  This paper introduces an expressive class of quotient-inductive types, called
QW-types. We show that in dependent type theory with uniqueness of identity
proofs, even the infinitary case of QW-types can be encoded using the
combination of inductive-inductive definitions involving strictly positive
occurrences of Hofmann-style quotient types, and Abel's size types. The latter,
which provide a convenient constructive abstraction of what classically would
be accomplished with transfinite ordinals, are used to prove termination of the
recursive definitions of the elimination and computation properties of our
encoding of QW-types. The development is formalized using the Agda theorem
prover.
