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doc-src/TutorialI/basics.tex

author | nipkow |

Mon, 28 Aug 2000 09:32:51 +0200 | |

changeset 9689 | 751fde5307e4 |

parent 9541 | d17c0b34d5c8 |

child 9792 | bbefb6ce5cb2 |

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\chapter{Basic Concepts} \section{Introduction} This is a tutorial on how to use Isabelle/HOL as a specification and verification system. Isabelle is a generic system for implementing logical formalisms, and Isabelle/HOL is the specialization of Isabelle for HOL, which abbreviates Higher-Order Logic. We introduce HOL step by step following the equation \[ \mbox{HOL} = \mbox{Functional Programming} + \mbox{Logic}. \] We assume that the reader is familiar with the basic concepts of both fields. For excellent introductions to functional programming consult the textbooks by Bird and Wadler~\cite{Bird-Wadler} or Paulson~\cite{paulson-ml2}. Although this tutorial initially concentrates on functional programming, do not be misled: HOL can express most mathematical concepts, and functional programming is just one particularly simple and ubiquitous instance. This tutorial introduces HOL via Isabelle/Isar~\cite{isabelle-isar-ref}, which is an extension of Isabelle~\cite{paulson-isa-book} with structured proofs.\footnote{Thus the full name of the system should be Isabelle/Isar/HOL, but that is a bit of a mouthful.} The most noticeable difference to classical Isabelle (which is the basis of another version of this tutorial) is the replacement of the ML level by a dedicated language for definitions and proofs. A tutorial is by definition incomplete. Currently the tutorial only introduces the rudiments of Isar's proof language. To fully exploit the power of Isar you need to consult the Isabelle/Isar Reference Manual~\cite{isabelle-isar-ref}. If you want to use Isabelle's ML level directly (for example for writing your own proof procedures) see the Isabelle Reference Manual~\cite{isabelle-ref}; for details relating to HOL see the Isabelle/HOL manual~\cite{isabelle-HOL}. All manuals have a comprehensive index. \section{Theories} \label{sec:Basic:Theories} Working with Isabelle means creating theories. Roughly speaking, a \bfindex{theory} is a named collection of types, functions, and theorems, much like a module in a programming language or a specification in a specification language. In fact, theories in HOL can be either. The general format of a theory \texttt{T} is \begin{ttbox} theory T = B\(@1\) + \(\cdots\) + B\(@n\): \(\textit{declarations, definitions, and proofs}\) end \end{ttbox} where \texttt{B}$@1$, \dots, \texttt{B}$@n$ are the names of existing theories that \texttt{T} is based on and \texttt{\textit{declarations, definitions, and proofs}} represents the newly introduced concepts (types, functions etc.) and proofs about them. The \texttt{B}$@i$ are the direct \textbf{parent theories}\indexbold{parent theory} of \texttt{T}. Everything defined in the parent theories (and their parents \dots) is automatically visible. To avoid name clashes, identifiers can be \textbf{qualified} by theory names as in \texttt{T.f} and \texttt{B.f}.\indexbold{identifier!qualified} Each theory \texttt{T} must reside in a \bfindex{theory file} named \texttt{T.thy}. This tutorial is concerned with introducing you to the different linguistic constructs that can fill \textit{\texttt{declarations, definitions, and proofs}} in the above theory template. A complete grammar of the basic constructs is found in the Isabelle/Isar Reference Manual. HOL's theory library is available online at \begin{center}\small \url{http://isabelle.in.tum.de/library/} \end{center} and is recommended browsing. Note that most of the theories in the library are based on classical Isabelle without the Isar extension. This means that they look slightly different than the theories in this tutorial, and that all proofs are in separate ML files. \begin{warn} HOL contains a theory \ttindexbold{Main}, the union of all the basic predefined theories like arithmetic, lists, sets, etc.