doc-src/Tutorial/basics.tex

+\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-ML}.  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.
+
+A tutorial is by definition incomplete. To fully exploit the power of the
+system you need to consult the Isabelle Reference Manual~\cite{Isa-Ref-Man}
+for details about Isabelle and the HOL chapter of the Logics
+manual~\cite{Isa-Logics-Man} for details relating to HOL. Both manuals have a
+comprehensive index.
+
+\section{Theories, proofs and interaction}
+\label{sec:Basic:Theories}
+
+Working with Isabelle means creating two different kinds of documents:
+theories and proof scripts. Roughly speaking, a \bfindex{theory} is a named
+collection of types and functions, much like a module in a programming
+language or a specification in a specification language. In fact, theories in
+HOL can be either. Theories must reside in files with the suffix
+\texttt{.thy}. The general format of a theory file \texttt{T.thy} is
+\begin{ttbox}
+T = B\(@1\) + \(\cdots\) + B\(@n\) +
+\({<}declarations{>}\)
+end
+\end{ttbox}
+where \texttt{B}$@1$, \dots, \texttt{B}$@n$ are the names of existing
+theories that \texttt{T} is based on and ${<}declarations{>}$ stands for the
+newly introduced concepts (types, functions etc). 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 qualified by
+theory names as in \texttt{T.f} and \texttt{B.f}. HOL's theory library is
+available online at \verb$http://www.in.tum.de/~isabelle/library/HOL/$ and is
+recommended browsing.
+\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}
+
+This tutorial is concerned with introducing you to the different linguistic
+constructs that can fill ${<}declarations{>}$ in the above theory template.
+A complete grammar of the basic constructs is found in Appendix~A
+of~\cite{Isa-Ref-Man}, for reference in times of doubt.
+
+The tutorial is also concerned with showing you how to prove theorems about
+the concepts in a theory. This involves invoking predefined theorem proving
+commands. Because Isabelle is written in the programming language
+ML,\footnote{Many concepts in HOL and ML are similar. Make sure you do not
+  confuse the two levels.} interacting with Isabelle means calling ML
+functions. Hence \bfindex{proof scripts} are sequences of calls to ML
+functions that perform specific theorem proving tasks. Nevertheless,
+familiarity with ML is absolutely not required.  All proof scripts for theory
+\texttt{T} (defined in file \texttt{T.thy}) should be contained in file
+\texttt{T.ML}. Theory and proof scripts are loaded (and checked!) by calling
+the ML function \ttindexbold{use_thy}:
+\begin{ttbox}
+use_thy "T";
+\end{ttbox}
+
+There are more advanced interfaces for Isabelle that hide the ML level from
+you and replace function calls by menu selection. There is even a special
+font with mathematical symbols. For details see the Isabelle home page.  This
+tutorial concentrates on the bare essentials and ignores such niceties.
+
+\section{Types, terms and formulae}
+\label{sec:TypesTermsForms}
+
+Embedded in the declarations of 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 \ttindex{bool}, the type of truth values,
+and \ttindex{nat}, the type of natural numbers.
+\item[type constructors,] in particular \ttindex{list}, the type of
+lists, and \ttindex{set}, the type of sets. Type constructors are written
+postfix, e.g.\ \texttt{(nat)list} is the type of lists whose elements are
+natural numbers. Parentheses around single arguments can be dropped (as in
+\texttt{nat list}), multiple arguments are separated by commas (as in
+\texttt{(bool,nat)foo}).
+\item[function types,] denoted by \ttindexbold{=>}. In HOL
+\texttt{=>} represents {\em total} functions only. As is customary,
+\texttt{$\tau@1$ => $\tau@2$ => $\tau@3$} means
+\texttt{$\tau@1$ => ($\tau@2$ => $\tau@3$)}. Isabelle also supports the
+notation \texttt{[$\tau@1,\dots,\tau@n$] => $\tau$} which abbreviates
+\texttt{$\tau@1$ => $\cdots$ => $\tau@n$ => $\tau$}.
+\item[type variables,] denoted by \texttt{'a}, \texttt{'b} etc, just like in
+ML. They give rise to polymorphic types like \texttt{'a => '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 flag \ttindexbold{show_types} that
+  tells Isabelle to display type information that is usually suppressed:
+  simply type
+\begin{ttbox}
+set show_types;
+\end{ttbox}
+
+\noindent
+at the ML-level. This can be reversed by \texttt{reset show_types;}.
+\end{warn}
+
+
+\textbf{Terms}\indexbold{term}
+are formed as in functional programming by applying functions to
+arguments. If \texttt{f} is a function of type \texttt{$\tau@1$ => $\tau@2$}
+and \texttt{t} is a term of type $\tau@1$ then \texttt{f~t} is a term of type
+$\tau@2$. HOL also supports infix functions like \texttt{+} and some basic
+constructs from functional programming:
+\begin{description}
+\item[\texttt{if $b$ then $t@1$ else $t@2$}]\indexbold{*if}
+means what you think it means and requires that
+$b$ is of type \texttt{bool} and $t@1$ and $t@2$ are of the same type.
+\item[\texttt{let $x$ = $t$ in $u$}]\indexbold{*let}
+is equivalent to $u$ where all occurrences of $x$ have been replaced by
+$t$. For example,
+\texttt{let x = 0 in x+x} means \texttt{0+0}. Multiple bindings are separated
+by semicolons: \texttt{let $x@1$ = $t@1$; \dots; $x@n$ = $t@n$ in $u$}.
