The metatheory of dependent types has seen a lot of progress in recent years. In particular, the development of categorical gluing finally lets us work with semantic presentations of type theory (such as categories with families) to establish fundamental properties of type theory such as canonicity and normalisation. However, proofs by gluing have yet to reach the stage of computer formalisation: formal proofs for the metatheory of dependent types are still stuck in the age of tedious syntactic proofs. The main reason for this is that semantic presentations of type theory are defined using sophisticated indexed inductive types, which are prone to “transport hell”. In this paper, we introduce a new technique to work with CwFs in intensional type theory without getting stuck in transport hell. More specifically, we construct an alternative presentation of the initial CwF which encodes the substitutions as metatheoretical functions. This has the effect of strictifying all the equations that are involved in the substitution calculus, which greatly reduces the need for transports. As an application, we use our strictified initial CwF to give a short and elegant proof of canonicity for a type theory with dependent products and booleans with large elimination. The resulting proof is fully formalised in Agda.

Some advantages of Cubical Type Theory, as implemented by Cubical Agda, over intensional Martin-Löf Type Theory include Quotient Inductive Types (QITs), which exist as instances of Higher Inductive Types, and functional extensionality, which is provable in Cubical Type Theory. However, HoTT features an infinite hierarchy of equalities that may become unwieldy in formalisations. Fortunately, QITs and functional extensionality are both preserved even if the equality levels of Cubical Type Theory are truncated to only homotopical Sets (h-Sets). In other words, removing the univalence axiom from Cubical Type Theory and instead postulating a conflicting axiom: the Uniqueness of Identity Proofs (UIP) postulate. Since univalence is proved in Cubical Type Theory from the so-called Glue Types, therefore, it is known that one can first remove the Glue Types (thus removing univalence) and then set-truncate all equalities (essentially assuming UIP), à la XTT. The result is a “h-Set Cubical Type Theory” that retains features such as functional extensionality and QITs.

However, in Cubical Agda, there are currently only two unsatisfying ways to achieve h-Set Cubical Type Theory. The first is to give up on the canonicity of the theory and simply postulate the UIP axiom, while the second way is to use a standard result stating “type formers preserve h-levels” to manually prove UIP for every defined type. The latter is, however, laborious work best suited for an automatic implementation by the proof assistant. In this project, we analyse formulations of UIP and detail their computation rules for Cubical Agda, and evaluate their suitability for implementation. We also implement a variant of Cubical Agda without Glue, which is already compatible with postulated UIP, in anticipation of a future implementation of UIP in Cubical Agda.

We present XTT, a version of Cartesian cubical type theory specialized for Bishop sets à la Coquand, in which every type enjoys a definitional version of the uniqueness of identity proofs. Using cubical notions, XTT reconstructs many of the ideas underlying Observational Type Theory, a version of intensional type theory that supports function extensionality. We prove the canonicity property of XTT (that every closed boolean is definitionally equal to a constant) using Artin gluing.

Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the “synthetic” development of homotopy theory within type theory, as well as in formalising ordinary set-level mathematics in type theory. In this paper, we construct models of a wide range of higher inductive types in a fairly wide range of settings. We introduce the notion of cell monad with parameters : a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category has weakly stable typal initial algebras for any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specialises to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction and general localisations. Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.

Higher inductive types (HITs) in Homotopy Type Theory allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types, and allow to define types with non-trivial higher equality types, such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define types satisfying uniqueness of equality proofs, such as the Cauchy reals, the partiality monad, and the well-typed syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs). Although there has been recent progress on a general theory of HITs, there is not yet a theoretical foundation for the combination of equality constructors and induction-induction, despite many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics. We further derive a section induction principle, stating that every algebra morphism into the algebra in question has a section, which is close to the intuitively expected elimination rules.