Here is Girard's `technical condition', then the reformulation by
Abramsky and Mellies (in their LICS'99 paper):

 (G)  if v is a weight generated by the weights occurring in \Theta,
      and x is a &-vertex, then v.~w(x) does not depend on the
      eigenvariable p_x.

 (AM) If v is a weight occuring in \Theta and x is a &-vertex,
      if v depends on p_x then v < w(L).

(In (G), ~z means "not z".)  The following is the simplest
(non-monomial) example I could come up with that satisfies (G) but not
(AM):

                   p|q  ~p~q
   _______________   1   1
  |   ____________|   \ /         p|q = "p or q"
  |  |  __________|    +          ~p~q = "(not p) and (not q)"
  |  \ /          |   /           1 = tensor unit
  |   &q           \ /            * = tensor
   \ /              *             &p = "&-vertex with eigenweight p"
    &p

(literals and remaining weights implicit).  Note that this is in fact
a (non-monomial) MALL proof net --- so (AM) is non-sensical ("too
strong") once the monomial constraint is removed.  Similarly, the
3-formula Gustave example fails (AM), but satisfies (G).  [To see that 
the Gustave example satisfies (G), observe that it satisfies

 (PRESENCE)  an axiom link z depends on a &-vertex x in a slice S 
             only if x is in S.

proved equivalent to (G) in one of the other FAQ answers.]

In summary:

  (1) The Gustave example satisfies Girard's technical condition.
  (2) Girard's technical condition does not imply softness.
  (3) The Abramsky-Mellies technical condition is strictly stronger than
      Girard's technical condition.
  (4) The Abramsky-Mellies technical condition becomes "too strong" upon
      relaxing the monomial constraint (cf. the example drawn above).


DJDH 2003/07/16