Implementing mathematics with the Nuprl proof development system
Implementing mathematics with the Nuprl proof development system
Programming in Martin-Lo¨f's type theory: an introduction
Programming in Martin-Lo¨f's type theory: an introduction
Computation and reasoning: a type theory for computer science
Computation and reasoning: a type theory for computer science
The ALF proof editor and its proof engine
TYPES '93 Proceedings of the international workshop on Types for proofs and programs
An Implementation of LF with Coercive Subtyping & Universes
Journal of Automated Reasoning
Pure Type Systems with Definitions
LFCS '94 Proceedings of the Third International Symposium on Logical Foundations of Computer Science
Open Proofs and Open Terms: A Basis for Interactive Logic
CSL '02 Proceedings of the 16th International Workshop and 11th Annual Conference of the EACSL on Computer Science Logic
Pure type systems with judgemental equality
Journal of Functional Programming
Coercive subtyping in lambda-free logical frameworks
Proceedings of the Fourth International Workshop on Logical Frameworks and Meta-Languages: Theory and Practice
ACM Transactions on Computational Logic (TOCL)
TYPES'02 Proceedings of the 2002 international conference on Types for proofs and programs
A type-theoretic framework for formal reasoning with different logical foundations
ASIAN'06 Proceedings of the 11th Asian computing science conference on Advances in computer science: secure software and related issues
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A lambda-free logical framework takes parameterisation and definitions as the basic notions to provide schematic mechanisms for specification of type theories and their use in practice. The framework presented here, PAL+, is a logical framework for specification and implementation of type theories, such as Martin-Löf's type theory or UTT. As in Martin-Löf's logical framework (Nordström et al., 1990), 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 (de Bruijn, 1980). 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 definitions (let-expressions). We claim that PAL+ is a correct meta-language for specifying type theories (e.g., dependent type theories), as it has the advantage of exactly capturing the intuitive concepts in object type theories, and that its implementation reflects 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.