The motivic Thom-Sebastiani theorem for regular and formal functions

Thanks to the work of Hrushovski and Loeser on motivic Milnor fibers, we give a model-theoretic proof for the motivic Thom-Sebastiani theorem in the case of regular functions. Moreover, slightly extending Hrushovski-Loeser's construction adjusted to Sebag, Loeser and Nicaise's motivic integration for formal schemes and rigid varieties, we formulate and prove an analogous result for formal functions. The latter is meaningful as it has been a crucial element of constructing Kontsevich-Soibelman's theory of motivic Donaldson-Thomas invariants.