\chapter{Basic Concepts}

\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 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}

\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 = B\(@1\) + \(\cdots\) + B\(@n\):

{\rmfamily\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 \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.

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. Note that most of the theories 

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 \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}%

\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 set the flag\index{flags}

  \isa{show_types}\index{*show_types (flag)}.  

  Isabelle will then 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,

which we introduce as we go along, can be set and reset in the same manner.%

\index{flags!setting and resetting}

\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 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 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{identifiers} may contain the characters \isa{_} 

and~\isa{'}.

\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 flag\index{flags}

\isa{show_brackets}\index{*show_brackets (flag)}:

\begin{ttbox}

ML "set show_brackets"; \(\dots\); ML "reset show_brackets";

\end{ttbox}

\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 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}

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 interaction 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{proofgeneral,Aspinall:TACAS:2000}.

Some interfaces (including the shell level) offer special fonts with

mathematical symbols. For those that do not, remember that \textsc{ascii}-equivalents

are shown in table~\ref{tab: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 \textsc{ascii} interface.  In addition you need to open an editor window to

create theory files.  While you are developing a theory, we recommend that you

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 by typing \texttt{Isabelle}.