Constructing InductiveInductive Types in Cubical Type Theory
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Abstract
Inductiveinductive types are a joint generalization of mutual inductive types and indexed inductive types. In extensional type theory, inductiveinductive types can be constructed from inductive types, and this construction has been conjectured to work in intensional type theory as well. In this paper, we show that the existing construction requires Uniqueness of Identity Proofs, and present a new construction (which we conjecture generalizes) of one particular inductiveinductive type in cubical type theory, which is compatible with homotopy type theory.
1 Introduction
These types have been studied extensively in Nordvall Forsberg [15]. There, in §5.3, inductiveinductive types with simple elimination rules (defined in op. cit. §3.2.5) are constructed from indexed inductive types in extensional type theory, and in §5.4 this is conjectured to work in intensional type theory as well.
 1.
In Sect. 2, we show that, in intensional type theory, if the types constructed by Nordvall Forsberg satisfy the simple elimination rules, then UIP holds (formalized in both Coq and Agda).
 2.
In Sect. 3, we give the construction of a particular inductiveinductive type with simple elimination rules in cubical type theory (formalized in cubical Agda).
1.1 Syntax and Conventions
We mostly mimic Agda syntax. The double bar symbol \(=\) is used for definitions directly and by pattern matching, and for equality of terms up to conversion. We write \((a : A) \rightarrow B\) for the dependent product type, and \(A \rightarrow B\) for the nondependent version. Functions are given by pattern matching \(f\; x = y\) or by lambda expressions \(f = \lambda x.y\). Similarly \((a : A) \times B\) is the dependent pair type, and \(A \times B\) the nondependent version. Pairs are (a, b), and projections are p.1 and p.2. The unit type is \(\top \), with unique inhabitant \(\star \). Identity types are \({x}\equiv _{X}{y}\) for the type of identifications of x with y in type X, and we write \(\texttt {refl}\) for a proof of reflexivity. We do not assume that axiom K holds for identity types. We write Type for a universe of types (where Agda uses \(\text {Set}\)). In Sect. 3 we work in cubical type theory, which will be explained there.
1.2 Running Example of an InductiveInductive Definition
For the purposes of this paper, we will focus on one relatively simple inductiveinductive definition (with only 5 clauses), parametrized by a type X, which is given in Fig. 1. We will use this definition to prove that Nordvall Forsberg’s construction implies UIP in Sect. 2 and as a running example to demonstrate our construction in cubical type theory in Sect. 3.
Our goal in this paper is to construct \((A,B,\eta ,\texttt {ext},\texttt {inj})\) of the types given in Fig. 1 such that the simple elimination rules hold without using UIP. But first, we will show why Nordvall Forsberg’s approach is not sufficient.
2 Deriving UIP
Uniqueness of Identity proofs (UIP) for a type X is the principle that, for all x : X, y : X, \(p : {x}\equiv _{X}{y}\), \(q : {x}\equiv _{X}{y}\), the type \({p}\equiv _{{x}\equiv _{X}{y}}{q}\) is inhabited. Equivalently, for all x : X, \(p : {x}\equiv _{X}{x}\), the type \({p}\equiv _{{x}\equiv _{X}{x}}{\texttt {refl}}\) is inhabited. It expresses that there is at most one proof of any equality. UIP is independent of standard intensional type theory [13], and is inconsistent with Homotopy Type Theory [18].
Nordvall Forsberg’s construction of inductiveinductive types is described in [15, §5.3]. In this section, we show that if the simple elimination rules hold for this construction of the inductiveinductive type in Fig. 1, then UIP holds for the type X (Theorem 1). This argument has been formalized in both Coq version 8.8.0 [8] (see UIP_from_Forsberg_II.v) and Agda using the –withoutK flag (see UIP_from_Forsberg_II.agda).
