Index Set Splitting of PUREs

(We use the notation of Saouter-Quinton-1203)
PURE: a Parametrized Uniform Recurrence Equation.
Index set splitting is a transformation that finds out if splitting a system of equations into individual sub-components can result in a set of SUREs. The scheduling of which can result in a better schedule.

What can be done better for PUNREs (Parametrized Unitarized Recurrence Equations).

Simple example for index set splitting:
Let the domain be 1 <= i,j <= N (the positive quadrant)

i <= j, X[i,j] = f(X[i,j-1])
i > j, X[I,J] = f(X[i-1,j])

The system can be broken down into two sub-domains
i <= j and i > j. Each of the subdomains have 1-d schedules.