@MISC{Altenkirch94consistencyin, author = {Thorsten Altenkirch}, title = {Consistency in ALF}, year = {1994} }

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Abstract

Introduction ALF is the implementation of a logical framework. This implies that ALF is an open system, we can use it for different approaches to Type Theory or even to encode conventional logic. ALF will only check whether our theories typecheck. We are responsible to make sure that the theory is consistent --- i.e. it is no problem to encode an inconsistent theory like System U in ALF. We could restrict ourselves to the monomorphic set theory as it is described in [NPS90], chapter 19. It is straightforward to implement this theory in ALF and to construct proof-objects by explicit definitions. However, this does not reflect the current usage of ALF, i.e.: ffl We want to be able to introduce new sets by giving a sequence of constructors. ffl We want to define non-canonical constants by pattern matching. Peter Dybjer has developed a notion of schemes to capture inductively defined sets [Dyb91, Dyb94]. Thierry Coq