Pal + : a Lambda-free Logical Framework 1 Motivations and Introduction 2.2 Judgement Forms

A lambda-free logical framework takes parameterisation and deenitions as the basic notions to provide schematic mechanisms for speciication of type theories and their use in practice. The framework presented here, PAL + , is a logical framework for speciication and implementation of type theories, such as Martin-LL of's type theory or UTT. As in Martin-LL of's logical framework NPS90], computational rules can be introduced and are used to give meanings to the declared constants. However, PAL + only allows one to talk about the concepts that are intuitively in the object type theories: types and their objects, and families of types and families of objects of types. In particular, in PAL + , one cannot directly represent families of families of entities, which could be done in other logical frameworks by means of lambda abstraction. PAL + is in the spirit of de Bruijn's PAL for Automath dB80]. Compared with PAL, PAL + allows one to represent parametric concepts such as families of types and families of non-parametric objects, which can be used by themselves as totalities as well as when they are fully instantiated. Such parametric objects are represented by local deenitions (let-expressions). We claim that PAL + is a correct metalanguage for specifying type theories (eg, dependent type theories), as it has the advantage of exactly capturing the intuitive concepts in object type theories, and that its implementation reeects the actual use of type theories in practice. We shall study the meta-theory of PAL + by developing its typed operational semantics and showing that it has nice meta-theoretic properties. This work is based on some notes I wrote in 1997 of the same title, which I did not nish, and a further development in my note in LFM'00 Luo00]. A lambda-free logical framework takes parameterisation and deenitions as the basic notions to provide schematic mechanisms for speciication of type theories and their use in practice. The reasons to consider lambda-free logical frameworks include: Parametric constants/deenitions and the associated operations of instan-tiation (substitution or cut) are more basic and arguably simpler no-tions/mechanisms than that of lambda-abstraction as found in other logical frameworks such as Martin-LL of's logical framework NPS90]. A para-metrically deened entity represents a family of entities, rather than a functional operation. The user of a proof system based on a lambda-free framework does not have to understand the meta-level lambda-abstraction that can be used to represent concepts …