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
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
47 \label{sec:Basic: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
57 imports B\(@1\) \(\ldots\) B\(@n\)
59 {\rmfamily\textit{declarations, definitions, and proofs}}
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}.
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.
86 HOL's theory collection is available online at
88 \url{http://isabelle.in.tum.de/library/HOL/}
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
95 For the more adventurous, there is the \emph{Archive of Formal Proofs},
96 a journal-like collection of more advanced Isabelle theories:
98 \url{http://afp.sourceforge.net/}
100 We hope that you will contribute to it yourself one day.%
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
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
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~$\tau@n$
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
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}
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:
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
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 $c@i$.
172 Terms may also contain
173 \isasymlambda-abstractions.\index{lambda@$\lambda$ expressions}
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.~$t$}.%
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}.
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.~$P$}.
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.~$P$}.%
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
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:
229 Remember that \isa{f t u} means \isa{(f t) u} and not \isa{f(t u)}!
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)}.
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.
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.
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
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}).
273 \label{sec: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.
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}\@.
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:
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}
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).
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.
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}
348 You can choose a different logic via the \pgmenu{Isabelle} $>$
349 \pgmenu{Logics} menu. For example, you may want to work in the real
350 numbers, an extension of HOL (see \S\ref{sec:real}).