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LJQ: a strongly focused calculus for intuitionistic logic
 COMPUTABILITY IN EUROPE 2006, VOLUME 3988 OF LNCS
, 2006
"... LJQ is a focused sequent calculus for intuitionistic logic, with a simple restriction on the first premisss of the usual left introduction rule for implication. We discuss its history (going back to about 1950, or beyond), present the underlying theory and its applications both to terminating proof ..."
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LJQ is a focused sequent calculus for intuitionistic logic, with a simple restriction on the first premisss of the usual left introduction rule for implication. We discuss its history (going back to about 1950, or beyond), present the underlying theory and its applications both to terminating proofsearch calculi and to callbyvalue reduction in lambda calculus.
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 construc ..."
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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 &quot;International Symposium on Logicbased Program Synthesis and Transformation 2004 (LOPSTR 2004)&quot;, 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
A sequent calculus for type theory
 CSL 2006. LNCS
, 2006
"... Based on natural deduction, Pure Type Systems (PTS) can express a wide range of type theories. In order to express proofsearch in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proofsearch and strongly related to natural ..."
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Cited by 4 (0 self)
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Based on natural deduction, Pure Type Systems (PTS) can express a wide range of type theories. In order to express proofsearch in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proofsearch and strongly related to natural deduction. PTSC are equipped with a normalisation procedure, adapted from Herbelin’s and defined by local rewrite rules as in Cutelimination, using explicit substitutions. It satisfies Subject Reduction and it is confluent. A PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising if and only if the latter is. We show how the conversion rules can be incorporated inside logical rules (as in syntaxdirected rules for type checking), so that basic proofsearch tactics in type theory are merely the rootfirst application of our inference rules.
Avoiding Equivariance in AlphaProlog
, 2004
"... Prolog is a logic programming language which is wellsuited for rapid prototyping of type systems and operational semantics of typed #calculi and many other languages involving bound names. In #Prolog, the nominal unification algorithm of Urban, Pitts and Gabbay is used instead of firstorder un ..."
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Cited by 4 (0 self)
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Prolog is a logic programming language which is wellsuited for rapid prototyping of type systems and operational semantics of typed #calculi and many other languages involving bound names. In #Prolog, the nominal unification algorithm of Urban, Pitts and Gabbay is used instead of firstorder unification.
A Sequent Calculus for Type Theory
, 2006
"... Abstract Based on natural deduction, Pure Type Systems (PTS) can express a wide range of type theories. In order to express proofsearch in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proofsearch and strongly related t ..."
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Abstract Based on natural deduction, Pure Type Systems (PTS) can express a wide range of type theories. In order to express proofsearch in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proofsearch and strongly related to natural deduction. PTSC are equipped with a normalisation procedure, adapted from Herbelin’s and defined by local rewrite rules as in Cutelimination, using explicit substitutions. It satisfies Subject Reduction and it is confluent. A PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising if and only if the latter is. We show how the conversion rules can be incorporated inside logical rules (as in syntaxdirected rules for type checking), so that basic proofsearch tactics in type theory are merely the rootfirst application of our inference rules.