  We propose a general framework for first-order functional logic programming,
supporting lazy functions, non-determinism and polymorphic datatypes whose data
constructors obey a set C of equational axioms. On top of a given C, we specify
a program as a set R of C-based conditional rewriting rules for defined
functions. We argue that equational logic does not supply the proper semantics
for such programs. Therefore, we present an alternative logic which includes
C-based rewriting calculi and a notion of model. We get soundness and
completeness for C-based rewriting w.r.t. models, existence of free models for
all programs, and type preservation results. As operational semantics, we
develop a sound and complete procedure for goal solving, which is based on the
combination of lazy narrowing with unification modulo C. Our framework is quite
expressive for many purposes, such as solving action and change problems, or
realizing the GAMMA computation model.
