Quotients, inductive types, and quotient inductive types

This paper introduces an expressive class of indexed quotient-inductive types, called QWI within the framework constructive type theory. They are initial algebras for families equational theories with possibly infinitary operators and equations. We prove that types can be derived from quotient inductive in theory toposes natural number object universes, provided those universes satisfy Weakly Initial Set Covers (WISC) axiom. do so by constructing as colimits a family approximations to them defined well-founded recursion over suitable notion size, whose definition involves WISC developed proof checked it using Agda theorem prover.