1 \chapter{Basic Concepts}
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
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
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
44 theory T = B\(@1\) + \(\cdots\) + B\(@n\):
45 \(\textit{declarations, definitions, and proofs}\)
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
66 \url{http://isabelle.in.tum.de/library/}
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.
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.
81 \section{Types, terms and formulae}
82 \label{sec:TypesTermsForms}
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
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
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~$\tau@n$
104 \item[type variables,]\indexbold{type variable}\indexbold{variable!type}
105 denoted by \isaindexbold{'a}, \isa{'b} etc., just like in ML. They give rise
106 to polymorphic types like \isa{'a \isasymFun~'a}, the type of the identity
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
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}
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
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
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~$e@n$}]
147 evaluates to $e@i$ if $e$ is of the form $c@i$.
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$} as in \isa{x < (y::nat)}. Note that
186 \ttindexboldpos{::}{$Isalamtc} 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.
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}:
203 ML "set show_brackets"; \(\dots\); ML "reset show_brackets";
209 Remember that \isa{f t u} means \isa{(f t) u} and not \isa{f(t u)}!
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)}.
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.
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{'}.
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
240 \label{sec:variables}
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.
261 If you use \isa{?}\index{$HOL0Ex@\texttt{?}} as an existential
262 quantifier, it needs to be followed by a space. Otherwise \isa{?x} is
263 interpreted as a schematic variable.
266 \section{Interaction and interfaces}
268 Interaction with Isabelle can either occur at the shell level or through more
269 advanced interfaces. To keep the tutorial independent of the interface we
270 have phrased the description of the intraction in a neutral language. For
271 example, the phrase ``to abandon a proof'' means to type \isacommand{oops} at the
272 shell level, which is explained the first time the phrase is used. Other
273 interfaces perform the same act by cursor movements and/or mouse clicks.
274 Although shell-based interaction is quite feasible for the kind of proof
275 scripts currently presented in this tutorial, the recommended interface for
276 Isabelle/Isar is the Emacs-based \bfindex{Proof
277 General}~\cite{Aspinall:TACAS:2000,proofgeneral}.
279 Some interfaces (including the shell level) offer special fonts with
280 mathematical symbols. For those that do not, remember that ASCII-equivalents
281 are shown in figure~\ref{fig:ascii} in the appendix.
283 Finally, a word about semicolons.\indexbold{$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}.