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doc-src/TutorialI/basics.tex

author | nipkow |

Wed Dec 06 13:22:58 2000 +0100 (2000-12-06) | |

changeset 10608 | 620647438780 |

parent 10538 | d1bf9ca9008d |

child 10695 | ffb153ef6366 |

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

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1 \chapter{Basic Concepts}

3 \section{Introduction}

5 This is a tutorial on how to use Isabelle/HOL as a specification and

6 verification system. Isabelle is a generic system for implementing logical

7 formalisms, and Isabelle/HOL is the specialization of Isabelle for

8 HOL, which abbreviates Higher-Order Logic. We introduce HOL step by step

9 following the equation

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

11 We assume that the reader is familiar with the basic concepts of both fields.

12 For excellent introductions to functional programming consult the textbooks

13 by Bird and Wadler~\cite{Bird-Wadler} or Paulson~\cite{paulson-ml2}. Although

14 this tutorial initially concentrates on functional programming, do not be

15 misled: HOL can express most mathematical concepts, and functional

16 programming is just one particularly simple and ubiquitous instance.

18 This tutorial introduces HOL via Isabelle/Isar~\cite{isabelle-isar-ref},

19 which is an extension of Isabelle~\cite{paulson-isa-book} with structured

20 proofs.\footnote{Thus the full name of the system should be

21 Isabelle/Isar/HOL, but that is a bit of a mouthful.} The most noticeable

22 difference to classical Isabelle (which is the basis of another version of

23 this tutorial) is the replacement of the ML level by a dedicated language for

24 definitions and proofs.

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

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

28 of Isar you need to consult the Isabelle/Isar Reference

29 Manual~\cite{isabelle-isar-ref}. If you want to use Isabelle's ML level

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

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

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

33 index.

35 \section{Theories}

36 \label{sec:Basic:Theories}

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

39 \bfindex{theory} is a named collection of types, functions, and theorems,

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

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

42 format of a theory \texttt{T} is

43 \begin{ttbox}

44 theory T = B\(@1\) + \(\cdots\) + B\(@n\):

45 \(\textit{declarations, definitions, and proofs}\)

46 end

47 \end{ttbox}

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

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

50 definitions, and proofs}} represents the newly introduced concepts

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

52 direct \textbf{parent theories}\indexbold{parent theory} of \texttt{T}.

53 Everything defined in the parent theories (and their parents \dots) is

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

55 \textbf{qualified} by theory names as in \texttt{T.f} and

56 \texttt{B.f}.\indexbold{identifier!qualified} Each theory \texttt{T} must

57 reside in a \bfindex{theory file} named \texttt{T.thy}.

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

60 constructs that can fill \textit{\texttt{declarations, definitions, and

61 proofs}} in the above theory template. A complete grammar of the basic

62 constructs is found in the Isabelle/Isar Reference Manual.

64 HOL's theory library is available online at

65 \begin{center}\small

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

67 \end{center}

68 and is recommended browsing. Note that most of the theories in the library

69 are based on classical Isabelle without the Isar extension. This means that

70 they look slightly different than the theories in this tutorial, and that all

71 proofs are in separate ML files.

73 \begin{warn}

74 HOL contains a theory \isaindexbold{Main}, the union of all the basic

75 predefined theories like arithmetic, lists, sets, etc.\ (see the online

76 library). Unless you know what you are doing, always include \isa{Main}

77 as a direct or indirect parent theory of all your theories.

78 \end{warn}

81 \section{Types, terms and formulae}

82 \label{sec:TypesTermsForms}

83 \indexbold{type}

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

86 logic whose type system resembles that of functional programming languages

87 like ML or Haskell. Thus there are

88 \begin{description}

89 \item[base types,] in particular \isaindex{bool}, the type of truth values,

90 and \isaindex{nat}, the type of natural numbers.

91 \item[type constructors,] in particular \isaindex{list}, the type of

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

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

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

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

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

97 \item[function types,] denoted by \isasymFun\indexbold{$IsaFun@\isasymFun}.

