wenzelm@12668: \chapter{The Basics} nipkow@8743: nipkow@8743: \section{Introduction} nipkow@8743: paulson@11405: This book is a tutorial on how to use the theorem prover Isabelle/HOL as a paulson@11405: specification and verification system. Isabelle is a generic system for paulson@11405: implementing logical formalisms, and Isabelle/HOL is the specialization paulson@11405: of Isabelle for HOL, which abbreviates Higher-Order Logic. We introduce paulson@11405: HOL step by step following the equation nipkow@8743: $\mbox{HOL} = \mbox{Functional Programming} + \mbox{Logic}.$ paulson@11456: We do not assume that you are familiar with mathematical logic. paulson@11456: However, we do assume that paulson@11456: you are used to logical and set theoretic notation, as covered paulson@11456: in a good discrete mathematics course~\cite{Rosen-DMA}, and paulson@11450: that you are familiar with the basic concepts of functional nipkow@11209: programming~\cite{Bird-Haskell,Hudak-Haskell,paulson-ml2,Thompson-Haskell}. nipkow@11209: Although this tutorial initially concentrates on functional programming, do nipkow@11209: not be misled: HOL can express most mathematical concepts, and functional nipkow@11209: programming is just one particularly simple and ubiquitous instance. nipkow@8743: nipkow@11205: Isabelle~\cite{paulson-isa-book} is implemented in ML~\cite{SML}. This has nipkow@11205: influenced some of Isabelle/HOL's concrete syntax but is otherwise irrelevant paulson@11450: for us: this tutorial is based on nipkow@11213: Isabelle/Isar~\cite{isabelle-isar-ref}, an extension of Isabelle which hides nipkow@11213: the implementation language almost completely. Thus the full name of the nipkow@11213: system should be Isabelle/Isar/HOL, but that is a bit of a mouthful. nipkow@11213: nipkow@11213: There are other implementations of HOL, in particular the one by Mike Gordon paulson@11450: \index{Gordon, Mike}% nipkow@11213: \emph{et al.}, which is usually referred to as the HOL system'' nipkow@11213: \cite{mgordon-hol}. For us, HOL refers to the logical system, and sometimes paulson@11450: its incarnation Isabelle/HOL\@. nipkow@8743: nipkow@8743: A tutorial is by definition incomplete. Currently the tutorial only nipkow@8743: introduces the rudiments of Isar's proof language. To fully exploit the power nipkow@11213: of Isar, in particular the ability to write readable and structured proofs, nipkow@15429: you should start with Nipkow's overview~\cite{Nipkow-TYPES02} and consult nipkow@15429: the Isabelle/Isar Reference Manual~\cite{isabelle-isar-ref} and Wenzel's nipkow@15429: PhD thesis~\cite{Wenzel-PhD} (which discusses many proof patterns) nipkow@15429: for further details. If you want to use Isabelle's ML level nipkow@8743: directly (for example for writing your own proof procedures) see the Isabelle nipkow@8743: Reference Manual~\cite{isabelle-ref}; for details relating to HOL see the nipkow@8743: Isabelle/HOL manual~\cite{isabelle-HOL}. All manuals have a comprehensive nipkow@8743: index. nipkow@8743: nipkow@8743: \section{Theories} nipkow@8743: \label{sec:Basic:Theories} nipkow@8743: paulson@11428: \index{theories|(}% nipkow@8743: Working with Isabelle means creating theories. Roughly speaking, a paulson@11428: \textbf{theory} is a named collection of types, functions, and theorems, nipkow@8743: much like a module in a programming language or a specification in a nipkow@8743: specification language. In fact, theories in HOL can be either. The general nipkow@8743: format of a theory \texttt{T} is nipkow@8743: \begin{ttbox} nipkow@15136: