We present an intrinsic representation of type theory in the proof assistant Cubical Agda, inspired by Awodey’s natural models of type theory. The initial natural model is defined as quotient inductive-inductive-recursive types, leading us to a syntax accepted by Cubical Agda without using any transports, postulates, or custom rewrite rules. We formalise some meta-properties such as the standard model, normalisation by evaluation for typed terms, and strictification constructions. Since our formalisation is carried out using Cubical Agda’s native support for quotient inductive types, all our constructions compute at a reasonable speed. When we try to develop more sophisticated metatheory, however, the ‘transport hell’ problem reappears. Ultimately, it remains a considerable struggle to develop the metatheory of type theory using an intrinsic representation that lacks strict equations. The effort required is about the same whether or not the notion of natural model is used.

Domain theory has been developed as a mathematical theory of computation and to give a denotational semantics to programming languages. It helps us to fix the meaning of language concepts, to understand how programs behave and to reason about programs. At the same time it serves as a great theory to model various algebraic effects such as non-determinism, partial functions, side effects and numerous other forms of computation. In the present paper, we present a general framework to construct algebraic effects in domain theory, where our domains are DCPOs: directed complete partial orders. We first describe so called DCPO algebras for a signature, where the signature specifies the operations on the DCPO and the inequational theory they obey. This provides a method to represent various algebraic effects, like partiality. We then show that initial DCPO algebras exist by defining them as so called Quotient Inductive-Inductive Types (QIITs), known from homotopy type theory. A quotient inductive-inductive type allows one to simultaneously define an inductive type and an inductive relation on that type, together with equations on the type. We illustrate our approach by showing that several well-known constructions of DCPOs fit our framework: coalesced sums, smash products and free DCPOs (partiality and power domains). Our work makes use of various features of homotopy type theory and is formalized in Cubical Agda.

We present a new design for an algebraic simplification library structured around concepts from universal algebra: theories, models, homomorphisms, and universal properties of free algebras and free extensions of algebras. The library’s dependently typed interface guarantees that both built-in and user-defined simplification modules are terminating, sound, and complete with respect to a well-specified class of equations. We have implemented the design in the Idris 2 and Agda dependently typed programming languages and shown that it supports modular extension to new theories, proof extraction and certification, goal extraction via reflection, and interactive development.

The conventional general syntax of indexed families in dependent type theories follow the style of “constructors returning a special case”, as in Agda, Lean, Idris, Coq, and probably many other systems. Fording is a method to encode indexed families of this style with index-free inductive types and an identity type. There is another trick that merges interleaved higher inductive-inductive types into a single big family of types. It makes use of a small universe as the index to distinguish the original types. In this paper, we show that these two methods can trivialize some very fancy-looking indexed families with higher inductive indices (which we refer to as higher indexed families).

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

We investigate containers and polynomial functors in Quantitative Type Theory, and give initial algebra semantics of inductive data types in the presence of linearity. We show that reasoning by induction is supported, and equivalent to initiality, also in the linear setting.

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.