In extensional type theory, Nordvall Forsberg proved that \({A}_\text {good}\;a\) is a mere proposition (all inhabitants are equal) [15, Lemma 5.37(ii)]. In intensional type theory as well, if function extensionality and UIP hold then \({A}_\text {good}\) is a mere proposition. This uniqueness of goodness proofs justifies having the definition of B ignore the goodness proof \({a}_\text {good}\), since \({a}_\text {good}\) can have at most one value.
Combining these results, we conclude that Nordvall Forsberg’s construction satisfies the simple elimination rules in intensional type theory only if UIP holds (Theorem 1).
2.1 Unique Goodness Implies UIP
Lemma 1
Proof
Lemma 2
(Unique goodness implies UIP). If \({A}_\text {good}\;t\) is a mere proposition for all \(t : {A}_\text {pre}\), then UIP holds for the type X.
Proof
2.2 Simple Elimination Rules Imply Unique Goodness
Now we prove that there are at least as many proofs of \(B\;({t}_\text {pre},{t}_\text {good})\) as there are of \({A}_\text {good}\;{t}_\text {pre}\).
Lemma 3
Proof
Lemma 4
(\(B\;a\) is contractible). Assuming the simple elimination rules from Fig. 1 hold for the \((A, B, \eta , \texttt {inj}, \texttt {ext})\) constructed above, for all a : A and \(b : B\;a\), \({\texttt {inj}\;a}\equiv _{B\;a}{b}\).
Proof
Referring to the simple elimination rules given in Fig. 1, we pattern match on B by giving motives \(( PA , PB )\) and methods \((P\eta , P\texttt {ext}, P\texttt {inj})\), and then using the resulting \( EB \).
We set \( PA \;a = \top \), and take \( PB \;a\;b = {\texttt {inj}\;a}\equiv _{B\;a}{b}\). Then we have \(P\eta \;x = \star \), and \(P\texttt {ext}\;a\;\star \;b\;H = \star \), and we take \(P\texttt {inj}\;a\;\star = \texttt {refl}: {\texttt {inj}\;a}\equiv _{B\;a}{\texttt {inj}\;a}\). The conclusion follows by \( EB : (a : A) \rightarrow (b : B\;a) \rightarrow {\texttt {inj}\;a}\equiv _{B\;a}{b}\).
Lemma 5
(Simple elimination rules imply unique goodness). If the simple eliminators hold for the \((A,B,\eta ,\texttt {inj},\texttt {ext})\) constructed above, then for all \(t : {A}_\text {pre}\), \({A}_\text {good}\;t\) is a mere proposition.
Proof
Assume that the simple elimination rules hold, and take \(t : {A}_\text {pre}\), and \(a_1\) and \(a_2\) in \({A}_\text {good}\;t\). We use the definition of f and g from Lemma 3 with \({t}_\text {pre} = t\) and \({t}_\text {good} = a_1\).
2.3 Simple Elimination Rules for Nordvall Forsberg’s Construction only if UIP
Theorem 1
If the simple elimination rules hold for Nordvall Forsberg’s construction, then UIP holds for the type X.
Therefore Nordvall Forsberg’s approach to constructing inductiveinductive types requires UIP. Since UIP is inconsistent with the Univalence axiom at the center of Homotopy Type Theory (HoTT) [18], we have an incentive to come up with a different construction which is consistent with HoTT.
3 Constructing an InductiveInductive Type in Cubical Type Theory
Cubical type theory [6] is a recently developed type theory which gives a constructive interpretation of the Univalence axiom of Homotopy Type Theory. It has an implementation as a mode for Agda [19], which we use to formalize the construction given in this section of the running example from Fig. 1.