theory T nipkow@15141: imports B$$@1$$ $$\ldots$$ B$$@n$$ nipkow@15136: begin paulson@11450: {\rmfamily\textit{declarations, definitions, and proofs}} nipkow@8743: end nipkow@15358: \end{ttbox}\cmmdx{theory}\cmmdx{imports} nipkow@15136: where \texttt{B}$@1$ \dots\ \texttt{B}$@n$ are the names of existing paulson@11450: theories that \texttt{T} is based on and \textit{declarations, paulson@11450: definitions, and proofs} represents the newly introduced concepts nipkow@8771: (types, functions etc.) and proofs about them. The \texttt{B}$@i$ are the paulson@11450: direct \textbf{parent theories}\indexbold{parent theories} of~\texttt{T}\@. paulson@11450: Everything defined in the parent theories (and their parents, recursively) is nipkow@8743: automatically visible. To avoid name clashes, identifiers can be paulson@11450: \textbf{qualified}\indexbold{identifiers!qualified} paulson@11450: by theory names as in \texttt{T.f} and~\texttt{B.f}. paulson@11450: Each theory \texttt{T} must paulson@11428: reside in a \textbf{theory file}\index{theory files} named \texttt{T.thy}. nipkow@8743: nipkow@8743: This tutorial is concerned with introducing you to the different linguistic paulson@11450: constructs that can fill the \textit{declarations, definitions, and paulson@11450: proofs} above. A complete grammar of the basic nipkow@12327: constructs is found in the Isabelle/Isar Reference nipkow@12327: Manual~\cite{isabelle-isar-ref}. nipkow@8743: nipkow@8743: \begin{warn} paulson@11428: HOL contains a theory \thydx{Main}, the union of all the basic paulson@10885: predefined theories like arithmetic, lists, sets, etc. paulson@10885: Unless you know what you are doing, always include \isa{Main} nipkow@10971: as a direct or indirect parent of all your theories. nipkow@12332: \end{warn} nipkow@16306: HOL's theory collection is available online at nipkow@16306: \begin{center}\small nipkow@16306: \url{http://isabelle.in.tum.de/library/HOL/} nipkow@16306: \end{center} nipkow@16359: and is recommended browsing. In subdirectory \texttt{Library} you find nipkow@16359: a growing library of useful theories that are not part of \isa{Main} nipkow@16359: but can be included among the parents of a theory and will then be nipkow@16359: loaded automatically. nipkow@16306: nipkow@16306: For the more adventurous, there is the \emph{Archive of Formal Proofs}, nipkow@16306: a journal-like collection of more advanced Isabelle theories: nipkow@16306: \begin{center}\small nipkow@16306: \url{http://afp.sourceforge.net/} nipkow@16306: \end{center} nipkow@16306: We hope that you will contribute to it yourself one day.% paulson@11428: \index{theories|)} nipkow@8743: nipkow@8743: paulson@10885: \section{Types, Terms and Formulae} nipkow@8743: \label{sec:TypesTermsForms} nipkow@8743: paulson@10795: Embedded in a theory are the types, terms and formulae of HOL\@. HOL is a typed nipkow@8771: logic whose type system resembles that of functional programming languages paulson@11450: like ML or Haskell. Thus there are paulson@11450: \index{types|(} nipkow@8743: \begin{description} paulson@11450: \item[base types,] paulson@11450: in particular \tydx{bool}, the type of truth values, paulson@11428: and \tydx{nat}, the type of natural numbers. paulson@11450: \item[type constructors,]\index{type constructors} paulson@11450: in particular \tydx{list}, the type of paulson@11428: lists, and \tydx{set}, the type of sets. Type constructors are written nipkow@8771: postfix, e.g.