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-\chapter{The Basics}
-
-\section{Introduction}
-
-This book is a tutorial on how to use the theorem prover 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 do not assume that you are familiar with mathematical logic.
-However, we do assume that
-you are used to logical and set theoretic notation, as covered
-in a good discrete mathematics course~\cite{Rosen-DMA}, and
-that you are familiar with the basic concepts of functional
-programming~\cite{Bird-Haskell,Hudak-Haskell,paulson-ml2,Thompson-Haskell}.
-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.
-
-Isabelle~\cite{paulson-isa-book} is implemented in ML~\cite{SML}. This has
-influenced some of Isabelle/HOL's concrete syntax but is otherwise irrelevant
-for us: this tutorial is based on
-Isabelle/Isar~\cite{isabelle-isar-ref}, an extension of Isabelle which hides
-the implementation language almost completely. Thus the full name of the
-system should be Isabelle/Isar/HOL, but that is a bit of a mouthful.
-
-There are other implementations of HOL, in particular the one by Mike Gordon
-\index{Gordon, Mike}%
-\emph{et al.}, which is usually referred to as ``the HOL system''
-\cite{mgordon-hol}. For us, HOL refers to the logical system, and sometimes
-its incarnation Isabelle/HOL\@.
-
-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, in particular the ability to write readable and structured proofs,
-you should start with Nipkow's overview~\cite{Nipkow-TYPES02} and consult
-the Isabelle/Isar Reference Manual~\cite{isabelle-isar-ref} and Wenzel's
-PhD thesis~\cite{Wenzel-PhD} (which discusses many proof patterns)
-for further details. 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}
-
-\index{theories|(}%
-Working with Isabelle means creating theories. Roughly speaking, a
-\textbf{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
-imports B\(@1\) \(\ldots\) B\(@n\)
-begin
-{\rmfamily\textit{declarations, definitions, and proofs}}
-end
-\end{ttbox}\cmmdx{theory}\cmmdx{imports}
-where \texttt{B}$@1$ \dots\ \texttt{B}$@n$ are the names of existing
-theories that \texttt{T} is based on and \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 theories} of~\texttt{T}\@.
-Everything defined in the parent theories (and their parents, recursively) is
-automatically visible. To avoid name clashes, identifiers can be
-\textbf{qualified}\indexbold{identifiers!qualified}
-by theory names as in \texttt{T.f} and~\texttt{B.f}.
-Each theory \texttt{T} must
-reside in a \textbf{theory file}\index{theory files} named \texttt{T.thy}.
-
-This tutorial is concerned with introducing you to the different linguistic
-constructs that can fill the \textit{declarations, definitions, and
- proofs} above. A complete grammar of the basic
-constructs is found in the Isabelle/Isar Reference
-Manual~\cite{isabelle-isar-ref}.
-
-\begin{warn}
- HOL contains a theory \thydx{Main}, the union of all the basic
- predefined theories like arithmetic, lists, sets, etc.
- Unless you know what you are doing, always include \isa{Main}
- as a direct or indirect parent of all your theories.
-\end{warn}
-HOL's theory collection is available online at
-\begin{center}\small
- \url{http://isabelle.in.tum.de/library/HOL/}
-\end{center}
-and is recommended browsing. In subdirectory \texttt{Library} you find
-a growing library of useful theories that are not part of \isa{Main}
-but can be included among the parents of a theory and will then be
-loaded automatically.
-
-For the more adventurous, there is the \emph{Archive of Formal Proofs},
-a journal-like collection of more advanced Isabelle theories:
-\begin{center}\small
- \url{http://afp.sourceforge.net/}
-\end{center}
-We hope that you will contribute to it yourself one day.%
-\index{theories|)}
-
-
-\section{Types, Terms and Formulae}
-\label{sec:TypesTermsForms}
-
-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
-\index{types|(}
-\begin{description}
-\item[base types,]
-in particular \tydx{bool}, the type of truth values,
-and \tydx{nat}, the type of natural numbers.
-\item[type constructors,]\index{type constructors}
- in particular \tydx{list}, the type of
-lists, and \tydx{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,]\index{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,]\index{type variables}\index{variables!type}
- denoted by \ttindexboldpos{'a}{$Isatype}, \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 that you ask Isabelle to display all type
- information via the Proof General menu item \pgmenu{Isabelle} $>$
- \pgmenu{Settings} $>$ \pgmenu{Show Types} (see \S\ref{sec:interface}
- for details).
-\end{warn}%
-\index{types|)}
-
-
-\index{terms|(}
-\textbf{Terms} 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, such as conditional expressions:
-\begin{description}
-\item[\isa{if $b$ then $t@1$ else $t@2$}]\index{*if expressions}
-Here $b$ is of type \isa{bool} and $t@1$ and $t@2$ are of the same type.
-\item[\isa{let $x$ = $t$ in $u$}]\index{*let expressions}
-is equivalent to $u$ where all free 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~$e@n$}]
-\index{*case expressions}
-evaluates to $e@i$ if $e$ is of the form $c@i$.
-\end{description}
-
-Terms may also contain
-\isasymlambda-abstractions.\index{lambda@$\lambda$ expressions}
-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.~$t$}.%
-\index{terms|)}
-
-\index{formulae|(}%
-\textbf{Formulae} are terms of type \tydx{bool}.
-There are the basic constants \cdx{True} and \cdx{False} and
-the usual logical connectives (in decreasing order of priority):
-\indexboldpos{\protect\isasymnot}{$HOL0not}, \indexboldpos{\protect\isasymand}{$HOL0and},
-\indexboldpos{\protect\isasymor}{$HOL0or}, and \indexboldpos{\protect\isasymimp}{$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\index{equality} is available in the form of the infix function
-\isa{=} 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. If $t@1$ and $t@2$ are of type
-\isa{bool} then \isa{=} acts as \rmindex{if-and-only-if}.
