theory Ind_Intro

imports Main

begin

chapter {* How to write a definitional package *}

section{* Introduction *}

text {*

\begin{flushright}

  {\em

    ``My thesis is that programming is not at the bottom of the intellectual \\

    pyramid, but at the top. It's creative design of the highest order. It \\

    isn't monkey or donkey work; rather, as Edsger Dijkstra famously \\

    claimed, it's amongst the hardest intellectual tasks ever attempted.''} \\[1ex]

    Richard Bornat, In defence of programming

\end{flushright}

\medskip

\noindent

Higher order logic, as implemented in Isabelle/HOL, is based on just a few primitive

constants, like equality, implication, and the description operator, whose properties are

described as axioms. All other concepts, such as inductive predicates, datatypes, or

recursive functions are \emph{defined} using these constants, and the desired

properties, for example induction theorems, or recursion equations are \emph{derived}

from the definitions by a \emph{formal proof}. Since it would be very tedious

for the average user to define complex inductive predicates or datatypes ``by hand''

just using the primitive operators of higher order logic, Isabelle/HOL already contains

a number of \emph{packages} automating such tedious work. Thanks to those packages,

the user can give a high-level specification, like a list of introduction rules or

constructors, and the package then does all the low-level definitions and proofs

behind the scenes. The packages are written in Standard ML, the implementation

language of Isabelle, and can be invoked by the user from within theory documents

written in the Isabelle/Isar language via specific commands. Most of the time,

when using Isabelle for applications, users do not have to worry about the inner

workings of packages, since they can just use the packages that are already part

of the Isabelle distribution. However, when developing a general theory that is

intended to be applied by other users, one may need to write a new package from

scratch. Recent examples of such packages include the verification environment

for sequential imperative programs by Schirmer \cite{Schirmer-LPAR04}, the

package for defining general recursive functions by Krauss \cite{Krauss-IJCAR06},

as well as the nominal datatype package by Berghofer and Urban \cite{Urban-Berghofer06}.

The scientific value of implementing a package should not be underestimated:

it is often more than just the automation of long-established scientific

results. Of course, a carefully-developed theory on paper is indispensable

as a basis. However, without an implementation, such a theory will only be of

very limited practical use, since only an implementation enables other users

to apply the theory on a larger scale without too much effort. Moreover,

implementing a package is a bit like formalizing a paper proof in a theorem

prover. In the literature, there are many examples of paper proofs that

turned out to be incomplete or even faulty, and doing a formalization is

a good way of uncovering such errors and ensuring that a proof is really

correct. The same applies to the theory underlying definitional packages.

For example, the general form of some complicated induction rules for nominal

datatypes turned out to be quite hard to get right on the first try, so

an implementation is an excellent way to find out whether the rules really

work in practice.

Writing a package is a particularly difficult task for users that are new

to Isabelle, since its programming interface consists of thousands of functions.

Rather than just listing all those functions, we give a step-by-step tutorial

for writing a package, using an example that is still simple enough to be

easily understandable, but at the same time sufficiently complex to demonstrate

enough of Isabelle's interesting features. As a running example, we have

chosen a rather simple package for defining inductive predicates. To keep

things simple, we will not use the general Knaster-Tarski fixpoint

theorem on complete lattices, which forms the basis of Isabelle's standard

inductive definition package originally due to Paulson \cite{Paulson-ind-defs}.

Instead, we will use a simpler \emph{impredicative} (i.e.\ involving

quantification on predicate variables) encoding of inductive predicates

suggested by Melham \cite{Melham:1992:PIR}. Due to its simplicity, this

package will necessarily have a reduced functionality. It does neither

support introduction rules involving arbitrary monotone operators, nor does

it prove case analysis (or inversion) rules. Moreover, it only proves a weaker

form of the rule induction theorem.

Reading this article does not require any prior knowledge of Isabelle's programming

interface. However, we assume the reader to already be familiar with writing

proofs in Isabelle/HOL using the Isar language. For further information on

this topic, consult the book by Nipkow, Paulson, and Wenzel

\cite{isa-tutorial}. Moreover, in order to understand the pieces of

code given in this tutorial, some familiarity with the basic concepts of the Standard

ML programming language, as described for example in the textbook by Paulson

\cite{paulson-ml2}, is required as well.

The rest of this article is structured as follows. In \S\ref{sec:ind-examples},

we will illustrate the ``manual'' definition of inductive predicates using

some examples. Starting from these examples, we will describe in

\S\ref{sec:ind-general-method} how the construction works in general.

The following sections are then dedicated to the implementation of a

package that carries out the construction of such inductive predicates.

First of all, a parser for a high-level notation for specifying inductive

predicates via a list of introduction rules is developed in \S\ref{sec:ind-interface}.

Having parsed the specification, a suitable primitive definition must be

added to the theory, which will be explained in \S\ref{sec:ind-definition}.

Finally, \S\ref{sec:ind-proofs} will focus on methods for proving introduction

and induction rules from the definitions introduced in \S\ref{sec:ind-definition}.

\nocite{Bornat-lecture}

*}

end