The most important difference between cubical type theory and standard intensional type theory as implemented by Coq or vanilla Agda is that the identity type \({x}\equiv _{X}{y}\) is represented (loosely speaking) by the type of functions p from an interval type \(\mathbb {I}\) with two endpoints \(i_0\) and \(i_1\) to X such that \(p\;i_0\) reduces to x and \(p\;i_1\) reduces to y. This allows, for example, a simple proof of function extensionality: if we have \(A :\,\text {Type}\), \(B : A\rightarrow \,\text {Type}\), f and g functions of type \((a : A) \rightarrow B\;a\), and \(h : (a : A) \rightarrow {f\;a}\equiv {g\;a}\), then we have \((\lambda i.\lambda a.h\;a\;i) : {f}\equiv {g}\). Taking \(\text {cong}\; f = \lambda p. \lambda i. f\; (p\;i) : {x}\equiv {y} \rightarrow {f\;x}\equiv {f\;y}\) and \(\circ \) for function composition, we also have nice properties such as \((\text {cong}\; f) \circ (\text {cong}\; g) = \text {cong}\; (f \circ g)\).

In Sect. 3.1, we approximate by dropping the indices, leaving a standard mutual inductive definition called the presyntax. This is the same as the presyntax given in Fig. 2.

In Sect. 3.2, we define goodness algebras, collections of predicates over the presyntax which define the index relationship (analogously to \({A}_\text {good}\) and \({B}_\text {good}\) from Sect. 2). We also show that a goodness algebra exists, and call it \(\mathbb {O}\).

In Sect. 3.3, we define a predicate nice on goodness algebras, such that if we have a nice goodness algebra, then we can construct the simple elimination rules. Being nice is similar to having proofs of goodness be unique as in Sect. 2.

In Sect. 3.4, we use pattern matching over the presyntax to define a function S from goodness algebras to goodness algebras.
 In Sect. 3.5, we define the limit of the sequenceand show that it is nice. This is the only section that utilizes the differences between cubical type theory and standard intentional type theory.$$\begin{aligned} \mathbb {O}, S\;\mathbb {O}, S\;(S\;\mathbb {O}), \dots , S^n\;\mathbb {O},\dots \end{aligned}$$
Given the nice goodness algebra in Sect. 3.5 we can then construct the simple elimination rules by Sect. 3.3. This construction has been formalized in Agda^{2} using the –cubical flag which implies –withoutK (see RunningExample.agda).
The intuition for our construction is that the Nordvall Forsberg’s approach of pairing an approximation with goodness predicates can be repeated, and each time the approximation gets better. Using HoTT terminology, we showed in Sect. 2 that one iteration suffices only if X has homotopy level 0 (is a homotopy set, satisfies UIP). In general, \(n+1\) iterations are sufficient if only if X has homotopy level n. The successor goodness algebra defined in Sect. 3.4 is a slightly simplified version of Nordvall Forsberg’s construction, and taking the limit (in Sect. 3.5) gives a construction which works for arbitrary homotopy levels.
3.1 Presyntax
The presyntax is the same as that used in Sect. 2, defined as a mutually inductive type in Fig. 2. The constructors of the presyntax have the same types as the constructors of the full inductiveinductive definition (given in Fig. 1), except we replace \(B\;a\) with \({B}_\text {pre}\) everywhere, ignoring the dependence of B on A.
We will use definitions by induction and by patternmatching on the presyntax in sections Sects. 3.3 and 3.4 respectively.
3.2 Goodness Algebras
 1.
For each sort X a type \(\text {Ix}\,X\) giving the data X depends on, and for each operation F constructing an element of sort X, a family \(\text {Arg}\,F : Y \rightarrow \text {Ix}\,X \rightarrow \,\text {Type}\) where Y collects the arguments of the operation in the presyntax, where \(\text {Arg}\,F\;y\;\phi \) gives the data needed to justify that presyntax constructed by \({F}_\text {pre}\) from y has index \(\phi \). In later sections we will also write \(\text {Ix}\,X\;\delta ^G\) and \(\text {Arg}\,F\;\delta ^G\) to specify which goodness algebra we are working in.
 2.
The type of the corresponding component in the goodness algebra. For sorts, this is a predicate relating \(\text {Ix}\) and the presyntax, while for the operations, this is a function witnessing that each element of \(\text {Arg}\) gives a goodness proof relating the index \(\phi \) to the presyntax.
