It is well known that multiplicative linear logic (MLL) is conservative over intuitionistic multiplicative linear logic (IMLL). In other words, if an IMLL formula is provable in MLL then it is already provable in IMLL.


Who first proved this, and how? It doesn’t seem to be in Girard’s original Linear Logic paper, yet I've never seen a reference given when this fact is referred to.



Conservativity of multiplicative linear logic over intuitionistic multiplicative linear logic
This is (a piece of) Proposition 3.8 in 


* Harold Schellinx, Some Syntactical Observations on Linear Logic, J. Logic Compututa., Vol. 1 No. 4, pp. 537-559, 1991. ([pdf at oxford journals](http://logcom.oxfordjournals.org/content/1/4/537.full.pdf))


His proof-theoretic argument is that the only way to use a sequent with multiple conclusions when giving a cut-free MLL proof of an intuitionistic sequent $\Gamma \Rightarrow C$ is as the left premise (of the form $\Gamma\_1 \Rightarrow A,C$) of the ${\multimap}L$ rule. But then the right premise (of the form $\Gamma\_2,B \Rightarrow \cdot$) must have an empty right hand side, and by induction it is easy to see that such sequents are unprovable in the absence of 0 or $\bot$. Hence any cut-free MLL proof of an intuitionistic sequent $\Gamma \Rightarrow C$ is limited to sequents with exactly one conclusion, and can be replayed as an IMLL proof.


