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Completing Herbelin’s programme
"... In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a CurryHoward interpretation of sequent calculus into a variant of the λcalculus, specifically a variant which manipulates formally “applicative contexts” and inverts t ..."
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In 1994 Herbelin started and partially achieved the programme of showing that, for intuitionistic implicational logic, there is a CurryHoward interpretation of sequent calculus into a variant of the λcalculus, specifically a variant which manipulates formally “applicative contexts” and inverts the associativity of “applicative terms”. Herbelin worked with a fragment of sequent calculus with constraints on left introduction. In this paper we complete Herbelin’s programme for full sequent calculus, that is, sequent calculus without the mentioned constraints, but where permutative conversions necessarily show up. This requires the introduction of a lambdalike calculus for full sequent calculus and an extension of natural deduction that gives meaning to “applicative contexts” and “applicative terms”. Such extension is a calculus with modus ponens and primitive substitution that refines von Plato’s natural deduction; it is also a “coercion calculus”, in the sense of Cervesato and Pfenning. The prooftheoretical outcome is noteworthy: the puzzling relationship between cut and substitution is settled; and cutelimination in sequent calculus is proven isomorphic to normalisation in the proposed natural deduction system. The isomorphism is the mapping that inverts the associativity of applicative terms.
On a localstep cutelimination procedure for the intuitionistic sequent calculus
 Proc. of the 13th Int. Conf. on Logic for Programming Artificial Intelligence and Reasoning (LPAR’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
Structural proof theory as rewriting
"... Abstract. The multiary version of the λcalculus with generalized applications integrates smoothly both a fragment of sequent calculus and the system of natural deduction of von Plato. It is equipped with reduction rules (corresponding to cutelimination/normalisation rules) and permutation rules, t ..."
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Abstract. The multiary version of the λcalculus with generalized applications integrates smoothly both a fragment of sequent calculus and the system of natural deduction of von Plato. It is equipped with reduction rules (corresponding to cutelimination/normalisation rules) and permutation rules, typical of sequent calculus and of natural deduction with generalised elimination rules. We argue that this system is a suitable tool for doing structural proof theory as rewriting. As an illustration, we investigate combinations of reduction and permutation rules and whether these combinations induce rewriting systems which are confluent and terminating. In some cases, the combination allows the simulation of nonterminating reduction sequences known from explicit substitution calculi. In other cases, we succeed in capturing interesting classes of derivations as the normal forms w.r.t. wellbehaved combinations of rules. We identify six of these “combined ” normal forms, among which are two classes, due to Herbelin and Mints, in bijection with normal, ordinary natural deductions. A computational explanation for the variety of “combined ” normal forms is the existence of three ways of expressing multiple application in the calculus. 1
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"... 1 Introduction In [2] we defined the generalised multiary y"calculus y"Jm, an extension of the y"calculus where application is generalised in two directions: (i) "generality", inthe sense of von Plato's generalised eliminations [7]; and (ii) &q ..."
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1 Introduction In [2] we defined the generalised multiary y&quot;calculus y&quot;Jm, an extension of the y&quot;calculus where application is generalised in two directions: (i) &quot;generality&quot;, inthe sense of von Plato's generalised eliminations [7]; and (ii) &quot;multiarity&quot;, i.e. the ability of applying functions to lists of arguments. The original motivationwas to extend Schwichtenberg's work on permutative conversions for intuitionistic cutfree sequent calculus [6]. y&quot;Jm comes equipped with a set of permutativeconversions for which the permutability theorem holds: two
Permutative Conversions in Intuitionistic Multiary Sequent Calculi with Cuts José Espírito Santo and Luís Pinto ⋆
"... Abstract. This work presents an extension with cuts of Schwichtenberg’s multiary sequent calculus. We identify a set of permutative conversions on it, prove their termination and confluence and establish the permutability theorem. We present our sequent calculus as the typing system of the generalis ..."
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Abstract. This work presents an extension with cuts of Schwichtenberg’s multiary sequent calculus. We identify a set of permutative conversions on it, prove their termination and confluence and establish the permutability theorem. We present our sequent calculus as the typing system of the generalised multiary λcalculus λJ m, a new calculus introduced in this work. λJ m corresponds to an extension of λcalculus with a notion of generalised multiary application, which may be seen as a function applied to a list of arguments and then explicitly substituted in another term. Prooftheoretically the corresponding typing rule encompasses, in a modular way, generalised eliminations of von Plato and Herbelin’s head cuts. 1