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

We examine ‘weak-distributivity’ as a rewriting rule ; WD defined on multiplicative proof-structures (so, in particular, on multiplicative proof-nets: MLL). This rewriting does not preserve the type of proofsnets, but does nevertheless preserve their correctness. The specific contribution of this paper, is to give a direct proof of completeness for ; WD : starting from a set of simple generators (proof-nets which are a n-ary of O-ized axioms), any mono-conclusion MLL proof-net can be reached by ; WD rewriting (up to and O associativity and commutativity). 1 Preliminaries 1.1 Multiplicative Linear Logic: sequent calculus and proof-nets The formulas of Multiplicative Linear Logic [1] are defined from the following grammar: A = X,X⊥, Y, Y ⊥, . . . | A A | AOA Atoms | Tensor | Par Negation (“orthogonal”) (.)⊥ is not a connective, but a defined unary operation over formulas, inductively defined by: (X)⊥ = X⊥, (X⊥)⊥ = X, (A B)⊥ = A⊥OB⊥, (AOB)⊥ = A⊥ B⊥ A sequent is a multiset Γ of formulas, written ` Γ . The rules of sequent calculusMLL (from whichMLL sequent calculus derivations is inductively defined as usual) are : ` A ,A⊥ cut ` Γ,A ` ∆,A⊥ ` Γ,∆ O ` Γ,A , B ` Γ,AOB ` Γ,A ` ∆,B ` Γ,∆ ,A B (Identity axiom)