  The Minimalist Foundation, for short MF, is a two-level foundation for
constructive mathematics ideated by Maietti and Sambin in 2005 and then fully
formalized by Maietti in 2009. MF serves as a common core among the most
relevant foundations for mathematics in the literature by choosing for each of
them the appropriate level of MF to be translated in a compatible way, namely
by preserving the meaning of logical and set-theoretical constructors. The
two-level structure consists of an intensional level, an extensional one, and
an interpretation of the latter in the former in order to extract intensional
computational contents from mathematical proofs involving extensional
constructions used in everyday mathematical practice. In 2013 a completely new
foundation for constructive mathematics appeared in the literature, called
Homotopy Type Theory, for short HoTT, which is an example of Voevodsky's
Univalent Foundations with a computational nature. So far no level of MF has
been proved to be compatible with any of the Univalent Foundations in the
literature. Here we show that both levels of MF are compatible with HoTT. This
result is made possible thanks to the peculiarities of HoTT which combines
intensional features of type theory with extensional ones by assuming
Voevodsky's Univalence Axiom and higher inductive quotient types. As a relevant
consequence, MF inherits entirely new computable models.