\ \isa{(nat)list} is the type of lists whose elements are nipkow@8743: natural numbers. Parentheses around single arguments can be dropped (as in nipkow@8771: \isa{nat list}), multiple arguments are separated by commas (as in nipkow@8771: \isa{(bool,nat)ty}). paulson@11450: \item[function types,]\index{function types} paulson@11450: denoted by \isasymFun\indexbold{$IsaFun@\isasymFun}. nipkow@8771: In HOL \isasymFun\ represents \emph{total} functions only. As is customary, nipkow@8771: \isa{$\tau@1$\isasymFun~$\tau@2$\isasymFun~$\tau@3$} means nipkow@8771: \isa{$\tau@1$\isasymFun~($\tau@2$\isasymFun~$\tau@3$)}. Isabelle also nipkow@8771: supports the notation \isa{[$\tau@1,\dots,\tau@n$] \isasymFun~$\tau$} nipkow@8771: which abbreviates \isa{$\tau@1$\isasymFun~$\cdots$\isasymFun~$\tau@n$nipkow@8743: \isasymFun~$\tau$}. paulson@11450: \item[type variables,]\index{type variables}\index{variables!type} paulson@10795: denoted by \ttindexboldpos{'a}{$Isatype}, \isa{'b} etc., just like in ML\@. They give rise nipkow@8771: to polymorphic types like \isa{'a \isasymFun~'a}, the type of the identity nipkow@8771: function. nipkow@8743: \end{description} nipkow@8743: \begin{warn} nipkow@8743: Types are extremely important because they prevent us from writing nipkow@16359: nonsense. Isabelle insists that all terms and formulae must be nipkow@16359: well-typed and will print an error message if a type mismatch is nipkow@16359: encountered. To reduce the amount of explicit type information that nipkow@16359: needs to be provided by the user, Isabelle infers the type of all nipkow@16359: variables automatically (this is called \bfindex{type inference}) nipkow@16359: and keeps quiet about it. Occasionally this may lead to nipkow@16359: misunderstandings between you and the system. If anything strange nipkow@16359: happens, we recommend that you ask Isabelle to display all type nipkow@16523: information via the Proof General menu item \pgmenu{Isabelle} $>$ nipkow@16523: \pgmenu{Settings} $>$ \pgmenu{Show Types} (see \S\ref{sec:interface} nipkow@16359: for details). paulson@11450: \end{warn}% paulson@11450: \index{types|)} nipkow@8743: nipkow@8743: paulson@11450: \index{terms|(} paulson@11450: \textbf{Terms} are formed as in functional programming by nipkow@8771: applying functions to arguments. If \isa{f} is a function of type nipkow@8771: \isa{$\tau@1$ \isasymFun~$\tau@2$} and \isa{t} is a term of type nipkow@8771: $\tau@1$ then \isa{f~t} is a term of type $\tau@2$. HOL also supports nipkow@8771: infix functions like \isa{+} and some basic constructs from functional paulson@11428: programming, such as conditional expressions: nipkow@8743: \begin{description} paulson@11450: \item[\isa{if $b$ then $t@1$ else $t@2$}]\index{*if expressions} paulson@11428: Here $b$ is of type \isa{bool} and $t@1$ and $t@2$ are of the same type. paulson@11450: \item[\isa{let $x$ = $t$ in $u$}]\index{*let expressions} nipkow@13814: is equivalent to $u$ where all free occurrences of $x$ have been replaced by nipkow@8743: $t$. For example, nipkow@8771: \isa{let x = 0 in x+x} is equivalent to \isa{0+0}. Multiple bindings are separated nipkow@13814: by semicolons: \isa{let $x@1$ = $t@1$;\dots; $x@n$ = $t@n$ in $u$}. nipkow@8771: \item[\isa{case $e$ of $c@1$ \isasymFun~$e@1$ |~\dots~| $c@n$ \isasymFun~$e@n$}] paulson@11450: \index{*case expressions} nipkow@8771: evaluates to $e@i$ if $e$ is of the form $c@i$. nipkow@8743: \end{description} nipkow@8743: nipkow@8743: Terms may also contain paulson@11450: \isasymlambda-abstractions.\index{lambda@$\lambda$ expressions} paulson@11450: For example, nipkow@8771: \isa{\isasymlambda{}x.