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**1 - 1**of**1**### An algebraical proof of the Contraction Criterion for Proof Nets

"... Abstract proof structures in multiplicative linear logic are graphs with some additional structure, and the class of proof nets is an inductively defined subclass. F. M'etayer established a correctness criterion by defining homology groups for proof structures, which characterize the proof nets amon ..."

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Abstract proof structures in multiplicative linear logic are graphs with some additional structure, and the class of proof nets is an inductively defined subclass. F. M'etayer established a correctness criterion by defining homology groups for proof structures, which characterize the proof nets among these. Using this result, we will present a completely algebraical proof of the traditional Contraction Criterion, due to Danos and Regnier. 1 Preliminaries 1.1 MLL The set of formulas of multiplicative linear logic (MLL) is defined as the smallest set containing atoms p 1 ; p 2 ; : : : and their formal negations p ? 1 ; p ? 2 ; : : : , and closed under the binary operations\Omega (tensor) and & (par ). Unary negation is defined by the (commutative) De Morgan laws, i.e. (a\Omega b) ? := a ? & b ? , etc. Derivable objects are sequents ) X, where X is a multiset of formulas. The rules of MLL read: axiom ) a ? ; a cut-rule ) X; a ) a ? ; Y ) X;Y tensor-rule ) X; a ) b; ...