+\item[\texttt{case $e$ of $c@1$ => $e@1$ | \dots | $c@n$ => $e@n$}]
+\indexbold{*case}
+evaluates to $e@i$ if $e$ is of the form
+$c@i$. See~\S\ref{sec:case-expressions} for details.
+\end{description}
+
+Terms may also contain $\lambda$-abstractions. For example, $\lambda
+x. x+1$ is the function that takes an argument $x$ and returns $x+1$. In
+Isabelle we write \texttt{\%x.~x+1}.\index{==>@{\tt\%}|bold}
+Instead of \texttt{\%x.~\%y.~\%z.~t} we can write \texttt{\%x~y~z.~t}.
+
+\textbf{Formulae}\indexbold{formula}
+are terms of type \texttt{bool}. There are the basic
+constants \ttindexbold{True} and \ttindexbold{False} and the usual logical
+connectives (in decreasing order of priority):
+\verb$~$\index{$HOL1@{\ttnot}|bold} (`not'),
+\texttt{\&}\index{$HOL2@{\tt\&}|bold} (`and'),
+\texttt{|}\index{$HOL2@{\ttor}|bold} (`or') and
+\texttt{-->}\index{$HOL2@{\tt-->}|bold} (`implies'),
+all of which (except the unary \verb$~$) associate to the right. In
+particular \texttt{A --> B --> C} means \texttt{A --> (B --> C)} and is thus
+logically equivalent with \texttt{A \& B --> C}
+(which is \texttt{(A \& B) --> C}).
+
+Equality is available in the form of the infix function
+\texttt{=} of type \texttt{'a => 'a => bool}. Thus \texttt{$t@1$ = $t@2$} 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 \texttt{bool}, \texttt{=} acts as if-and-only-if.
+
+The syntax for quantifiers is
+\texttt{!~$x$.$\,P$}\index{$HOLQ@{\ttall}|bold} (`for all $x$') and
+\texttt{?~$x$.$\,P$}\index{$HOLQ@{\tt?}|bold} (`exists $x$').
+There is even \texttt{?!~$x$.$\,P$}\index{$HOLQ@{\ttuniquex}|bold}, which
+means that there exists exactly one $x$ that satisfies $P$. Instead of
+\texttt{!} and \texttt{?} you may also write \texttt{ALL} and \texttt{EX}.
+Nested quantifications can be abbreviated:
+\texttt{!$x~y~z$.$\,P$} means \texttt{!$x$.~!$y$.~!$z$.$\,P$}.
+
+Despite type inference, it is sometimes necessary to attach explicit
+\bfindex{type constraints} to a term.  The syntax is \texttt{$t$::$\tau$} as
+in \texttt{x < (y::nat)}. Note that \texttt{::} binds weakly and should
+therefore be enclosed in parentheses: \texttt{x < y::nat} is ill-typed
+because it is interpreted as \texttt{(x < y)::nat}. The main reason for type
+constraints are overloaded functions like \texttt{+}, \texttt{*} and
+\texttt{<}. (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 flag
+\ttindexbold{show_brackets}:
+\begin{ttbox}
+set show_brackets; \(\dots\); reset show_brackets;
+\end{ttbox}
+\end{warn}
+
+\begin{itemize}
+\item
+Remember that \texttt{f t u} means \texttt{(f t) u} and not \texttt{f(t u)}!
+\item
+Isabelle allows infix functions like \texttt{+}. The prefix form of function
+application binds more strongly than anything else and hence \texttt{f~x + y}
+means \texttt{(f~x)~+~y} and not \texttt{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, \verb$~ ~ P = P$ means \verb$~ ~(P = P)$ and
+not \verb$(~ ~P) = P$. When using \texttt{=} to mean logical equivalence,
+enclose both operands in parentheses, as in \texttt{(A \& B) = (B \& 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 \texttt{f = (\%x.~x)}. This includes
+\ttindex{if}, \ttindex{let}, \ttindex{case}, \verb$%$ and quantifiers.
+\item
+Never write \texttt{\%x.x} or \texttt{!x.x=x} because \texttt{x.x} is always
+read as a single qualified identifier that refers to an item \texttt{x} in
+theory \texttt{x}. Write \texttt{\%x.~x} and \texttt{!x.~x=x} instead.
+\end{itemize}
+
+\section{Variables}
+\label{sec:variables}
+
+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} or \bfindex{unknown}, which starts with a \texttt{?}.  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 \texttt{?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.
+\begin{warn}
+  The existential quantifier \texttt{?}\index{$HOLQ@{\tt?}} needs to be
+  followed by a space. Otherwise \texttt{?x} is interpreted as a schematic
+  variable.
+\end{warn}
+
+\section{Getting started}
+
+Assuming you have installed Isabelle, you start it by typing \texttt{isabelle
+  HOL} in a shell window.\footnote{Since you will always want to use HOL when
+  studying this tutorial, you can set the shell variable
+  \texttt{ISABELLE_LOGIC} to \texttt{HOL} once and for all and simply execute
+  \texttt{isabelle}.} This presents you with Isabelle's most basic ASCII
+interface. In addition you need to open an editor window to create theories
+(\texttt{.thy} files) and proof scripts (\texttt{.ML} files). While you are
+developing a proof, we recommend to type each proof command into the ML-file
+first and then enter it into Isabelle by copy-and-paste, thus ensuring that
+you have a complete record of your proof.