\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.

\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.

\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{;}} Some interfaces,

for example Proof General, require each command to be terminated by a

semicolon, whereas others, for example the shell level, do not. But even 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. Note that for readibility

reasons we often drop the final semicolon in the text.

\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.