control.
We can interpret this from a traditional viewpoint as the following theorem.
Theorem 1.2 The key formulas in Theorems 1,3,4 above can be derived with an idealized strategy.
We use the name 1-Decompose to also refer
to the Decompose operation,
we use the name 1-motivated unknown to
refer to motivated unknown and we use the name
idealized 1-strategy to refer to an idealized strategy.
For the 1-Decompose operation,
the key was to find a polynomial k in
the knowns and
one unknown
and a polynomial q such that when q is substituted
for the one unknown of k, one obtains p.
Likewise, we could consider 
-decompositions
which would consist of finding a polynomial k
in unknowns and polynomials
such that when 
is substituted
for the j-th unknown for 
, one
obtains p. An -Decompose operation would then
be an operation which found all j-motivated unknowns
for and an idealized -strategy would allow
the use of the -Decompose operation.
A variant on 1-Decompose which we shall use
frequently is called symmetric 1-Decompose.
This applies in an algebra with involution, 
for
all w,
for example, a matrix algebra with adjoints or transposes.
Symmetric
1-Decompose applied to p yields
a 2-decomposition of p as
where the second polynomial 
is the adjoint of the first.
We were not able to derive the key formulas of
the theorem of item 2 above
with a 1-strategy, but
they
can be derived with a symmetrized
2-strategy.
The use of a symmetrized 2-strategy
forces human intervention,
but it is small because the
2-decomposition required is
easy to recognize from the output of the NCProcess1 command.
(Note that it originally took 5 years to
discover the theorem).