# (no subject)

```[----- The Types Forum, http://www.cis.upenn.edu/~bcpierce/types -----]

I have a basic question about the system LU, as introduced by:

J.-Y. Girard, On The Unity of Logic,
Annals of Pure and Applied Logic 59:201--217, 1993.

I need here only a small fragment of the system, and do not require
polarities.  Write G for Gamma, D for Delta, L for Lambda, P for Pi.

Sequents have the form:

G; G' |- D'; D

Weakening and contraction apply to G' and D', but not G and D.

Here are some of the rules:

------Id
A;|-;A

G;G'|-D';D,A   A,L;G'|-D';P
---------------------------Cut
G,L;G'|-D';D,P

G,A;G'|-D';D
------------Dereliction
G;A,G'|-D';D

;G'|-D';A
----------!R
;G'|-D';!A

G;A,G'|-D';D
-------------!L
G,!A;G'|-D';D

Consider the following proof

------Id              ------Id
B;|-;B                B;|-;B
------Dereliction     ------Dereliction
;B|-;B                ;B|-;B
-------!R             -------!R
;B|-;!B               ;B|-;!B
---------------------------------otimes-R
A, A-o!B;|-;!B          ;B|-;!B otimes !B
------------------------------------------Cut
A, A-o!B;|-;!B

How does one eliminate the Cut from this proof?

This problem is closely related to one identified in the natural
deduction
formulation of linear logic by Hyland et al., and also discussed in my
paper,
"There's no substitute for linear logic".

Cheers,  -- P

Philip Wadler, Professor of Theoretical Computer Science
Informatics, University of Edinburgh