theory Ind_Intro

imports Main 

begin

chapter {* How to Write a Definitional Package\label{chp:package} *}

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 \cite{Bornat-lecture}

  \end{flushright}

  \medskip

  HOL is based on just a few primitive constants, like equality and

  implication, whose properties are described by axioms. All other concepts,

  such as inductive predicates, datatypes, or recursive functions have to be defined

  in terms of those constants, and the desired properties, for example

  induction theorems, or recursion equations have to be derived from the definitions

  by a formal proof. Since it would be very tedious for a user to define

  complex inductive predicates or datatypes ``by hand'' just using the

  primitive operators of higher order logic, \emph{definitional packages} have

  been implemented automating such work. Thanks to those packages, the user

  can give a high-level specification, for example a list of introduction

  rules or constructors, and the package then does all the low-level

  definitions and proofs behind the scenes. In this chapter we explain how

  such a package can be implemented.

  As a running example, we have chosen a rather simple package for defining

  inductive predicates. To keep things really 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.  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

  induction principle for inductive predicates.

*}

end