Constructing Infinitary Quotient-Inductive Types

Abstract

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, 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.

A class of higher inductive types in Zermelo‐Fraenkel set theory

“…We now define our class of higher inductive types that we will construct in $$\mathsf {Set}$$. It will be clear by the definition that this is a special case of $$QW$$‐types [8]. Definition A $$\emph {polynomial}$$ is a set A together with a family of sets $$(B_a)_{a \in A}$$.…”

“…In fact our examples of interest fall within a smaller class with a simpler definition and clearer semantics. This class was studied by Blass [5] under the name free algebras subject to identities and by Fiore, Pitts and Steenkamp in [8] under the name $$QW$$ ‐types (we will refer to them by the latter name). A well known example of such a type is that of “unordered countably branching trees.” We modify the definition of T above to get the higher inductive type $$T_{\operatorname{Sym}}$$ by adding equations as follows, where we write $$\operatorname{Sym}(\omega )$$ for the type of permutations $$\omega \rightarrow \omega$$.…”

“…As they remark, the obvious construction of $$T_{\operatorname{Sym}}$$ as a quotient of T requires the axiom of choice. Fiore, Pitts and Steenkamp showed that in fact it is an example of a $$QW$$‐type [8, Example 2].…”

“…Fiore, Pitts and Steenkamp [8] gave an electronically verified proof that $$QW$$‐types can be constructed in type theory using Agda sized types and universes closed under inductive‐inductive types. We can see from Blass' counterexample that some combination of these assumptions for $$\mathsf {Set}$$ must lead to the existence of uncountable regular cardinals.…”

Free Commutative Monoids in Homotopy Type Theory

“…While Cubical Agda [39] accepts all three HIT definitions, the status of defining HITs in HoTT and Cubical type theories in general is an open research question. Various schemas for defining HITs have been proposed [35,14,11,22,23]. HITs have also been used in other computer science applications [5,24,6,4].…”