\ (see the online library). Unless you know what you are doing, always include \texttt{Main} as a direct or indirect parent theory of all your theories. \end{warn} \section{Types, terms and formulae} \label{sec:TypesTermsForms} \indexbold{type} Embedded in a theory are the types, terms and formulae of HOL. HOL is a typed logic whose type system resembles that of functional programming languages like ML or Haskell. Thus there are \begin{description} \item[base types,] in particular \isaindex{bool}, the type of truth values, and \isaindex{nat}, the type of natural numbers. \item[type constructors,] in particular \isaindex{list}, the type of lists, and \isaindex{set}, the type of sets. Type constructors are written postfix, e.g.\ \isa{(nat)list} is the type of lists whose elements are natural numbers. Parentheses around single arguments can be dropped (as in \isa{nat list}), multiple arguments are separated by commas (as in \isa{(bool,nat)ty}). \item[function types,] denoted by \isasymFun\indexbold{$IsaFun@\isasymFun}. In HOL \isasymFun\ represents \emph{total} functions only. As is customary, \isa{$\tau@1$ \isasymFun~$\tau@2$ \isasymFun~$\tau@3$} means \isa{$\tau@1$ \isasymFun~($\tau@2$ \isasymFun~$\tau@3$)}. Isabelle also supports the notation \isa{[$\tau@1,\dots,\tau@n$] \isasymFun~$\tau$} which abbreviates \isa{$\tau@1$ \isasymFun~$\cdots$ \isasymFun~$\tau@n$ \isasymFun~$\tau$}. \item[type variables,]\indexbold{type variable}\indexbold{variable!type} denoted by \isaindexbold{'a}, \isa{'b} etc., just like in ML. They give rise to polymorphic types like \isa{'a \isasymFun~'a}, the type of the identity function. \end{description} \begin{warn} Types are extremely important because they prevent us from writing nonsense. Isabelle insists that all terms and formulae must be well-typed and will print an error message if a type mismatch is encountered. To reduce the amount of explicit type information that needs to be provided by the user, Isabelle infers the type of all variables automatically (this is called \bfindex{type inference}) and keeps quiet about it. Occasionally this may lead to misunderstandings between you and the system. If anything strange happens, we recommend to set the \rmindex{flag} \ttindexbold{show_types} that tells Isabelle to display type information that is usually suppressed: simply type \begin{ttbox} ML "set show_types" \end{ttbox} \noindent This can be reversed by \texttt{ML "reset show_types"}. Various other flags can be set and reset in the same manner.\indexbold{flag!(re)setting} \end{warn} \textbf{Terms}\indexbold{term} are formed as in functional programming by applying functions to arguments. If \isa{f} is a function of type \isa{$\tau@1$ \isasymFun~$\tau@2$} and \isa{t} is a term of type $\tau@1$ then \isa{f~t} is a term of type $\tau@2$. HOL also supports infix functions like \isa{+} and some basic constructs from functional programming: \begin{description} \item[\isa{if $b$ then $t@1$ else $t@2$}]\indexbold{*if} means what you think it means and requires that $b$ is of type \isa{bool} and $t@1$ and $t@2$ are of the same type. \item[\isa{let $x$ = $t$ in $u$}]\indexbold{*let} is equivalent to $u$ where all occurrences of $x$ have been replaced by $t$. For example, \isa{let x = 0 in x+x} is equivalent to \isa{0+0}. Multiple bindings are separated by semicolons: \isa{let $x@1$ = $t@1$; \dots; $x@n$ = $t@n$ in $u$}. \item[\isa{case $e$ of $c@1$ \isasymFun~$e@1$ |~\dots~| $c@n$ \isasymFun~$e@n$}] \indexbold{*case} evaluates to $e@i$ if $e$ is of the form $c@i$. \end{description} Terms may also contain \isasymlambda-abstractions\indexbold{$Isalam@\isasymlambda}. For example, \isa{\isasymlambda{}x.~x+1} is the function that takes an argument \isa{x} and returns \isa{x+1}. Instead of \isa{\isasymlambda{}x.\isasymlambda{}y.\isasymlambda{}z.