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

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

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

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

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

103 \isasymFun~$\tau$}.

104 \item[type variables,]\indexbold{type variable}\indexbold{variable!type}

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

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

107 function.

108 \end{description}

109 \begin{warn}

110 Types are extremely important because they prevent us from writing

111 nonsense. Isabelle insists that all terms and formulae must be well-typed

112 and will print an error message if a type mismatch is encountered. To

113 reduce the amount of explicit type information that needs to be provided by

114 the user, Isabelle infers the type of all variables automatically (this is

115 called \bfindex{type inference}) and keeps quiet about it. Occasionally

116 this may lead to misunderstandings between you and the system. If anything

117 strange happens, we recommend to set the \rmindex{flag}

118 \isaindexbold{show_types} that tells Isabelle to display type information

119 that is usually suppressed: simply type

120 \begin{ttbox}

121 ML "set show_types"

122 \end{ttbox}

124 \noindent

125 This can be reversed by \texttt{ML "reset show_types"}. Various other flags

126 can be set and reset in the same manner.\indexbold{flag!(re)setting}

127 \end{warn}

130 \textbf{Terms}\indexbold{term} are formed as in functional programming by

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

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

133 $\tau@1$ then \isa{f~t} is a term of type $\tau@2$. HOL also supports

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

135 programming:

136 \begin{description}

137 \item[\isa{if $b$ then $t@1$ else $t@2$}]\indexbold{*if}

138 means what you think it means and requires that

139 $b$ is of type \isa{bool} and $t@1$ and $t@2$ are of the same type.

140 \item[\isa{let $x$ = $t$ in $u$}]\indexbold{*let}

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

142 $t$. For example,

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

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

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

146 \indexbold{*case}

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

148 \end{description}

150 Terms may also contain

151 \isasymlambda-abstractions\indexbold{$Isalam@\isasymlambda}. For example,

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

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

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

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

157 \textbf{Formulae}\indexbold{formula} are terms of type \isaindexbold{bool}.

158 There are the basic constants \isaindexbold{True} and \isaindexbold{False} and

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

160 \indexboldpos{\isasymnot}{$HOL0not}, \indexboldpos{\isasymand}{$HOL0and},

161 \indexboldpos{\isasymor}{$HOL0or}, and \indexboldpos{\isasymimp}{$HOL0imp},

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

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

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

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

167 Equality is available in the form of the infix function

168 \isa{=}\indexbold{$HOL0eq@\texttt{=}} of type \isa{'a \isasymFun~'a

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

170 and $t@2$ are terms of the same type. In case $t@1$ and $t@2$ are of type

171 \isa{bool}, \isa{=} acts as if-and-only-if. The formula

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

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

175 The syntax for quantifiers is

176 \isa{\isasymforall{}x.~$P$}\indexbold{$HOL0All@\isasymforall} and

177 \isa{\isasymexists{}x.~$P$}\indexbold{$HOL0Ex@\isasymexists}. There is

178 even \isa{\isasymuniqex{}x.~$P$}\index{$HOL0ExU@\isasymuniqex|bold}, which

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

180 quantifications can be abbreviated: \isa{\isasymforall{}x~y~z.~$P$} means

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

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

184 \textbf{type constraints}\indexbold{type constraint} to a term. The syntax is

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

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

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

188 \isa{(x < y)::nat}. The main reason for type constraints are overloaded

189 functions like \isa{+}, \isa{*} and \isa{<}. See {\S}\ref{sec:overloading} for

190 a full discussion of overloading.