 3.
A way to combine the goodness algebra with the presyntax to form an inductiveinductive object. For sorts, we pair the presyntax with a goodness proof, while for operations we apply the operation given by the goodness algebra, mimicking the construction in Fig. 3.
3.3 Niceness
In this section, we identify a property niceness that is sufficient for a goodness algebra to produce an inductiveinductive object \((A, B, \eta , \texttt {ext}, \texttt {inj})\) which satisfies the simple elimination rules, as given in Fig. 1.
To define niceness, we use the concept of equivalence, as defined in Univalent Foundations Program [18] (§4.4 Contractible fibers). Given a function \(f : A \rightarrow B\), we write \(\text {isEquiv}\,f\) (leaving A and B implicit) to denote that f is an equivalence between A and B. We will also write \(A \simeq B\) for the type of pairs of a function f with a proof that f is an equivalence.
Lemma 6
(Nice goodness algebras give simple elimination rules). Given a goodness algebra \(\delta ^G\) with proof of niceness \(\delta ^N\), the inductiveinductive object \((A, B, \eta , \texttt {ext}, \texttt {inj})\) produced from \(\delta ^G\) as specified in Sect. 3.2 satisfies the simple induction rules given in Fig. 1.
Proof
The proof is formalized in RunningExample.agda. The main idea of the proof is to induct on the presyntax, and exploit the equivalences provided by niceness \(\delta ^N\). In the \(\texttt {inj}\) case for example, we have a proof of \(\delta ^G.B\;\phi \;({\texttt {inj}}_\text {pre}\;a)\). But without loss of generality, we can replace that goodness proof with \(\delta ^G.\texttt {inj}\) applied to an element of \(\text {Arg}\,\texttt {inj}\;a\;\phi \), which contains both a proof \({a}_\text {good} : \delta ^G.A\;\star \;a\) and a proof that \({(a, {a}_\text {good})}\equiv {\phi }\). Using J to eliminate that equality leaves a goal to which the provided simple induction step for \(\texttt {inj}\) applies. This proof does not use cubical type theory in any essential way.
3.4 Successor Goodness Algebra
We are trying to create a nice goodness algebra by taking the limit of successive approximations, so we need a step function, which we will call S, that takes a goodness algebra \(\delta ^G\) and returns a new goodness algebra \(S\;\delta ^G\), which is closer in some sense to being nice. We do so by pattern matching on the presyntax to unroll one level of the recurrence equations niceness encodes.
The projection functions \((\delta ^\pi \;\delta ^G)\) consist of applying the map given by the second component of \((E\;\delta ^G)\) everywhere in sight. The sorts are then defined by the first component of \((E\;\delta ^G)\), while the operations can be defined to be the corresponding projection function itself.
Concretely, for the sort B, we define \((\delta ^\pi \;\delta ^G).B\) to map between \(\text {Ix}\,B\; (S \delta ^G)\) and \(\text {Ix}\,B\;\delta ^G\). This consists of applying the function \(((E\;\delta ^G).A\;{a}_\text {pre}\;\star \;.2)\) which we defined by pattern matching above to \({a}_\text {good}\). Then, since \((S\;\delta ^G).B\) gets an inductive index \(\phi \) in \((S\;\delta ^G)\) but \(((E\;\delta ^G)\;b\;\phi \;.1)\) is expecting an inductive index in \(\delta ^G\), we span the gap with the projection function \((\delta ^\pi \;\delta ^G).B\) just defined. The definition of A follows the same pattern, but \((\delta ^\pi \;\delta ^G).A\) is even simpler because \(\text {Ix}\,A\;\delta ^G = \top \) regardless of what goodness algebra we are working in.
