form $c@i$.

170 \end{description}

172 Terms may also contain

173 \isasymlambda-abstractions.\index{lambda@$\lambda$ expressions}

174 For example,

175 \isa{\isasymlambda{}x.~x+1} is the function that takes an argument \isa{x} and

176 returns \isa{x+1}. Instead of

177 \isa{\isasymlambda{}x.\isasymlambda{}y.\isasymlambda{}z.~$t$} we can write

178 \isa{\isasymlambda{}x~y~z.~$t$}.%

179 \index{terms|)}

181 \index{formulae|(}%

182 \textbf{Formulae} are terms of type \tydx{bool}.

183 There are the basic constants \cdx{True} and \cdx{False} and

184 the usual logical connectives (in decreasing order of priority):

185 \indexboldpos{\protect\isasymnot}{$HOL0not}, \indexboldpos{\protect\isasymand}{$HOL0and},

186 \indexboldpos{\protect\isasymor}{$HOL0or}, and \indexboldpos{\protect\isasymimp}{$HOL0imp},

187 all of which (except the unary \isasymnot) associate to the right. In

188 particular \isa{A \isasymimp~B \isasymimp~C} means \isa{A \isasymimp~(B

189 \isasymimp~C)} and is thus logically equivalent to \isa{A \isasymand~B

190 \isasymimp~C} (which is \isa{(A \isasymand~B) \isasymimp~C}).

192 Equality\index{equality} is available in the form of the infix function

193 \isa{=} of type \isa{'a \isasymFun~'a

194 \isasymFun~bool}. Thus \isa{$t@1$ = $t@2$} is a formula provided $t@1$

195 and $t@2$ are terms of the same type. If $t@1$ and $t@2$ are of type

196 \isa{bool} then \isa{=} acts as \rmindex{if-and-only-if}.

197 The formula

198 \isa{$t@1$~\isasymnoteq~$t@2$} is merely an abbreviation for

199 \isa{\isasymnot($t@1$ = $t@2$)}.

201 Quantifiers\index{quantifiers} are written as

202 \isa{\isasymforall{}x.~$P$} and \isa{\isasymexists{}x.~$P$}.

203 There is even

204 \isa{\isasymuniqex{}x.~$P$}, which

205 means that there exists exactly one \isa{x} that satisfies \isa{$P$}.

206 Nested quantifications can be abbreviated:

207 \isa{\isasymforall{}x~y~z.~$P$} means

208 \isa{\isasymforall{}x.\isasymforall{}y.\isasymforall{}z.~$P$}.%

209 \index{formulae|)}

211 Despite type inference, it is sometimes necessary to attach explicit

212 \bfindex{type constraints} to a term. The syntax is

213 \isa{$t$::$\tau$} as in \isa{x < (y::nat)}. Note that

214 \ttindexboldpos{::}{$Isatype} binds weakly and should therefore be enclosed

215 in parentheses. For instance,

216 \isa{x < y::nat} is ill-typed because it is interpreted as

217 \isa{(x < y)::nat}. Type constraints may be needed to disambiguate

218 expressions

219 involving overloaded functions such as~\isa{+},

220 \isa{*} and~\isa{<}. Section~\ref{sec:overloading}

221 discusses overloading, while Table~\ref{tab:overloading} presents the most

222 important overloaded function symbols.

224 In general, HOL's concrete \rmindex{syntax} tries to follow the conventions of

225 functional programming and mathematics. Here are the main rules that you

226 should be familiar with to avoid certain syntactic traps:

227 \begin{itemize}

228 \item

229 Remember that \isa{f t u} means \isa{(f t) u} and not \isa{f(t u)}!

230 \item

231 Isabelle allows infix functions like \isa{+}. The prefix form of function

232 application binds more strongly than anything else and hence \isa{f~x + y}

233 means \isa{(f~x)~+~y} and not \isa{f(x+y)}.

234 \item Remember that in HOL if-and-only-if is expressed using equality. But

235 equality has a high priority, as befitting a relation, while if-and-only-if

236 typically has the lowest priority. Thus, \isa{\isasymnot~\isasymnot~P =

237 P} means \isa{\isasymnot\isasymnot(P = P)} and not

238 \isa{(\isasymnot\isasymnot P) = P}. When using \isa{=} to mean

239 logical equivalence, enclose both operands in parentheses, as in \isa{(A

240 \isasymand~B) = (B \isasymand~A)}.

241 \item

242 Constructs with an opening but without a closing delimiter bind very weakly

243 and should therefore be enclosed in parentheses if they appear in subterms, as

244 in \isa{(\isasymlambda{}x.~x) = f}. This includes

245 \isa{if},\index{*if expressions}

246 \isa{let},\index{*let expressions}

247 \isa{case},\index{*case expressions}

248 \isa{\isasymlambda}, and quantifiers.

249 \item

250 Never write \isa{\isasymlambda{}x.x} or \isa{\isasymforall{}x.x=x}

251 because \isa{x.x} is always taken as a single qualified identifier. Write

252 \isa{\isasymlambda{}x.~x} and \isa{\isasymforall{}x.~x=x} instead.

