this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multi-sets are avoided altogether in these proofs, and termination measures are replaced by three nested structural inductions. Parameters are treated as variables bound in derivations, thus naturally capturing occurrence conditions. A starting point for the proofs is Kleene's sequent system G 3 [Kle52], which we derive systematically from the point of view that a sequent calculus should be a calculus of proof search for natural deductions. It can easily be related to Gentzen's original and other sequent calculi. We augment G 3 with proof terms that are stable under weakening. These proof terms enable the structural induction and furthermore form the basis of the representation of the proof in LF. The most closely related work on cut elimination is Martin-Lo# f 's proof of admissibility [ML68]. In Martin-Lo# f 's system the cut rule incorporates aspects of both weakening and contraction which enables a structural induction argument closely related to ours. However, without the introduction of proof terms, the implicit weakening in the cut rule makes it difficult to implement this proof directly. Herbelin [Her95] restates this proof and proceeds by assigning proof terms only to restricted sequent calculi LJT and LKT which correspond more immediately to

This work is dedicated to my parents. Acknowledgments Firstly, and foremost, I would like to thank my principal advisor, Frank Pfenning, for his patience with me, and for teaching me most of what I know about logic and type theory. I would also like to acknowledge some useful discussions with Kevin Watkins which led me to simplify some of this work. Finally, I would like to thank my other advisor, John Reynolds, for all his kindness and support over the last five years. Abstract This thesis introduces a new logical system, ordered linear logic, which combines reasoning with unrestricted, linear, and ordered hypotheses. The logic conservatively extends (intuitionistic) linear logic, which contains both unrestricted and linear hypotheses, with a notion of ordered hypotheses. Ordered hypotheses must be used exactly once, subject to the order in which they were assumed (i.e., their order cannot be changed during the course of a derivation). This ordering constraint allows for logical representations of simple data structures such as stacks and queues. We construct ordered linear logic in the style of Martin-L"of from the basic notion of a hypothetical judgement. We then show normalization for the system by constructing a sequent calculus presentation and proving cut-elimination of the sequent system.

We present a sequent calculus for intuitionistic non-commutative linear logic (INCLL) , show that it satisfies cut elimination, and investigate its relationship to a natural deduction system for the logic. We show how normal natural deductions correspond to cut-free derivations, and arbitrary natural deductions to sequent derivations with cut. This gives us a syntactic proof of normalization for a rich system of non-commutative natural deduction and its associated -calculus. INCLL conservatively extends linear logic with means to express sequencing, which has applications in functional programming, logical frameworks, logic programming, and natural language parsing. 1 Introduction Linear logic [11] has been described as a logic of state because it views linear hypotheses as resources which may be consumed in the course of a deduction. It thereby significantly extends the expressive power of both classical and intuitionistic logics, yet it does not offer means to express sequencing. Th...

this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call N-systems, are symbolic logics generally given via introduction and elimination rules for the logical connectives which operate on the right, i.e., they manipulate the succedent formula. Examples are Gentzen's NJ and NK (Gentzen 1935). Logical deduction systems are given via left-introduction and right-introduction rules for the logical connectives. Although others have called these systems "sequent calculi", we call them L-systems to avoid confusion with other systems given in sequent style. Examples are Gentzen's LK and LJ (Gentzen 1935). In this paper we are primarily interested in L-systems. The advantage of N-systems is that they seem closer to actual reasoning, while L-systems on the other hand seem to have an easier proof theory. L-systems are often extended with a "cut" rule as part of showing that for a given L-system and N-system, the derivations of each system can be encoded in the other. For example, NK proves the same as LK + cut (Gentzen 1935). Proof Normalization. A system is consistent when it is impossible to prove false, i.e., derive absurdity from zero assumptions. A system is analytic (has the analycity property) when there is an e#ective method to decompose any conclusion sequent into simpler premise sequents from which the conclusion can be obtained by some rule in the system such that the conclusion is derivable i# the premises are derivable (Maenpaa 1993). To achieve the goals of consistency and analycity, it has been customary to consider

Based on natural deduction, Pure Type Systems (PTS) can express a wide range of type theories. In order to express proof-search in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proof-search 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 Cut-elimination, 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 syntax-directed rules for type checking), so that basic proofsearch tactics in type theory are merely the root-first application of our inference rules.

Abstract Based on natural deduction, Pure Type Systems (PTS) can express a wide range of type theories. In order to express proof-search in such theories, we introduce the Pure Type Sequent Calculi (PTSC) by enriching a sequent calculus due to Herbelin, adapted to proof-search 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 Cut-elimination, 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 syntax-directed rules for type checking), so that basic proofsearch tactics in type theory are merely the root-first application of our inference rules.

In existing game models, total functionals have no simple characterization neither in term of game strategies, nor in term of the total set-theoretical functionals they define. We show that the situation changes if we extend the usual notion of game by allowing infinite plays. Total functionals are

The typed -calculus and the natural deduction are isomorph. There is a mapping (the Curry-Howard's isomorphism) between each others. This paper shows that the typed calculi with explicit substitutions are isomorph to a logical deduction system which is equivalent to the natural deduction. This logical formalism inherits the explicit substitutions properties: confluence on open terms, first-order calculus and no PSN. 1 Introduction What is called in mathematics a proof or a demonstration is in fact a more or less informal speech which is founded on some axioms and which follows the greek logic tradition. The aim of the demonstration theory is to supply a formal framework in order to describe a raisoning; the proof becomes a rigorous object. It can be handled like any other mathematical object. In a proof, we have to distinguish what is proved from the demonstration itself. In the following, we will use formalisms where the proved objects are propositional formulas 1 . More precisely,...

Abstract Superdeduction is a systematic way to extend a deduction system like the sequent calculus by new deduction rules computed from the user theory. We show how this could be done in a systematic, correct and complete way. We prove in detail the strong normalization of a proof term language that models appropriately superdeduction. We finaly examplify on several examples, including equality and noetherian induction, the usefulness of this approach which is implemented in the lemuridæ system, written in TOM. 1

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