-The formula
-\isa{$t@1$~\isasymnoteq~$t@2$} is merely an abbreviation for
-\isa{\isasymnot($t@1$ = $t@2$)}.
-
-Quantifiers\index{quantifiers} are written as
-\isa{\isasymforall{}x.~$P$} and \isa{\isasymexists{}x.~$P$}.
-There is even
-\isa{\isasymuniqex{}x.~$P$}, 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.~$P$}.%
-\index{formulae|)}
-
-Despite type inference, it is sometimes necessary to attach explicit
-\bfindex{type constraints} to a term. The syntax is
-\isa{$t$::$\tau$} as in \isa{x < (y::nat)}. Note that
-\ttindexboldpos{::}{$Isatype} binds weakly and should therefore be enclosed
-in parentheses. For instance,
-\isa{x < y::nat} is ill-typed because it is interpreted as
-\isa{(x < y)::nat}. Type constraints may be needed to disambiguate
-expressions
-involving overloaded functions such as~\isa{+},
-\isa{*} and~\isa{<}. Section~\ref{sec:overloading}
-discusses overloading, while Table~\ref{tab:overloading} presents the most
-important overloaded function symbols.
-
-In general, HOL's concrete \rmindex{syntax} tries to follow the conventions of
-functional programming and mathematics. Here are the main rules that you
-should be familiar with to avoid certain syntactic traps:
-\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{(\isasymlambda{}x.~x) = f}. This includes
-\isa{if},\index{*if expressions}
-\isa{let},\index{*let expressions}
-\isa{case},\index{*case expressions}
-\isa{\isasymlambda}, and quantifiers.
-\item
-Never write \isa{\isasymlambda{}x.x} or \isa{\isasymforall{}x.x=x}
-because \isa{x.x} is always taken as a single qualified identifier. Write
-\isa{\isasymlambda{}x.~x} and \isa{\isasymforall{}x.~x=x} instead.
-\item Identifiers\indexbold{identifiers} may contain the characters \isa{_}
-and~\isa{'}, except at the beginning.
-\end{itemize}
-
-For the sake of readability, we use the usual mathematical symbols throughout
-the tutorial. Their \textsc{ascii}-equivalents are shown in table~\ref{tab:ascii} in
-the appendix.
-
-\begin{warn}
-A particular problem for novices can be the priority of operators. If
-you are unsure, use additional 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 the Proof General flag \pgmenu{Isabelle} $>$
-\pgmenu{Settings} $>$ \pgmenu{Show Brackets} (see \S\ref{sec:interface}).
-\end{warn}
-
-
-\section{Variables}
-\label{sec:variables}
-\index{variables|(}
-
-Isabelle distinguishes free and bound variables, 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 \textbf{schematic
- variable}\index{variables!schematic} or \textbf{unknown}\index{unknowns},
-which must have a~\isa{?} as its first character.
-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
-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}
- For historical reasons, Isabelle accepts \isa{?} as an ASCII representation
- of the \(\exists\) symbol. However, the \isa{?} character must then be followed
- by a space, as in \isa{?~x. f(x) = 0}. Otherwise, \isa{?x} is
- interpreted as a schematic variable. The preferred ASCII representation of
- the \(\exists\) symbol is \isa{EX}\@.
-\end{warn}%
-\index{variables|)}
-
-\section{Interaction and Interfaces}
-\label{sec:interface}
-
-The recommended interface for Isabelle/Isar is the (X)Emacs-based
-\bfindex{Proof General}~\cite{proofgeneral,Aspinall:TACAS:2000}.
-Interaction with Isabelle at the shell level, although possible,
-should be avoided. Most of the tutorial is independent of the
-interface and is phrased in a neutral language. For example, the
-phrase ``to abandon a proof'' corresponds to the obvious
-action of clicking on the \pgmenu{Undo} symbol in Proof General.
-Proof General specific information is often displayed in paragraphs
-identified by a miniature Proof General icon. Here are two examples:
-\begin{pgnote}
-Proof General supports a special font with mathematical symbols known
-as ``x-symbols''. All symbols have \textsc{ascii}-equivalents: for
-example, you can enter either \verb!&! or \verb!\<and>! to obtain
-$\land$. For a list of the most frequent symbols see table~\ref{tab:ascii}
-in the appendix.
-
-Note that by default x-symbols are not enabled. You have to switch
-them on via the menu item \pgmenu{Proof-General} $>$ \pgmenu{Options} $>$
-\pgmenu{X-Symbols} (and save the option via the top-level
-\pgmenu{Options} menu).
-\end{pgnote}
-
-\begin{pgnote}
-Proof General offers the \pgmenu{Isabelle} menu for displaying
-information and setting flags. A particularly useful flag is
-\pgmenu{Isabelle} $>$ \pgmenu{Settings} $>$ \pgdx{Show Types} which
-causes Isabelle to output the type information that is usually
-suppressed. This is indispensible in case of errors of all kinds
-because often the types reveal the source of the problem. Once you
-have diagnosed the problem you may no longer want to see the types
-because they clutter all output. Simply reset the flag.
-\end{pgnote}
-
-\section{Getting Started}
-
-Assuming you have installed Isabelle and Proof General, you start it by typing
-\texttt{Isabelle} in a shell window. This launches a Proof General window.
-By default, you are in HOL\footnote{This is controlled by the
-\texttt{ISABELLE_LOGIC} setting, see \emph{The Isabelle System Manual}
-for more details.}.
-
-\begin{pgnote}
-You can choose a different logic via the \pgmenu{Isabelle} $>$
-\pgmenu{Logics} menu.
-\end{pgnote}