253 \item Identifiers\indexbold{identifiers} may contain the characters \isa{_}

254 and~\isa{'}, except at the beginning.

255 \end{itemize}

257 For the sake of readability, we use the usual mathematical symbols throughout

258 the tutorial. Their \textsc{ascii}-equivalents are shown in table~\ref{tab:ascii} in

259 the appendix.

261 \begin{warn}

262 A particular problem for novices can be the priority of operators. If

263 you are unsure, use additional parentheses. In those cases where

264 Isabelle echoes your input, you can see which parentheses are dropped

265 --- they were superfluous. If you are unsure how to interpret

266 Isabelle's output because you don't know where the (dropped)

267 parentheses go, set the Proof General flag \pgmenu{Isabelle} $>$

268 \pgmenu{Settings} $>$ \pgmenu{Show Brackets} (see \S\ref{sec:interface}).

269 \end{warn}

272 \section{Variables}

273 \label{sec:variables}

274 \index{variables|(}

276 Isabelle distinguishes free and bound variables, as is customary. Bound

277 variables are automatically renamed to avoid clashes with free variables. In

278 addition, Isabelle has a third kind of variable, called a \textbf{schematic

279 variable}\index{variables!schematic} or \textbf{unknown}\index{unknowns},

280 which must have a~\isa{?} as its first character.

281 Logically, an unknown is a free variable. But it may be

282 instantiated by another term during the proof process. For example, the

283 mathematical theorem $x = x$ is represented in Isabelle as \isa{?x = ?x},

284 which means that Isabelle can instantiate it arbitrarily. This is in contrast

285 to ordinary variables, which remain fixed. The programming language Prolog

286 calls unknowns {\em logical\/} variables.

288 Most of the time you can and should ignore unknowns and work with ordinary

289 variables. Just don't be surprised that after you have finished the proof of

290 a theorem, Isabelle will turn your free variables into unknowns. It

291 indicates that Isabelle will automatically instantiate those unknowns

292 suitably when the theorem is used in some other proof.

293 Note that for readability we often drop the \isa{?}s when displaying a theorem.

294 \begin{warn}

295 For historical reasons, Isabelle accepts \isa{?} as an ASCII representation

296 of the \(\exists\) symbol. However, the \isa{?} character must then be followed

297 by a space, as in \isa{?~x. f(x) = 0}. Otherwise, \isa{?x} is

298 interpreted as a schematic variable. The preferred ASCII representation of

299 the \(\exists\) symbol is \isa{EX}\@.

300 \end{warn}%

301 \index{variables|)}

303 \section{Interaction and Interfaces}

304 \label{sec:interface}

306 The recommended interface for Isabelle/Isar is the (X)Emacs-based

307 \bfindex{Proof General}~\cite{proofgeneral,Aspinall:TACAS:2000}.

308 Interaction with Isabelle at the shell level, although possible,

309 should be avoided. Most of the tutorial is independent of the

310 interface and is phrased in a neutral language. For example, the

311 phrase ``to abandon a proof'' corresponds to the obvious

312 action of clicking on the \pgmenu{Undo} symbol in Proof General.

313 Proof General specific information is often displayed in paragraphs

314 identified by a miniature Proof General icon. Here are two examples:

315 \begin{pgnote}

316 Proof General supports a special font with mathematical symbols known

317 as ``x-symbols''. All symbols have \textsc{ascii}-equivalents: for

318 example, you can enter either \verb!&! or \verb!\<and>! to obtain

319 $\land$. For a list of the most frequent symbols see table~\ref{tab:ascii}

320 in the appendix.

322 Note that by default x-symbols are not enabled. You have to switch

323 them on via the menu item \pgmenu{Proof-General} $>$ \pgmenu{Options} $>$

324 \pgmenu{X-Symbols} (and save the option via the top-level

325 \pgmenu{Options} menu).

326 \end{pgnote}

328 \begin{pgnote}

329 Proof General offers the \pgmenu{Isabelle} menu for displaying

330 information and setting flags. A particularly useful flag is

331 \pgmenu{Isabelle} $>$ \pgmenu{Settings} $>$ \pgdx{Show Types} which

332 causes Isabelle to output the type information that is usually

333 suppressed. This is indispensible in case of errors of all kinds

334 because often the types reveal the source of the problem. Once you

335 have diagnosed the problem you may no longer want to see the types

336 because they clutter all output. Simply reset the flag.

337 \end{pgnote}

339 \section{Getting Started}

341 Assuming you have installed Isabelle and Proof General, you start it by typing

342 \texttt{Isabelle} in a shell window. This launches a Proof General window.

343 By default, you are in HOL\footnote{This is controlled by the

344 \texttt{ISABELLE_LOGIC} setting, see \emph{The Isabelle System Manual}

345 for more details.}.

347 \begin{pgnote}

348 You can choose a different logic via the \pgmenu{Isabelle} $>$

349 \pgmenu{Logics} menu.

350 \end{pgnote}