is leads to an almost exponential increase in the number of regions, as each previously generated region spawns more regions in the higher dimensions. This would result in a stark increase in running time.

One can conclude that number of terms and exponents contribute equally to the running time of CAD.

This was conducted for a 2 variable case, with a maximum of $5$ terms per polynomial and a maximum of 2 polynomials per clause. The number of clauses was varied from $2$ to $6$. The results were averaged over 50 randomly generated prenex formulae. An almost exponential increase is observed in this case, as shown in Fig. 4.

One could infer from this decrease in gradient that the total number of terms in a collection of projected polynomials depends to a greater extent on the number of terms per polynomial, than on the number of polynomials.

On Validity of the Preprocessor
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As the analysis is performed on the assumption that the input is in the CNF format, with each prenex formula containing two or more clauses, it is necessary to give an account on the percentage of randomly generated prenex formulae, that can be converted to a CNF 