Over the past two decades, numerous systems have brought some of the benefits of dependent typing to a wide variety of new programming languages, often by restricting which terms can appear inside types. Such techniques are known as refinement types, occurrence typing, liquid types, and path dependent types, among others. However, the restrictions adopted by these systems often break the substitution property, because they explicitly disallow the ability to substitute arbitrary terms for variables inside types. This leads to significant complexity in the design and metatheory of these systems, increasing the possibility of significant errors. We consider a specific line of work on occurrence typing, namely, the calculus underlying Typed Racket due to Tobin-Hochstadt and Felleisen 2010. We show that the fundamental challenge of substitution into types resulted in multiple flaws in the formalism and the syntactic type soundness theorem of this work. These flaws are replicated in several other papers building on this work, and also surface as a soundness bug in Typed Racket itself. We identify and repair these problems, revising the core calculus of Typed Racket and giving a semantic type soundness proof using step-indexed logical relations, formalized in Lean. We argue that this approach is simpler than it may seem, and easily scales to handle the complexity of the occurrence typing in Typed Racket.

We give a new syntax-independent account of finitely-presented generalized algebraic theories (GATs) as finite cell complexes in the category of categories with families (CwFs), in which GATs are constructed by successive pushouts along the CwF morphisms generically postulating a sort, an operation, or an equation. Inspired by the fat small object argument of Makkai, Rosický, and Vokřínek, we introduce fat GAT presentations, thereby allowing infinite presentations with non-linear dependency structure. Then, motivated by wanting our GATs to self-describe, we extend presentations to admit infinitary arities, including infinitely deep dependency chains. Finally, we verify that these generalized GATs satisfy expected semantic properties including Frey’s Gabriel–Ulmer duality.

We present a dependently-typed cross-linguistic framework for analyzing the telicity and culminativity of events, accompanied by examples of using our framework to model English sentences. Our framework consists of two parts. In the nominal domain, we model the boundedness of noun phrases and its relationship to subtyping, delimited quantities, and adjectival modification. In the verbal domain, we define a dependent event calculus, modeling telic events as those whose undergoer is bounded, culminating events as telic events that achieve their inherent endpoint, and consider adverbial modification. In both domains, we pay particular attention to associated entailments. Our framework is defined as an extension of intensional Martin-Löf dependent type theory, and the rules and examples in this paper have been formalized in the Agda proof assistant.

We present a new way to control the unfolding of definitions in dependent type theory. Traditionally, proof assistants require users to fix whether each definition will or will not be unfolded in the remainder of a development; unfolding definitions is often necessary in order to reason about them, but an excess of unfolding can result in brittle proofs and intractably large proof goals. In our system, definitions are by default not unfolded, but users can selectively unfold them in a local manner. We justify our mechanism by means of elaboration to a core theory with extension types – a connective first introduced in the context of homotopy type theory – and by establishing a normalization theorem for our core calculus. We have implemented controlled unfolding in the proof assistant, inspiring an independent implementation in Agda.

We present a novel formulation of universe polymorphism in dependent type theory in terms of monads on the category of strict partial orders, and a novel algebraic structure, displacement algebras, on top of which one can implement a generalized form of McBride’s “crude but effective stratification” scheme for lightweight universe polymorphism. We give some examples of exotic but consistent universe hierarchies, and prove that every universe hierarchy in our sense can be embedded in a displacement algebra and hence implemented via our generalization of McBride’s scheme. Many of our technical results are mechanized in Agda, and we have an OCaml library for universe levels based on displacement algebras, for use in proof assistant implementations.

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.

We present a cubical type theory based on the Cartesian cube category (faces, degeneracies, symmetries, diagonals, but no connections or reversal) with univalent universes, each containing Π, Σ, path, identity, natural number, boolean, suspension, and glue (equivalence extension) types. The type theory includes a syntactic description of a uniform Kan operation, along with judgmental equality rules defining the Kan operation on each type. The Kan operation uses both a different set of generating trivial cofibrations and a different set of generating cofibrations than the Cohen, Coquand, Huber, and Mörtberg (CCHM) model. Next, we describe a constructive model of this type theory in Cartesian cubical sets. We give a mechanized proof, using Agda as the internal language of cubical sets in the style introduced by Orton and Pitts, that glue, Π, Σ, path, identity, boolean, natural number, suspension types, and the universe itself are Kan in this model, and that the universe is univalent. An advantage of this formal approach is that our construction can also be interpreted in a range of other models, including cubical sets on the connections cube category and the De Morgan cube category, as used in the CCHM model, and bicubical sets, as used in directed type theory.