~x+1} is the function that takes an argument \isa{x} and nipkow@8771: returns \isa{x+1}. Instead of nipkow@8771: \isa{\isasymlambda{}x.\isasymlambda{}y.\isasymlambda{}z.~$t$} we can write paulson@11450: \isa{\isasymlambda{}x~y~z.~$t$}.% paulson@11450: \index{terms|)} nipkow@8743: paulson@11450: \index{formulae|(}% paulson@11450: \textbf{Formulae} are terms of type \tydx{bool}. paulson@11428: There are the basic constants \cdx{True} and \cdx{False} and nipkow@8771: the usual logical connectives (in decreasing order of priority): paulson@11420: \indexboldpos{\protect\isasymnot}{$HOL0not}, \indexboldpos{\protect\isasymand}{$HOL0and}, paulson@11420: \indexboldpos{\protect\isasymor}{$HOL0or}, and \indexboldpos{\protect\isasymimp}{$HOL0imp}, nipkow@8743: all of which (except the unary \isasymnot) associate to the right. In nipkow@8771: particular \isa{A \isasymimp~B \isasymimp~C} means \isa{A \isasymimp~(B nipkow@8771: \isasymimp~C)} and is thus logically equivalent to \isa{A \isasymand~B nipkow@8771: \isasymimp~C} (which is \isa{(A \isasymand~B) \isasymimp~C}). nipkow@8743: paulson@11450: Equality\index{equality} is available in the form of the infix function paulson@11450: \isa{=} of type \isa{'a \isasymFun~'a nipkow@8771: \isasymFun~bool}. Thus \isa{$t@1$ = $t@2$} is a formula provided $t@1$ paulson@11450: and $t@2$ are terms of the same type. If $t@1$ and $t@2$ are of type paulson@11450: \isa{bool} then \isa{=} acts as \rmindex{if-and-only-if}. paulson@11450: The formula nipkow@8771: \isa{$t@1$~\isasymnoteq~$t@2$} is merely an abbreviation for nipkow@8771: \isa{\isasymnot($t@1$ = $t@2$)}. nipkow@8743: paulson@11450: Quantifiers\index{quantifiers} are written as paulson@11450: \isa{\isasymforall{}x.~$P$} and \isa{\isasymexists{}x.~$P$}. paulson@11420: There is even paulson@11450: \isa{\isasymuniqex{}x.~$P$}, which paulson@11420: means that there exists exactly one \isa{x} that satisfies \isa{$P$}. paulson@11420: Nested quantifications can be abbreviated: paulson@11420: \isa{\isasymforall{}x~y~z.~$P$} means paulson@11450: \isa{\isasymforall{}x.\isasymforall{}y.\isasymforall{}z.~$P$}.% paulson@11450: \index{formulae|)} nipkow@8743: nipkow@8743: Despite type inference, it is sometimes necessary to attach explicit paulson@11428: \bfindex{type constraints} to a term. The syntax is nipkow@8771: \isa{$t$::$\tau$} as in \isa{x < (y::nat)}. Note that nipkow@10538: \ttindexboldpos{::}{$Isatype} binds weakly and should therefore be enclosed paulson@11450: in parentheses. For instance, paulson@11450: \isa{x < y::nat} is ill-typed because it is interpreted as paulson@11450: \isa{(x < y)::nat}. Type constraints may be needed to disambiguate paulson@11450: expressions paulson@11450: involving overloaded functions such as~\isa{+}, paulson@11450: \isa{*} and~\isa{<}. Section~\ref{sec:overloading} paulson@11450: discusses overloading, while Table~\ref{tab:overloading} presents the most nipkow@10695: important overloaded function symbols. nipkow@8743: paulson@11450: In general, HOL's concrete \rmindex{syntax} tries to follow the conventions of paulson@11450: functional programming and mathematics. Here are the main rules that you paulson@11450: should be familiar with to avoid certain syntactic traps: nipkow@8743: \begin{itemize} nipkow@8743: \item nipkow@8771: Remember that \isa{f t u} means \isa{(f t) u} and not \isa{f(t u)}! nipkow@8743: \item nipkow@8771: Isabelle allows infix functions like \isa{+}. The prefix form of function nipkow@8771: application binds more strongly than anything else and hence \isa{f~x + y} nipkow@8771: means \isa{(f~x)~+~y} and not \isa{f(x+y)}. nipkow@8743: \item Remember that in HOL if-and-only-if is expressed using equality. But nipkow@8743: equality has a high priority, as befitting a relation, while if-and-only-if nipkow@8771: typically has the lowest priority. Thus, \isa{\isasymnot~\isasymnot~P = nipkow@8771: P} means \isa{\isasymnot\isasymnot(P = P)} and not nipkow@8771: \isa{(\isasymnot\isasymnot P) = P}. When using \isa{=} to mean nipkow@8771: logical equivalence, enclose both operands in parentheses, as in \isa{(A nipkow@8743: \isasymand~B) = (B \isasymand~A)}. nipkow@8743: \item nipkow@8743: Constructs with an opening but without a closing delimiter bind very weakly nipkow@8743: and should therefore be enclosed in parentheses if they appear in subterms, as paulson@11450: in \isa{(\isasymlambda{}x.~x) = f}. This includes paulson@11450: \isa{if},\index{*if expressions} paulson@11450: \isa{let},\index{*let expressions} paulson@11450: \isa{case},\index{*case expressions} paulson@11450: \isa{\isasymlambda}, and quantifiers. nipkow@8743: \item nipkow@8771: Never write \isa{\isasymlambda{}x.x} or \isa{\isasymforall{}x.x=x} nipkow@12327: because \isa{x.x} is always taken as a single qualified identifier. Write nipkow@8771: \isa{\isasymlambda{}x.~x} and \isa{\isasymforall{}x.~x=x} instead. paulson@11450: \item Identifiers\indexbold{identifiers} may contain the characters \isa{_} nipkow@12327: and~\isa{'}, except at the beginning. nipkow@8743: \end{itemize} nipkow@8743: paulson@11450: For the sake of readability, we use the usual mathematical symbols throughout nipkow@10983: the tutorial. Their \textsc{ascii}-equivalents are shown in table~\ref{tab:ascii} in nipkow@8771: the appendix. nipkow@8771: paulson@11450: \begin{warn} nipkow@16359: A particular problem for novices can be the priority of operators. If nipkow@16359: you are unsure, use additional parentheses. In those cases where nipkow@16359: Isabelle echoes your input, you can see which parentheses are dropped nipkow@16359: --- they were superfluous. If you are unsure how to interpret nipkow@16359: Isabelle's output because you don't know where the (dropped) nipkow@16523: parentheses go, set the Proof General flag \pgmenu{Isabelle}$>$nipkow@16523: \pgmenu{Settings}$>$\pgmenu{Show Brackets} (see \S\ref{sec:interface}). paulson@11450: \end{warn} paulson@11450: nipkow@8743: nipkow@8743: \section{Variables} nipkow@8743: \label{sec:variables} paulson@11450: \index{variables|(} nipkow@8743: paulson@11450: Isabelle distinguishes free and bound variables, as is customary. Bound nipkow@8743: variables are automatically renamed to avoid clashes with free variables. In paulson@11428: addition, Isabelle has a third kind of variable, called a \textbf{schematic paulson@11428: variable}\index{variables!schematic} or \textbf{unknown}\index{unknowns}, nipkow@13439: which must have a~\isa{?} as its first character. paulson@11428: Logically, an unknown is a free variable. But it may be nipkow@8743: instantiated by another term during the proof process. For example, the nipkow@8771: mathematical theorem$x = x$is represented in Isabelle as \isa{?x = ?x}, nipkow@8743: which means that Isabelle can instantiate it arbitrarily. This is in contrast nipkow@8743: to ordinary variables, which remain fixed. The programming language Prolog nipkow@8743: calls unknowns {\em logical\/} variables. nipkow@8743: nipkow@8743: Most of the time you can and should ignore unknowns and work with ordinary nipkow@8743: variables. Just don't be surprised that after you have finished the proof of paulson@11450: a theorem, Isabelle will turn your free variables into unknowns. It nipkow@8743: indicates that Isabelle will automatically instantiate those unknowns nipkow@8743: suitably when the theorem is used in some other proof. nipkow@9689: Note that for readability we often drop the \isa{?