3.5 Limit of Goodness Algebras
We will now construct a nice goodness algebra by taking the limit of the sequence \(S^n\;\mathbb {O}\) and showing that it is nice, where \(S^n\;\mathbb {O}\) is defined by recursion on n with \(S^0\mathbb {O}= \mathbb {O}, S^{1+n}\mathbb {O}= S(S^n\;\mathbb {O})\). But first, we consider the limit of a chain of types.
Limit of Types. This subsection Limit of Types is formalized in Chain.agda.
This construction commutes with most type formers: dependent function types, dependent pair types, identity types, and constants. We also note a dependent version of the fact that the limit of a chain is equivalent to the limit of a shifted chain to substitute for Ahrens et al. [1, Lemma 12].
Lemma 7
We only use Lemma 7 when \(Y_1\;n\;(\pi _X\;n\;x) = Y_0\;(1+n)\;x\), so we may take g to be the identity, leaving \(t^+\) the shifted chain of t up to X arguments.
Limit of Goodness Algebras. Now we use the lemmas about chains to construct a nice goodness algebra, and then conclude by constructing an inductiveinductive object \((A,B,\eta ,\texttt {ext},\texttt {inj})\) that satisfies the simple elimination rules.
Lemma 8
A nice goodness algebra exists.
Proof
The sorts of the limit goodness algebra are defined as a chain, and operations act pointwise on each component of the chain. To prove that the operations are equivalences, we compose a proof that \(\text {Arg}\) commutes with chains (given by combining the lemmas about chains commuting with type formers) with a proof that for each sort, the chain given by the \((E\;(S^n\;\mathbb {O}))\) is equivalent to the chain given by \((S^n\;\mathbb {O})\) (given by Lemma 7). Since \((E\;(S^n\;\mathbb {O}))\) is defined by pattern matching to reduce to \(\text {Arg}\), the right and left sides of these equivalences agree, and we find that the operations are indeed nice. See the formalization for details.
Theorem 2
There exists an inductiveinductive object \((A, B, \eta , \texttt {ext}, \texttt {inj})\) that satisfies the simple elimination rules as defined in Fig. 1.
Proof
A nice goodness algebra exists by Lemma 8, therefore we can construct \((A, B, \eta , \texttt {ext}, \texttt {inj})\) satisfying the simple elimination rules by Lemma 6.
We have therefore succeeded. In cubical type theory, the inductiveinductive definition from Fig. 1 is constructible.
4 Related Work
The principle of simultaneously defining a type and a family over that type has been used many times before. Danielsson [9] used an inductiveinductiverecursive definition to define the syntax of dependent type theory, and Chapman [5] used an inductiveinductive definition for the same purpose. Conway’s surreal numbers [7] are given (up to a defined equivalence relation) by the inductiveinductive definition of number and less than, where less than is a relation indexed by two numbers [15, §7.1]. The HoTT book §11.3 gives a definition of the Cauchy reals as a higher inductiveinductive definition [18].
In his thesis and previous papers [15, 16, 17], Nordvall Forsberg studies the general theory of inductiveinductive types, axiomatizing a limited class of such definitions, and giving a set theoretic model showing that they are consistent. He also considers various extensions such as allowing a third type indexed by the first two, allowing the second type to be indexed by two elements of the first, or combining inductiveinductive definitions with inductiverecursive definitions from Dybjer and Setzer [10].
There have been several attempts to define a general class of inductiveinductive types larger than that in Nordvall Forsberg’s thesis. Kaposi and Kovács [14] gives an external syntactic description of a class which includes higher inductiveinductive types, and Altenkirch et al. [2] gives a semantic description of a class including quotient inductiveinductive types, but neither gives a type of codes that can be reasoned about internally. Working with UIP, Altenkirch et al. [4] propose a class of quotient inductiveinductive types.
Nordvall Forsberg’s thesis [15] appears to give the best previously known reduction of inductiveinductive types to regular inductive types known. As we have shown, Nordvall Forsberg’s approach can only be applied to intensional type theory if UIP holds. Furthermore, the equations for both Nordvall Forsberg’s approach and our approach only hold propositionally.
