  We present two of the three major steps in the construction of motivic
integration, that is, a homomorphism between Grothendieck semigroups that are
associated with a first-order theory of algebraically closed valued fields, in
the fundamental work of Hrushovski and Kazhdan. We limit our attention to a
simple major subclass of V-minimal theories of the form ACVF_S(0, 0), that is,
the theory of algebraically closed valued fields of pure characteristic $0$
expanded by a (VF, Gamma)-generated substructure S in the language L_RV. The
main advantage of this subclass is the presence of syntax. It enables us to
simplify the arguments with many different technical details while following
the major steps of the Hrushovski-Kazhdan theory.
