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}

     2

     3 \section{Introduction}

     4

     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.

    20

    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.

    27

    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\@.

    33

    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.

    45

    46 \section{Theories}

    47 \label{sec:Basic:Theories}

    48

    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}.

    73

    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}.

    79

    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.

    94

    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|)}

   102

   103

   104 \section{Types, Terms and Formulae}

   105 \label{sec:TypesTermsForms}

   106

   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|)}

   150

   151

   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}

   171

   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|)}

   180

   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}).

   191

   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($t@1$ = $t@2$)}.

   200

   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|)}

   210

   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.   223   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}   256   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.   260   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}   270   271   272 \section{Variables}   273 \label{sec:variables}   274 \index{variables|(}   275   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.   287   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|)}   302   303 \section{Interaction and Interfaces}   304 \label{sec:interface}   305   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.   321   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}   327   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}   338   339 \section{Getting Started}   340   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.}.   346   347 \begin{pgnote}   348 You can choose a different logic via the \pgmenu{Isabelle}$>\$

   349 \pgmenu{Logics} menu.

   350 \end{pgnote}