Many other structures have been reduced to plain inductive types. Our construction of inductiveinductive types can be seen as an adaptation of the technique in Ahrens et al. [1], where coinductive types are constructed from \(\mathbb {N}\) by taking a limit. Indexed inductive types (which are used in Nordvall Forsberg’s construction) are constructed from plain inductive types in Altenkirch et al. [3], with good computational properties (provided an identity type that satisfies J strictly). And small inductionrecursion is reduced to plain indexed inductive types in Hancock et al. [11].
5 Conclusions and Future Work
 1.
Shown that the construction of inductiveinductive types given by Nordvall Forsberg implies UIP.
 2.
Given an alternative construction of one particular inductiveinductive type in cubical type theory, which is compatible with Homotopy Type Theory.
We claim that the construction of our specific running example is straightforwardly generalizable to other inductiveinductive types, and have formalized the construction of a number of other examples including types with nonfinitary constructors and indices to support this claim (see the GitHub repository referenced in the introduction).

An internal definition of inductiveinductive specifications in HoTT. Early experiments suggest that this requires surmounting difficulties related to increasingly complex coherence conditions similar to those encountered when defining semisimplicial sets, c.f. Herbelin [12].

Extending the proof given here to construct the general elimination rules. The general elimination rules were defined in Nordvall Forsberg [15], but that formulation they relies on either strict computation rules or extensional type theory to be well typed. Kaposi and Kovács [14] give equivalent rules which are well typed in intensional type theory.

Identifying what needs to be added for the simple elimination rules to have the expected computational behavior. Given the similar construction method, this hopefully also allows the construction of coinductive types with nice computational behavior, c.f. Ahrens et al. [1].

In the opposite direction from the previous point, rewriting the construction given here in Coq + Function Extensionality. While the elimination rules will have poor computational behavior, this would make using inductiveinductive types in Coq possible without requiring any change to Coq itself, while being compatible with HoTT. In particular, using cubical type theory makes the proofs in Sect. 3.5 simpler, but we speculate that axiomatic function extensionality is sufficient.
Footnotes
 1.
The formalization can be found at https://github.com/jashug/ConstructingII.
 2.
Agda version 2.6.0 commit bd338484d.
Notes
Acknowledgements
I would like to thank Talia Ringer and Dan Grossman from the UW PLSE lab, for their invaluable feedback throughout the revision process. I also thank Pavel Panchekha, John Leo, Remy Wang, and Fredrik Nordvall Forsberg for their comments.
Some of this work was completed while studying at Tokyo Institute of Technology under Professor Ryo Kashima. I would like to thank Professor Kashima, as well as my fellow lab members and mentors Asami and Maniwa for making my stay both productive and enjoyable.