}s when displaying a theorem. nipkow@8743: \begin{warn} paulson@11450: For historical reasons, Isabelle accepts \isa{?} as an ASCII representation paulson@11450: of the $$\exists$$ symbol. However, the \isa{?} character must then be followed paulson@11450: by a space, as in \isa{?~x. f(x) = 0}. Otherwise, \isa{?x} is paulson@11450: interpreted as a schematic variable. The preferred ASCII representation of paulson@11450: the $$\exists$$ symbol is \isa{EX}\@. paulson@11450: \end{warn}% paulson@11450: \index{variables|)} nipkow@8743: paulson@10885: \section{Interaction and Interfaces} nipkow@16306: \label{sec:interface} nipkow@8771: nipkow@16359: The recommended interface for Isabelle/Isar is the (X)Emacs-based nipkow@16359: \bfindex{Proof General}~\cite{proofgeneral,Aspinall:TACAS:2000}. nipkow@16359: Interaction with Isabelle at the shell level, although possible, nipkow@16359: should be avoided. Most of the tutorial is independent of the nipkow@16359: interface and is phrased in a neutral language. For example, the nipkow@16359: phrase to abandon a proof'' corresponds to the obvious nipkow@16523: action of clicking on the \pgmenu{Undo} symbol in Proof General. nipkow@16359: Proof General specific information is often displayed in paragraphs nipkow@16359: identified by a miniature Proof General icon. Here are two examples: nipkow@16359: \begin{pgnote} nipkow@16359: Proof General supports a special font with mathematical symbols known nipkow@16359: as x-symbols''. All symbols have \textsc{ascii}-equivalents: for nipkow@16359: example, you can enter either \verb!&! or \verb!\! to obtain nipkow@16359:$\land$. For a list of the most frequent symbols see table~\ref{tab:ascii} nipkow@16359: in the appendix. nipkow@8771: nipkow@16359: Note that by default x-symbols are not enabled. You have to switch nipkow@16523: them on via the menu item \pgmenu{Proof-General}$>$\pgmenu{Options}$>$nipkow@16523: \pgmenu{X-Symbols} (and save the option via the top-level nipkow@16523: \pgmenu{Options} menu). nipkow@16306: \end{pgnote} nipkow@8771: nipkow@16306: \begin{pgnote} nipkow@16523: Proof General offers the \pgmenu{Isabelle} menu for displaying nipkow@16359: information and setting flags. A particularly useful flag is nipkow@16523: \pgmenu{Isabelle}$>$\pgmenu{Settings}$>$\pgdx{Show Types} which nipkow@16359: causes Isabelle to output the type information that is usually nipkow@16306: suppressed. This is indispensible in case of errors of all kinds nipkow@16359: because often the types reveal the source of the problem. Once you nipkow@16359: have diagnosed the problem you may no longer want to see the types nipkow@16359: because they clutter all output. Simply reset the flag. nipkow@16306: \end{pgnote} nipkow@8771: paulson@10885: \section{Getting Started} nipkow@8743: nipkow@16359: Assuming you have installed Isabelle and Proof General, you start it by typing nipkow@16359: \texttt{Isabelle} in a shell window. This launches a Proof General window. nipkow@16359: By default, you are in HOL\footnote{This is controlled by the nipkow@16359: \texttt{ISABELLE_LOGIC} setting, see \emph{The Isabelle System Manual} nipkow@16359: for more details.}. nipkow@16359: nipkow@16359: \begin{pgnote} nipkow@16523: You can choose a different logic via the \pgmenu{Isabelle}$>\$ nipkow@38432: \pgmenu{Logics} menu. nipkow@16359: \end{pgnote}