We present the definition of the logical framework TF, the Type Framework. TF is a lambda-free lo... more We present the definition of the logical framework TF, the Type Framework. TF is a lambda-free logical framework; it does not include lambda-abstraction or product kinds. We give formal proofs of several results in the metatheory of TF, and show how it can be conservatively embedded in the logical framework LF: its judgements can be seen as the judgements of LF that are in beta-normal, eta-long normal form. We show how several properties, such as adequacy theorems for object theories and the injectivity of constants, can be proven more easily in TF, and then `lifted' to LF.
Proceedings of the Fourth International Workshop on Logical Frameworks and Meta-Languages Theory and Practice - LFMTP '09, 2009
Coercive subtyping is a powerful approach to subtyping in dependent type theories, but its theore... more Coercive subtyping is a powerful approach to subtyping in dependent type theories, but its theoretical properties are often difficult to prove. Lambda-free logical frameworks such as TF have shown themselves to be a powerful tool for investigating the theory of logical frameworks, thanks to the close correspondance between a lambda-free frame and a traditional framework such as LF. We show how a type theory with coercive subtyping may be defined within TF. An operation of typecasting plays the role that coercive application plays in LF. We show that the resulting systems in TF and LF are equivalent, and how several results may be proven more easily in TF and then lifted to LF.
It is possible to represent the terms of a syntax with binding constructors by a family of types,... more It is possible to represent the terms of a syntax with binding constructors by a family of types, indexed by the free variables that may occur. This approach has been used several times for the study of syntax and substitution, but never for the formalization of the metatheory of a typing system. We describe a recent formalization of the metatheory of Pure Type Systems in Coq as an example of such a formalization. In general, careful thought is required as to how each definition and theorem should be stated, usually in an unfamiliar 'big-step' form; but, once the correct form has been found, the proofs are very elegant and direct.
We present a programme of research for pluralist formalisations, that is, formalisations that inv... more We present a programme of research for pluralist formalisations, that is, formalisations that involve proving results in more than one foundation.A foundation consists of two parts: a logical part, which provides a notion of inference, and a non-logical part, which provides the entities to be reasoned about. An LTT is a formal system composed of two such separate parts. We show how LTTs may be used as the basis for a pluralist formalisation.We show how different foundations may be formalised as LTTs, and also describe a new method for proof reuse. If we know that a translation Φ exists between two logic-enriched type theories (LTTs) S and T, and we have formalised a proof of a theorem α in S, we may wish to make use of the fact that Φ(α) is a theorem of T. We show how this is sometimes possible by writing a proof script MΦ. For any proof script Mα that proves a theorem α in S, if we change Mα so it first imports MΦ, the resulting proof script will still parse, and will be a proof of...
A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for formi... more A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named LTT 0 and LTT * 0 , which we claim correspond closely to the classical predicative systems of second order arithmetic ACA 0 and ACA. We justify this claim by translating each second-order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics. The two LTTs we construct are subsystems of the logic-enriched type theory LTT W , which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system ACA 0 has also been claimed to correspond to Weyl's foundation. By casting ACA 0 and ACA as LTTs, we are able to compare them with LTT W. It is a consequence of the work in this paper that LTT W is strictly stronger than ACA 0. The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.
We construct a logic-enriched type theory LTTW that corresponds closely to the predicative system... more We construct a logic-enriched type theory LTTW that corresponds closely to the predicative system of foundations presented by Hermann Weyl in Das Kontinuum. We formalize many results from that book in LTTW, including Weyl's definition of the cardinality of a set and several results from real analysis, using the proof assistant Plastic that implements the logical framework LF. This case
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