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Strong normalisation of Herbelin's explicit substitution calculus with substitution propagation
"... . Herbelin presented (at CSL'94) a simple sequent calculus for minimal implicational logic, extensible to full rstorder intuitionistic logic, with a complete system of cutreduction rules which is both conuent and strongly normalising. Some of the cut rules may be regarded as rules to construct exp ..."
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. Herbelin presented (at CSL'94) a simple sequent calculus for minimal implicational logic, extensible to full rstorder intuitionistic logic, with a complete system of cutreduction rules which is both conuent and strongly normalising. Some of the cut rules may be regarded as rules to construct explicit substitutions. He observed that the addition of a cut permutation rule, for propagation of such substitutions, breaks the proof of strong normalisation; the implicit conjecture is that the rule may be added without breaking strong normalisation. We prove this conjecture, thus showing how to model betareduction in his calculus (extended with rules to allow cut permutations). 1 Introduction Herbelin gave in [5] a calculus for minimal implicational logic, using a notation for proof terms that, in contrast to the usual lambdacalculus notation for natural deduction, brings head variables to the surface. It is thus a sequent calculus, with the nice feature that its cutfree terms are in ...
GraphBased Proof Counting and Enumeration with Applications for Program Fragment Synthesis
 in "International Symposium on Logicbased Program Synthesis and Transformation 2004 (LOPSTR 2004)", S. ETALLE (editor)., Lecture Notes in Computer Science
, 2004
"... Abstract. For use in earlier approaches to automated module interface adaptation, we seek a restricted form of program synthesis. Given some typing assumptions and a desired result type, we wish to automatically build a number of program fragments of this chosen typing, using functions and values av ..."
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Abstract. For use in earlier approaches to automated module interface adaptation, we seek a restricted form of program synthesis. Given some typing assumptions and a desired result type, we wish to automatically build a number of program fragments of this chosen typing, using functions and values available in the given typing environment. We call this problem term enumeration. To solve the problem, we use the CurryHoward correspondence (propositionsastypes, proofsasprograms) to transform it into a proof enumeration problem for an intuitionistic logic calculus. We formally study proof enumeration and counting in this calculus. We prove that proof counting is solvable and give an algorithm to solve it. This in turn yields a proof enumeration algorithm. 1
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 .
Proof Search in Lax Logic
, 2000
"... This paper describes two new sequent calculi for Lax Logic. One calculus is for proof enumeration for quanti ed Lax Logic, the other calculus is for theorem proving in propositional Lax Logic ..."
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This paper describes two new sequent calculi for Lax Logic. One calculus is for proof enumeration for quanti ed Lax Logic, the other calculus is for theorem proving in propositional Lax Logic
Two Loop Detection Mechanisms: a Comparison
, 1997
"... . In order to compare two loop detection mechanisms we describe two calculi for theorem proving in intuitionistic propositional logic. We call them both MJ Hist , and distinguish between them by description as `Swiss' or `Scottish'. These calculi combine in different ways the ideas on focused pro ..."
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. In order to compare two loop detection mechanisms we describe two calculi for theorem proving in intuitionistic propositional logic. We call them both MJ Hist , and distinguish between them by description as `Swiss' or `Scottish'. These calculi combine in different ways the ideas on focused proof search of Herbelin and Dyckhoff & Pinto with the work of Heuerding et al on loop detection. The Scottish calculus detects loops earlier than the Swiss calculus but at the expense of modest extra storage in the history. A comparison of the two approaches is then given, both on a theoretic and on an implementational level. 1 Introduction The main interest of this paper is the comparison of the two loop detection mechanisms described below. In order to do this we illustrate their use on the permutationfree sequent calculus MJ for the propositional fragment of intuitionistic logic. This gives calculi whose implementations are suitable for theorem proving. Backwards proof search and theorem ...
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.
On a localstep cutelimination procedure for the intuitionistic sequent calculus
 Proc. of the 13th Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning (LPARâ€™06), volume 4246 of LNCS
, 2006
"... Abstract. In this paper we investigate, for intuitionistic implicational logic, the relationship between normalization in natural deduction and cutelimination in a standard sequent calculus. First we identify a subset of proofs in the sequent calculus that correspond to proofs in natural deduction. ..."
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Abstract. In this paper we investigate, for intuitionistic implicational logic, the relationship between normalization in natural deduction and cutelimination in a standard sequent calculus. First we identify a subset of proofs in the sequent calculus that correspond to proofs in natural deduction. Then we define a reduction relation on those proofs that exactly corresponds to normalization in natural deduction. The reduction relation is simulated soundly and completely by a cutelimination procedure which consists of local proof transformations. It follows that the sequent calculus with our cutelimination procedure is a proper extension that is conservative over natural deduction with normalization. 1
A Permutationfree Calculus for Lax Logic
, 1998
"... this paper the same `permutationfree' techniques used to develop MJ are applied to Lax Logic, giving a `permutationfree' calculus for Lax Logic. As our starting point we take the above cited papers of Fairtlough & Mendler and of Benton, Bierman & de Paiva. 2 Natural Deduction ..."
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this paper the same `permutationfree' techniques used to develop MJ are applied to Lax Logic, giving a `permutationfree' calculus for Lax Logic. As our starting point we take the above cited papers of Fairtlough & Mendler and of Benton, Bierman & de Paiva. 2 Natural Deduction
Bidirectional Decision Procedures for the Intuitionistic Propositional Modal Logic IS4
"... Abstract. We present a multicontext focused sequent calculus whose derivations are in bijective correspondence with normal natural deductions in the propositional fragment of the intuitionistic modal logic IS4. This calculus, suitable for the enumeration of normal proofs, is the starting point for ..."
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Abstract. We present a multicontext focused sequent calculus whose derivations are in bijective correspondence with normal natural deductions in the propositional fragment of the intuitionistic modal logic IS4. This calculus, suitable for the enumeration of normal proofs, is the starting point for the development of a sequent calculusbased bidirectional decision procedure for propositional IS4. In this system, relevant derived inference rules are constructed in a forward direction prior to proof search, while derivations constructed using these derived rules are searched over in a backward direction. We also present a variant which searches directly over normal natural deductions. Experimental results show that on most problems, the bidirectional prover is competitive with both conventional backward provers using loopdetection and inverse method provers, significantly outperforming them in a number of cases. 1
Metatheory in the HigherOrder Logic Framework Isabelle
, 1996
"... Isabelle [Pau94] is a generic theorem proving environment. It is written in ML, and is part of the LCF [GMW79] family of tacticbased theorem provers. The core system uses a metalogic consisting of the implicational/universal fragment of an intuitionistic higherorder logic with equality. Within thi ..."
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Isabelle [Pau94] is a generic theorem proving environment. It is written in ML, and is part of the LCF [GMW79] family of tacticbased theorem provers. The core system uses a metalogic consisting of the implicational/universal fragment of an intuitionistic higherorder logic with equality. Within this system, a large number object logics can be represented. A general method of encoding a sequent calculus as an object logic in a form suitable for proving properties of the calculus is presented. Some suggestions about general use of Isabelle in this area are made. 1 Introduction As with most Logical Frameworks, Isabelle uses an ASCII notation for the symbols of logic not available on a keyboard. Appendix A gives a basic introduction to this, but throughout this paper standard logical notations will be used for ease of reading. Isabelle represents a highly modular system with many incompatible object logics having been developed based on a single core, the Pure core system. The Pure syst...