Let be an o-minimal expansion of the real field, and let be the language consisting of all nested Rolle leaves over . We call a set nested sub-Pfaffian over if it is the projection of a positive Boolean combination of definable sets and nested Rolle leaves over . Assuming that admits analytic cell decomposition, we prove that the complement of a nested sub-Pfaffian set over is again a nested sub-Pfaffian set over . As a corollary, we obtain that if admits analytic cell decomposition, then the Pfaffian closure of is obtained by adding to all nested Rolle leaves over , a one-stage process, and that is model complete in the language .
Jean-Marie Lion. Patrick Speissegger. "The theorem of the complement for nested sub-Pfaffian sets." Duke Math. J. 155 (1) 35 - 90, 1 October 2010. https://doi.org/10.1215/00127094-2010-050