~$t$} we can write \isa{\isasymlambda{}x~y~z.~$t$}. \textbf{Formulae}\indexbold{formula} are terms of type \isaindexbold{bool}. There are the basic constants \isaindexbold{True} and \isaindexbold{False} and the usual logical connectives (in decreasing order of priority): \indexboldpos{\isasymnot}{$HOL0not}, \indexboldpos{\isasymand}{$HOL0and}, \indexboldpos{\isasymor}{$HOL0or}, and \indexboldpos{\isasymimp}{$HOL0imp}, all of which (except the unary \isasymnot) associate to the right. In particular \isa{A \isasymimp~B \isasymimp~C} means \isa{A \isasymimp~(B \isasymimp~C)} and is thus logically equivalent to \isa{A \isasymand~B \isasymimp~C} (which is \isa{(A \isasymand~B) \isasymimp~C}). Equality is available in the form of the infix function \isa{=}\indexbold{$HOL0eq@\texttt{=}} of type \isa{'a \isasymFun~'a \isasymFun~bool}. Thus \isa{$t@1$ = $t@2$} is a formula provided $t@1$ and $t@2$ are terms of the same type. In case $t@1$ and $t@2$ are of type \isa{bool}, \isa{=} acts as if-and-only-if. The formula \isa{$t@1$~\isasymnoteq~$t@2$} is merely an abbreviation for \isa{\isasymnot($t@1$ = $t@2$)}. The syntax for quantifiers is \isa{\isasymforall{}x.~$P$}\indexbold{$HOL0All@\isasymforall} and \isa{\isasymexists{}x.~$P$}\indexbold{$HOL0Ex@\isasymexists}. There is even \isa{\isasymuniqex{}x.~$P$}\index{$HOL0ExU@\isasymuniqex|bold}, which means that there exists exactly one \isa{x} that satisfies \isa{$P$}. Nested quantifications can be abbreviated: \isa{\isasymforall{}x~y~z.~$P$} means \isa{\isasymforall{}x.\isasymforall{}y.\isasymforall{}z.~$P$}. Despite type inference, it is sometimes necessary to attach explicit \textbf{type constraints}\indexbold{type constraint} to a term. The syntax is \isa{$t$::$\tau$} as in \isa{x < (y::nat)}. Note that \ttindexboldpos{::}{$Isalamtc} binds weakly and should therefore be enclosed in parentheses: \isa{x < y::nat} is ill-typed because it is interpreted as \isa{(x < y)::nat}. The main reason for type constraints are overloaded functions like \isa{+}, \isa{*} and \isa{<}. (See \S\ref{sec:TypeClasses} for a full discussion of overloading.) \begin{warn} In general, HOL's concrete syntax tries to follow the conventions of functional programming and mathematics. Below we list the main rules that you should be familiar with to avoid certain syntactic traps. A particular problem for novices can be the priority of operators. If you are unsure, use more rather than fewer parentheses. In those cases where Isabelle echoes your input, you can see which parentheses are dropped---they were superfluous. If you are unsure how to interpret Isabelle's output because you don't know where the (dropped) parentheses go, set (and possibly reset) the \rmindex{flag} \ttindexbold{show_brackets}: \begin{ttbox} ML "set show_brackets"; \(\dots\); ML "reset show_brackets"; \end{ttbox} \end{warn} \begin{itemize} \item Remember that \isa{f t u} means \isa{(f t) u} and not \isa{f(t u)}! \item Isabelle allows infix functions like \isa{+}. The prefix form of function application binds more strongly than anything else and hence \isa{f~x + y} means \isa{(f~x)~+~y} and not \isa{f(x+y)}. \item Remember that in HOL if-and-only-if is expressed using equality. But equality has a high priority, as befitting a relation, while if-and-only-if typically has the lowest priority. Thus, \isa{\isasymnot~\isasymnot~P = P} means \isa{\isasymnot\isasymnot(P = P)} and not \isa{(\isasymnot\isasymnot P) = P}. When using \isa{=} to mean logical equivalence, enclose both operands in parentheses, as in \isa{(A \isasymand~B) = (B \isasymand~A)}. \item Constructs with an opening but without a closing delimiter bind very weakly and should therefore be enclosed in parentheses if they appear in subterms, as in \isa{f = (\isasymlambda{}x.~x)}. This includes \isaindex{if}, \isaindex{let}, \isaindex{case}, \isa{\isasymlambda}, and quantifiers. \item Never write \isa{\isasymlambda{}x.x} or \isa{\isasymforall{}x.x=x} because \isa{x.x} is always read as a single qualified identifier that refers to an item \isa{x} in theory \isa{x}. Write \isa{\isasymlambda{}x.