We show that the class of L-constructible functions is closed under integration for any P -minimal expansion of a p-adic field (K, L). This generalizes results previously known for semi-algebraic and sub-analytic structures. As part of the proof, we obtain a weak version of cell decomposition and function preparation for P -minimal structures, a result which is independent of the existence of Skolem functions. A direct corollary is that Denef's results on the rationality of Poincaré series hold in any P -minimal expansion of a p-adic field (K, L).

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We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, p p -adically closed fields, real closed and algebraically closed valued fields and closed ordered differential fields. Furthermore, we prove pro-definability of other distinguished subspaces, some of which have an interesting geometric interpretation. Our general strategy consists of showing that definable types are uniformly definable, a property which implies pro-definability using an argument due to E. Hrushovski and F. Loeser. Uniform definability of definable types is finally achieved by studying classes of stably embedded pairs.

We prove that in a P -minimal structure, every definable set can be partitioned as a finite union of classical cells and regular clustered cells. This is a generalization of previously known cell decomposition results by Denef and Mourgues, which were dependent on the existence of definable Skolem functions. Clustered cells have the same geometric structure as classical, Denef-type cells, but do not have a definable function as center. Instead, the center is given by a definable set whose fibers are finite unions of balls.