We present a technical introduction to mechanizing language definitions and meta-theory using LF and Twelf. LF is a logical framework designed for representing languages that are specified by inductively-defined judgements. Twelf is an implementation of LF that includes additional support for checking meta-theorems about represented languages. In this article, we first summarize a canonical-forms presentation of LF, following the treatment of CLF by Watkins et al. Next, we use the simply-typed #-calculus as a running example of mechanization in LF and Twelf: we show how to adequately encode the simply-typed #-calculus in LF, and then we prove type preservation and strengthening as examples of Twelf meta-theory.

Over the years, logical framework research has produced various type theories designed primarily for the representation of deductive systems. Reasoning about these representations requires expressive special purpose meta logics, that are in general not part of the logical framework.

The logical framework LF is a type theory defined by Harper, Honsell and Plotkin. It is wellsuited to serve as a meta language to represent deductive systems. LF and its logic programming implementation Elf are also well-suited to represent meta-theoretic proofs and their computational content, but search for such proofs lies outside the framework. This thesis proposes a computational meta logic (MLF) for the Horn fragment of LF. The Horn fragment is a significant restriction of LF but it is powerful enough to represent non-trivial problems. This thesis demonstrates how MLF can be used for the problem of compiler verification. It also discusses some theoretical properties of MLF. Contents 1 Introduction 1 2 Motivation 3 2.1 An Example : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.1 A Toy Language : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.2 Natural Semantics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :...

rule). This is also the case for NJ and LJ as defined in this formalisation. This is due to the particular nature of the logics in question, and does not necessarily generalise to other logics. In particular, a formalisation of linear logic would not work in this fashion, and a more complex variable-referencing mechanism would be required. See Section 6 for a further discussion of this problem. Other operations, such as substitutions (sub in Table 2) and weakening, require lift and drop operations as defined in [27] to ensure the correctness of the de Bruijn indexing.

We define an equivalent variant LK sp of the Gentzen sequent calculus LK. In LK sp weakenings or contractions can be performed in parallel. This modification allows us to interpret a symmetrical system of mix elimination rules by a finite rewriting system; the termination of this rewriting system can be machine checked. We give also a self-contained strong normalization proof by structural induction. We give another strong normalization proof by a strictly monotone subrecursive interpretation; this interpretation gives subrecursive bounds for the length of derivations. We give a strong normalization proof by applying orthogonal term rewriting results for a confluent restriction of the mix elimination system .

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We present a new proof of cut elimination for linear logic which proceeds by three nested structural inductions, avoiding the explicit use of multi-sets and termination measures on sequent derivations. The computational content of this proof is a non-deterministic algorithm for cut elimination which is amenable to an elegant implementation in Elf. We show this implementation in detail. This work was supported by NSF Grant CCR-9303383 The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of NSF or the U.S. government. Keywords: Linear Logic, Cut Elimination, Logical Framework Contents 1 Introduction 1 2 CLL: A Traditional Sequent Calculus for Linear Logic 1 3 LV: Another Sequent Calculus for Linear Logic 3 4 Linear Proof Terms 8 5 Representation in LF 11 5.1 Formulas : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 5.2 Pro...

views and conclusions contained in this document are those of the author, and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government, or any other entity. Keywords: refinement types, logical frameworks, subtyping, intersection types, The logical framework LF and its metalogic Twelf can be used to encode and reason about a wide variety of logics, languages, and other deductive systems in a formal, machine-checkable way. Recent studies have shown that ML-like languages can profitably be extended with a notion of subtyping called refinement types. A refinement type discipline uses an extra layer of term classification above the usual type system to more accurately capture certain properties of terms. I propose that adding refinement types to LF is both useful and practical. To support the claim, I exhibit an extension of LF with refinement types called LFR, work out important details of its metatheory, delineate a practical algorithm for refinement type reconstruction, and present several case studies