~x} and \isa{\isasymforall{}x.~x=x} instead. \item Identifiers\indexbold{identifier} may contain \isa{_} and \isa{'}. \end{itemize} For the sake of readability the usual mathematical symbols are used throughout the tutorial. Their ASCII-equivalents are shown in figure~\ref{fig:ascii} in the appendix. \section{Variables} \label{sec:variables} \indexbold{variable} Isabelle distinguishes free and bound variables just as is customary. Bound variables are automatically renamed to avoid clashes with free variables. In addition, Isabelle has a third kind of variable, called a \bfindex{schematic variable}\indexbold{variable!schematic} or \bfindex{unknown}, which starts with a \isa{?}. Logically, an unknown is a free variable. But it may be instantiated by another term during the proof process. For example, the mathematical theorem $x = x$ is represented in Isabelle as \isa{?x = ?x}, which means that Isabelle can instantiate it arbitrarily. This is in contrast to ordinary variables, which remain fixed. The programming language Prolog calls unknowns {\em logical\/} variables. Most of the time you can and should ignore unknowns and work with ordinary variables. Just don't be surprised that after you have finished the proof of a theorem, Isabelle will turn your free variables into unknowns: it merely indicates that Isabelle will automatically instantiate those unknowns suitably when the theorem is used in some other proof. Note that for readability we often drop the \isa{?}s when displaying a theorem. \begin{warn} If you use \isa{?}\index{$HOL0Ex@\texttt{?}} as an existential quantifier, it needs to be followed by a space. Otherwise \isa{?x} is interpreted as a schematic variable. \end{warn} \section{Interaction and interfaces} Interaction with Isabelle can either occur at the shell level or through more advanced interfaces. To keep the tutorial independent of the interface we have phrased the description of the intraction in a neutral language. For example, the phrase ``to abandon a proof'' means to type \isacommand{oops} at the shell level, which is explained the first time the phrase is used. Other interfaces perform the same act by cursor movements and/or mouse clicks. Although shell-based interaction is quite feasible for the kind of proof scripts currently presented in this tutorial, the recommended interface for Isabelle/Isar is the Emacs-based \bfindex{Proof General}~\cite{Aspinall:TACAS:2000,proofgeneral}. Some interfaces (including the shell level) offer special fonts with mathematical symbols. For those that do not, remember that ASCII-equivalents are shown in figure~\ref{fig:ascii} in the appendix. Finally, a word about semicolons.\indexbold{$Isar@\texttt{;}} Commands may but need not be terminated by semicolons. At the shell level it is advisable to use semicolons to enforce that a command is executed immediately; otherwise Isabelle may wait for the next keyword before it knows that the command is complete. \section{Getting started} Assuming you have installed Isabelle, you start it by typing \texttt{isabelle -I HOL} in a shell window.\footnote{Simply executing \texttt{isabelle -I} starts the default logic, which usually is already \texttt{HOL}. This is controlled by the \texttt{ISABELLE_LOGIC} setting, see \emph{The Isabelle System Manual} for more details.} This presents you with Isabelle's most basic ASCII interface. In addition you need to open an editor window to create theory files. While you are developing a theory, we recommend to type each command into the file first and then enter it into Isabelle by copy-and-paste, thus ensuring that you have a complete record of your theory. As mentioned above, Proof General offers a much superior interface. If you have installed Proof General, you can start it with \texttt{Isabelle}.