We shall start with defining exprepl for bodyreplicators. Let us consider
a bodyreplicator 
, where
, 
, 
. We
point out that, for all replicators considered, exprepl
(replicator) means that only replicator is expanded and not any
other replicator which may be generated by the expansion of
replicator. We define:
where 
, and the name j does not occur in
spqbr.
Let us apply this formalism to the following bodyreplicator:
In the above symbolism:
,
,
, so the expansion is given by:
which yields:
The expansion of a concatenator 
where 
, sep is either ``;'' or ``,'', is
defined as:
where , and the name j does not occur in
mseq.
For example consider 
. Its
expansion is given by:
The second kind of sequence replicators, an imbricator
is expanded as follows:
where 
are the following strings:
where , and the name j occurs in neither
p nor
q nor t.
In the case of the inbricator:
we have:
and the expansion is given by
The expansion of both left- and right-replicators can be defined in
a similar
way. Let 
and
, where either 
, or pattern=p @ t @ q, be a left- and right-replicator
respectively.
Let . We define:
A replicator in which in=inc=1 and fi is an integer is said to be in normal form. The following result shows that if in, inc, fi are all integers in some replicator, then there is a normal form for the replicator which expands to the same basic expression.
Proof.
This follows immediately from the fact that in exprepl for every
replicator,
the argument of COPY is always subject to substitution by
or 
. 