In their usual form, representation independence metatheorems provide an external guarantee that two implementations of an abstract interface are interchangeable when they are related by an operation-preserving correspondence. If our programming language is dependently-typed, however, we would like to appeal to such invariance results within the language itself, in order to obtain correctness theorems for complex implementations by transferring them from simpler, related implementations. Recent work in proof assistants has shown that Voevodsky’s univalence principle allows transferring theorems between isomorphic types, but many instances of representation independence in programming involve non-isomorphic representations. In this paper, we develop techniques for establishing internal relational representation independence results in dependent type theory, by using higher inductive types to simultaneously quotient two related implementation types by a heterogeneous correspondence between them. The correspondence becomes an isomorphism between the quotiented types, thereby allowing us to obtain an equality of implementations by univalence. We illustrate our techniques by considering applications to matrices, queues, and finite multisets. Our results are all formalized in Cubical Agda, a recent extension of Agda which supports univalence and higher inductive types in a computationally well-behaved way.

We prove normalization for (univalent, Cartesian) cubical type theory, closing the last major open problem in the syntactic metatheory of cubical type theory. Our normalization result is reduction-free, in the sense of yielding a bijection between equivalence classes of terms in context and a tractable language of β/η-normal forms. As corollaries we obtain both decidability of judgmental equality and the injectivity of type constructors.

We contribute XTT, a cubical reconstruction of Observational Type Theory [Altenkirch et al., 2007] which extends Martin-Löf’s intensional type theory with a dependent equality type that enjoys function extensionality and a judgmental version of the unicity of identity proofs principle (UIP): any two elements of the same equality type are judgmentally equal. Moreover, we conjecture that the typing relation can be decided in a practical way. In this paper, we establish an algebraic canonicity theorem using a novel extension of the logical families or categorical gluing argument inspired by Coquand and Shulman [Coquand, 2018; Shulman, 2015]: every closed element of boolean type is derivably equal to either true or false.

RedPRL is an experimental proof assistant based on Cartesian cubical computational type theory, a new type theory for higher-dimensional constructions inspired by homotopy type theory. In the style of Nuprl, RedPRL users employ tactics to establish behavioral properties of cubical functional programs embodying the constructive content of proofs. Notably, RedPRL implements a two-level type theory, allowing an extensional, proof-irrelevant notion of exact equality to coexist with a higher-dimensional proof-relevant notion of paths.

We present a dependent type theory organized around a Cartesian notion of cubes (with faces, degeneracies, and diagonals), supporting both fibrant and non-fibrant types. The fibrant fragment validates Voevodsky’s univalence axiom and includes a circle type, while the non-fibrant fragment includes exact (strict) equality types satisfying equality reflection. Our type theory is defined by a semantics in cubical partial equivalence relations, and is the first two-level type theory to satisfy the canonicity property: all closed terms of boolean type evaluate to either true or false.

Homotopy type theory is an extension of Martin-Löf type theory, based on a correspondence with homotopy theory and higher category theory. In homotopy type theory, the propositional equality type is proof-relevant, and corresponds to paths in a space. This allows for a new class of datatypes, called higher inductive types, which are specified by constructors not only for points but also for paths. In this paper, we consider a programming application of higher inductive types. Version control systems such as Darcs are based on the notion of patches—syntactic representations of edits to a repository. We show how patch theory can be developed in homotopy type theory. Our formulation separates formal theories of patches from their interpretation as edits to repositories. A patch theory is presented as a higher inductive type. Models of a patch theory are given by maps out of that type, which, being functors, automatically preserve the structure of patches. Several standard tools of homotopy theory come into play, demonstrating the use of these methods in a practical programming context.