192 \begin{warn}

193 In general, HOL's concrete syntax tries to follow the conventions of

194 functional programming and mathematics. Below we list the main rules that you

195 should be familiar with to avoid certain syntactic traps. A particular

196 problem for novices can be the priority of operators. If you are unsure, use

197 more rather than fewer parentheses. In those cases where Isabelle echoes your

198 input, you can see which parentheses are dropped---they were superfluous. If

199 you are unsure how to interpret Isabelle's output because you don't know

200 where the (dropped) parentheses go, set (and possibly reset) the \rmindex{flag}

201 \isaindexbold{show_brackets}:

202 \begin{ttbox}

203 ML "set show_brackets"; \(\dots\); ML "reset show_brackets";

204 \end{ttbox}

205 \end{warn}

207 \begin{itemize}

208 \item

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

210 \item

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

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

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

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

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

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

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

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

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

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

221 \item

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

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

224 in \isa{f = (\isasymlambda{}x.~x)}. This includes \isaindex{if},

225 \isaindex{let}, \isaindex{case}, \isa{\isasymlambda}, and quantifiers.

226 \item

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

228 because \isa{x.x} is always read as a single qualified identifier that

229 refers to an item \isa{x} in theory \isa{x}. Write

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

231 \item Identifiers\indexbold{identifier} may contain \isa{_} and \isa{'}.

232 \end{itemize}

234 For the sake of readability the usual mathematical symbols are used throughout

235 the tutorial. Their ASCII-equivalents are shown in figure~\ref{fig:ascii} in

236 the appendix.

239 \section{Variables}

240 \label{sec:variables}

241 \indexbold{variable}

243 Isabelle distinguishes free and bound variables just as is customary. Bound

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

245 addition, Isabelle has a third kind of variable, called a \bfindex{schematic

246 variable}\indexbold{variable!schematic} or \bfindex{unknown}, which starts

247 with a \isa{?}. Logically, an unknown is a free variable. But it may be

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

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

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

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

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

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

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

256 a theorem, Isabelle will turn your free variables into unknowns: it merely

257 indicates that Isabelle will automatically instantiate those unknowns

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

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

260 \begin{warn}

261 If you use \isa{?}\index{$HOL0Ex@\texttt{?}} as an existential

262 quantifier, it needs to be followed by a space. Otherwise \isa{?x} is

263 interpreted as a schematic variable.

264 \end{warn}

266 \section{Interaction and interfaces}

268 Interaction with Isabelle can either occur at the shell level or through more

269 advanced interfaces. To keep the tutorial independent of the interface we

270 have phrased the description of the intraction in a neutral language. For

271 example, the phrase ``to abandon a proof'' means to type \isacommand{oops} at the

272 shell level, which is explained the first time the phrase is used. Other

273 interfaces perform the same act by cursor movements and/or mouse clicks.

274 Although shell-based interaction is quite feasible for the kind of proof

275 scripts currently presented in this tutorial, the recommended interface for

276 Isabelle/Isar is the Emacs-based \bfindex{Proof

277 General}~\cite{Aspinall:TACAS:2000,proofgeneral}.

279 Some interfaces (including the shell level) offer special fonts with

280 mathematical symbols. For those that do not, remember that ASCII-equivalents

281 are shown in figure~\ref{fig:ascii} in the appendix.

283 Finally, a word about semicolons.\indexbold{$Isar@\texttt{;}}

284 Commands may but need not be terminated by semicolons.

285 At the shell level it is advisable to use semicolons to enforce that a command

286 is executed immediately; otherwise Isabelle may wait for the next keyword

287 before it knows that the command is complete.

290 \section{Getting started}

292 Assuming you have installed Isabelle, you start it by typing \texttt{isabelle

293 -I HOL} in a shell window.\footnote{Simply executing \texttt{isabelle -I}

294 starts the default logic, which usually is already \texttt{HOL}. This is

295 controlled by the \texttt{ISABELLE_LOGIC} setting, see \emph{The Isabelle

296 System Manual} for more details.} This presents you with Isabelle's most

297 basic ASCII interface. In addition you need to open an editor window to

298 create theory files. While you are developing a theory, we recommend to

299 type each command into the file first and then enter it into Isabelle by

300 copy-and-paste, thus ensuring that you have a complete record of your theory.

301 As mentioned above, Proof General offers a much superior interface.

302 If you have installed Proof General, you can start it with \texttt{Isabelle}.