2

R.VOREADOU

It is clear, however, that [11] excludes too much. There are many

situations, even trivial ones, of pairs (h,h') of equal allowable natural

transformations between shapes which are not both proper; for example,

1[AI], ^ A * 1 ^ : [A,I] -•[A,I]

bI(eSl)a"1, e(l$bA) : [A,I]B(AHI) -- I

where A is a non-constant shape.

Moreover, one can easily check that Proposition 7.6 of [11] can be

stated and proved without the hypothesis about propriety of shapes (this

is Proposition B of [18,§1]). In Proposition 7.8, however, propriety of

shapes is more essentially involved.

So, our work has started from a careful examination of the proof of

Proposition 7.8 of [11],with respect to the point where the restriction

to "proper" shapes is essentially needed there. It is easily seen that

this point is in Case (ii) ([11,p.136]), where, for the inductive argu-

ment to work, the possibility that (in the notation of [11,p.136]) [B,C]

be associated via x with a prime factor of P must be excluded, and this

can be done only by introducing some special hypothesis.

In [11], this special hypothesis is the hypothesis that the shapes

are proper.

In [18],we take the class % of all the allowable graphs, which

are precisely of the form that (produces the case which) we want to ex-

clude, and we make the special hypothesis that the graphs of the natural

transformations in question are not in Wl , with lf[ being a class containing

7)f and defined so that it works well in the inductive proofs [more precise-

ly, lt[ is so defined, that the property "l%" of the graph of an allowable

natural transformation is inherited by the graphs of the allowable natural

transformations of lower rank, when we apply the cut elimination theorem].

With this hypothesis (replacing the hypothesis of [11] that the shapes are

proper), the method of [11] leads to the improved results of [18,§1],

In the following, we make the most direct (iQe. the weakest), with

respect to our purpose, special hypothesis: we suppose from the beginning

that, in the situation of Proposition 7.8 of [11] (which will be replaced

by Proposition 2 below), the allowable natural transformation h does not

admit the case we want to exclude; thus we form Proposition 2 below, which