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Structural Cut Elimination  I. Intuitionistic and Classical Logic
 Information and Computation
, 2000
"... 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]. Multisets are avoided altogether in these proofs, and termination measures are replaced b ..."
Abstract

Cited by 53 (17 self)
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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]. Multisets 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 MartinLo# f 's proof of admissibility [ML68]. In MartinLo# 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
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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Cited by 3 (0 self)
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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.