References
 1.Ahrens, B., Capriotti, P., Spadotti, R.: Nonwellfounded trees in homotopy type theory. In: TLCA (2015)Google Scholar
 2.Altenkirch, T., Capriotti, P., Dijkstra, G., Kraus, N., Nordvall Forsberg, F.: Quotient inductiveinductive types. In: Baier, C., Dal Lago, U. (eds.) FoSSaCS 2018. LNCS, vol. 10803, pp. 293–310. Springer, Cham (2018). https://doi.org/10.1007/9783319893662_16CrossRefGoogle Scholar
 3.Altenkirch, T., Ghani, N., Hancock, P., McBride, C., Morris, P.: Indexed containers. J. Funct. Program. 25, e5 (2015)MathSciNetCrossRefGoogle Scholar
 4.Altenkirch, T., Kaposi, A., Kovács, A.: Constructing quotient inductiveinductive types. Proc. ACM Program. Lang. 3(POPL), 2:1–2:24 (2019). http://doi.acm.org/10.1145/3290315Google Scholar
 5.Chapman, J.: Type theory should eat itself. Electron. Notes Theoret. Comput. Sci. 228, 21–36 (2009). http://www.sciencedirect.com/science/article/pii/S157106610800577X. Proceedings of the International Workshop on Logical Frameworks and Metalanguages: Theory and Practice (LFMTP 2008)
 6.Cohen, C., Coquand, T., Huber, S., Mörtberg, A.: Cubical Type Theory: a constructive interpretation of the univalence axiom. IfCoLog J. Logics Appl. 4(10), 3127–3169 (2017). http://www.collegepublications.co.uk/journals/ifcolog/?00019zbMATHGoogle Scholar
 7.Conway, J.: On Numbers and Games. AK Peters Series. Taylor & Francis, Milton Park (2000). https://books.google.com/books?id=tXiVo8qA5PQCCrossRefGoogle Scholar
 8.The Coq Development Team: The Coq proof assistant, version 8.8.0, April 2018. https://doi.org/10.5281/zenodo.1219885
 9.Danielsson, N.A.: A formalisation of a dependently typed language as an inductiverecursive family. In: Altenkirch, T., McBride, C. (eds.) TYPES 2006. LNCS, vol. 4502, pp. 93–109. Springer, Heidelberg (2007). https://doi.org/10.1007/9783540744641_7CrossRefGoogle Scholar
 10.Dybjer, P., Setzer, A.: A finite axiomatization of inductiverecursive definitions. In: Girard, J.Y. (ed.) TLCA 1999. LNCS, vol. 1581, pp. 129–146. Springer, Heidelberg (1999). https://doi.org/10.1007/3540489592_11CrossRefGoogle Scholar
 11.Hancock, P., McBride, C., Ghani, N., Malatesta, L., Altenkirch, T.: Small induction recursion. In: Hasegawa, M. (ed.) TLCA 2013. LNCS, vol. 7941, pp. 156–172. Springer, Heidelberg (2013). https://doi.org/10.1007/9783642389467_13CrossRefGoogle Scholar
 12.Herbelin, H.: A dependentlytyped construction of semisimplicial types. Math. Struct. Comput. Sci. 25(5), 1116–1131 (2015)MathSciNetCrossRefGoogle Scholar
 13.Hofmann, M., Streicher, T.: The groupoid interpretation of type theory. In: TwentyFive Years of Constructive Type Theory (Venice, 1995), Oxford Logic Guides, vol. 36, pp. 83–111. Oxford University Press, New York (1998)Google Scholar
 14.Kaposi, A., Kovács, A.: A syntax for higher inductiveinductive types. In: Kirchner, H. (ed.) 3rd International Conference on Formal Structures for Computation and Deduction, FSCD 2018. Leibniz International Proceedings in Informatics, LIPIcs, vol. 108, pp. 20:1–20:18. Schloss Dagstuhl–LeibnizZentrum fuer Informatik, Dagstuhl, Germany (2018). http://drops.dagstuhl.de/opus/volltexte/2018/9190
 15.Nordvall Forsberg, F.: Inductiveinductive definitions. Ph.D. thesis. Swansea University (2013)Google Scholar
 16.Nordvall Forsberg, F., Setzer, A.: Inductiveinductive definitions. In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 454–468. Springer, Heidelberg (2010). https://doi.org/10.1007/9783642152054_35CrossRefGoogle Scholar
 17.Nordvall Forsberg, F., Setzer, A.: A finite axiomatisation of inductiveinductive definitions. In: Berger, U., Hannes, D., Schuster, P., Seisenberger, M. (eds.) Logic, Construction, Computation, Ontos Mathematical Logic, vol. 3, pp. 259–287. Ontos Verlag (2012)Google Scholar
 18.The Univalent Foundations Program: Homotopy Type Theory: Univalent Foundations of Mathematics, Institute for Advanced Study (2013). https://homotopytypetheory.org/book
 19.Vezzosi, A.: Adding cubes to agda (2017). https://hottuf.github.io/2017/abstracts/cubicalagda.pdf
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