Completeness of MLL proof-nets w.r.t. weak distributivity

Abstract

We examine ‘weak-distributivity’ as a rewriting rule $\underset{\leadsto{}}{WD}$ defined on multiplicative proof-structures (so, in particular, on multiplicative proof-nets: MLL). This rewriting does not preserve the type of proof-nets, but does nevertheless preserve their correctness. The specific contribution of this paper, is to give a direct proof of completeness for $\underset{\leadsto{}}{WD}$: starting from a set of simple generators (proof-nets which are a $n$-ary $\otimes$ of $\parr$-ized axioms), any mono-conclusion MLL proof-net can be reached by $\underset{\leadsto{}}{WD}$ rewriting (up to $\otimes$ and $\parr$ associativity and commutativity).