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Mechanizing Language Definitions
, 2006
"... We present a technical introduction to mechanizing language definitions and metatheory using LF and Twelf. LF is a logical framework designed for representing languages that are specified by inductivelydefined judgements. Twelf is an implementation of LF that includes additional support for checki ..."
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We present a technical introduction to mechanizing language definitions and metatheory using LF and Twelf. LF is a logical framework designed for representing languages that are specified by inductivelydefined judgements. Twelf is an implementation of LF that includes additional support for checking metatheorems about represented languages. In this article, we first summarize a canonicalforms presentation of LF, following the treatment of CLF by Watkins et al. Next, we use the simplytyped #calculus as a running example of mechanization in LF and Twelf: we show how to adequately encode the simplytyped #calculus in LF, and then we prove type preservation and strengthening as examples of Twelf metatheory.
A Meta Linear Logical Framework
 In 4th International Workshop on Logical Frameworks and MetaLanguages (LFM’04
, 2003
"... 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. ..."
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Cited by 5 (1 self)
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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.
Strong Normalization Proofs for Cut Elimination in Gentzen's Sequent Calculi
 Banach Center Publication
, 1999
"... 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 ..."
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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 selfcontained 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 .
Refinement Types for Logical Frameworks
, 2010
"... 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, machinecheckable way. Recent studies have shown that MLlike languages can profitably be extended with a notion of subtyping called r ..."
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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, machinecheckable way. Recent studies have shown that MLlike 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 that highlight the utility of refinement types for formalized mathematics. In the end I find that refinement types and LF are a match made in heaven: refinements enable many rich new modes of expression, and the simplicity of
A Formalisation Of Weak Normalisation (With Respect To Permutations) Of Sequent Calculus Proofs
, 1999
"... 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 ..."
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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 variablereferencing 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.
A Computational Meta Logic for the Horn Fragment of LF
, 1995
"... 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 wellsuited to represent metatheoretic proofs and their computational content, but ..."
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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 wellsuited to represent metatheoretic 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 nontrivial 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 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :...
Proving Isomorphism of FirstOrder Logic Proof Systems in HOL
, 1998
"... We prove in HOL that three proof systems for classical rstorder predicate logic, the Hilbertian axiomatization, the system of natural deduction, and a variant of sequent calculus, are isomorphic. The isomorphism is in the sense that provability of a conclusion from hypotheses in one of these proof ..."
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We prove in HOL that three proof systems for classical rstorder predicate logic, the Hilbertian axiomatization, the system of natural deduction, and a variant of sequent calculus, are isomorphic. The isomorphism is in the sense that provability of a conclusion from hypotheses in one of these proof systems is equivalent to provability ofthis conclusion from the same hypotheses in the others. Proving isomorphism of these three proof systems allows us to guarantee that metalogical provability properties about one of them would also hold in relation to the others. We prove the deduction, monotonicity, and compactness theorems for Hilbertian axiomatization, and the substitution theorem for the system of natural deduction. Then we show how these properties can be translated between the proof systems. Besides, by proving a theorem which states that provability in attened sequent calculus implies provability in standard sequent calculus, we show how some metalogical results about Hilbertian axiomatization and natural deduction can be translated to sequent calculus.
Machinechecked Cutelimination for Display Logic
, 2006
"... Belnap’s Display Logic is a generalised sequentstyle framework which is able to capture many different logics in one uniform setting. Its main attractions are twofold: a “display property ” which allows every formula to be introduced as the whole of the left or right side of a sequent, and a single ..."
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Belnap’s Display Logic is a generalised sequentstyle framework which is able to capture many different logics in one uniform setting. Its main attractions are twofold: a “display property ” which allows every formula to be introduced as the whole of the left or right side of a sequent, and a single cutelimination theorem which works for all “proper ” display calculi. Belnap’s original proof of this theorem is short and succinct, but it has been criticised by some as being too informal. The only other published proof turns out to contain an error. Modern interactive proof assistants like Nqthm, HOL, Isabelle, Coq and PVS invariably trace their origins to the simplytyped lambdacalculus of Church (although Mizar is an exception). Such tools have matured to the point where deep theorems in logic and mathematics like Gödel’s Incompleteness Theorem and the Four Colour Theorem can be expressed and proved by expert users using these systems.
Abstract LFM 2004 Preliminary Version A Meta Linear Logical Framework
"... Logical frameworks serve as metalanguages to represent deductive systems, sometimes requiring special purpose meta logics to reason about the representations. In this work, we describe L + ω, meta logic for the linear logical framework LLF [CP96] and illustrate its use via a proof of the admissibil ..."
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Logical frameworks serve as metalanguages to represent deductive systems, sometimes requiring special purpose meta logics to reason about the representations. In this work, we describe L + ω, meta logic for the linear logical framework LLF [CP96] and illustrate its use via a proof of the admissibility of cut in the sequent calculus for the tensor fragment of linear logic. L + ω is firstorder, intuitionistic, and not linear. The soundness of L + ω is shown. 1
Proving Metatheorems with Twelf
, 2006
"... Notes to proofreaders: • I haven’t had time to do many of the adequacy proofs yet. If you know of a way of streamlining these, I’d love to hear it. I wonder if moving to a Kevinstyle presentation would help? • The theorem in Appendix A stating that equivalent worlds have the same canonical forms is ..."
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Notes to proofreaders: • I haven’t had time to do many of the adequacy proofs yet. If you know of a way of streamlining these, I’d love to hear it. I wonder if moving to a Kevinstyle presentation would help? • The theorem in Appendix A stating that equivalent worlds have the same canonical forms is true (in the sense that I wrote out the proof on paper, but I haven’t had a chance to type it up yet). I haven’t seen this written up anywhere, so please let me know if it exists. These notes are a rough draft; please alert me to any errors or omissions!