k -k.
 k A.

Now suppose that a new row is added. The number of columns becomes:

 U.+A.-L.+k -k
 Ck+1 + 1 3-4
 1
 = k +

so there is a new column which can be assigned as the output for the new

row. Theorem 3-1 is proved.

Corollary: It is always possible to specify the index outputs as

offsets from the index values in an IDM.

Proof: For the list choose any list element as the index output offset.

For the range note that for L.=U. the output can be expressed as an
 7 1
offset from the function index value, say k Now assume that for

L.<U. all index outputs have been defined as offsets, say i+k Suppose
3-- 1 U
U. is increased by A.. A new function index, the new U.,
 1 1
has been introduced. At the same time a new variable index value,

U. +k not previously assigned as an output is introduced, which is

offset from U. by the same factor as all previous outputs. The

corollary is proved.

 The theorem and its corollary not only guarantee that index

outputs are possible for arbitrary limits, but also that there is an

index output set assignment which makes the determination of output

variable index values easily obtainable from the function index values.

3.3.4 Index Decisions

 In an IDM the columns not assigned as index outputs are termed

index decisions. Index decisions result from the lower limit offset

for a variable index range being less than the upper limit offset, or