Implementing Mathematics with The Nuprl Proof Development System
This chapter will explain how to write definitions and statements with the Nuprl system, and in doing so it will introduce a minimal set of system commands to the beginning user and explain how the type theory introduced in chapter 2 can be used to express mathematical propositions. The first four sections of the chapter concentrate on discussing the system commands and some relevant syntactic features of the Nuprl logic. The rest of the chapter describes the conceptual issues involved in using type theory as a general-purpose language for expressing mathematics. These conceptual issues are discussed via examples drawn from four areas of mathematics: logic (both constructive and classical), number theory, algebra and set theory. (Chapter 11 contains a summary of definitions and theorems from other areas of mathematics.) It is intended that this chapter be read while experimenting with the system. Throughout this chapter the typewriter font represents text that is actually used in the Nuprl system, while the is used in general discussions.
The Nuprl system provides an interactive medium which is accessed via a screen divided into various windows3.1and a keyboard/mouse which allows communication with the system. The windows represent regions of the system with specialized roles in the interactive process; the different kinds of windows are listed in figure 3.1.