Categorical gluing is a powerful technique for proving meta-theorems of type theories such as canonicity and normalization. Synthetic Tait Computability (STC) provides an abstract treatment of the complex gluing models by internalizing the gluing category into a modal dependent type theory with a phase distinction. This work presents a mechanization of STC in the Istari proof assistant. Istari is a Martin-Löf-style extensional type theory with equality reflection, which avoids much of the explicit transport reasoning typically found in intensional proof assistants. This work develops a reusable library for synthetic phase distinction, including modalities, extension types, and strict glue types, and applies it to two case studies: (1) a canonicity model for dependent type theory with dependent products and booleans with large elimination, and (2) a Kripke canonicity model for the cost-aware logical framework. Our results demonstrate that the core STC constructions can be formalized essentially verbatim in Istari, preserving the elegance of the on-paper arguments while ensuring machine-checked correctness.

Many proof assistants allow the use of features and axioms that increase their expressive power. However, these extensions must be used with care, as some combinations are known to lead to logical inconsistencies. Therefore, proof assistants include mechanisms that track which extensions are used in a proof development or module, ensuring that incompatible extensions are not used simultaneously. Unfortunately, existing extension tracking mechanisms are external to the type system. This means that we cannot specify precisely which extensions a definition depends on. Having the ability to write more precise specifications means we are not picking an overapproximation of the extensions needed, which prevents reusing definitions in the presence of incompatible extensions. Furthermore, we cannot refer to definitions that use incompatible extensions even if they are never used in inconsistent ways. The reasoning principles of one extension therefore cannot be used as a metatheory to reason about the properties of an incompatible extension. In this report, I explore the use of the Dependent Calculus of Indistinguishability (DCOI) by Liu et al. for extension tracking. DCOI is a dependent type system with dependency tracking, where terms and variables are assigned dependency levels alongside their types. These dependency levels form a lattice that describes which levels are permitted to access what. To instead track extensions, each set of extensions would correspond to a dependency level, and the lattice would describe how extensions are permitted to interact.

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 propose an enhancement to inductive types and records in a dependent type theory, namely (co)conditions. With a primitive interval type, conditions generalize the cubical syntax of higher inductive types in homotopy type theory, while coconditions generalize the cubical path type. (Co)conditions are also useful without an interval type. The duality between conditions and coconditions is presented in an interesting way: The elimination principles of inductive types with conditions can be internalized with records with coconditions and vice versa. However, we do not develop the metatheory of conditions and coconditions in this paper. Instead, we only present the type checking.

We present a domain-specific type theory for constructions and proofs in category theory. The type theory axiomatizes notions of category, functor, profunctor and a generalized form of natural transformations. The type theory imposes an ordered linear restriction on standard predicate logic, which guarantees that all functions between categories are functorial, all relations are profunctorial, and all transformations are natural by construction, with no separate proofs necessary. Important category-theoretic proofs such as the Yoneda lemma and Co-yoneda lemma become simple type-theoretic proofs about the relationship between unit, tensor and (ordered) function types, and can be seen to be ordered refinements of theorems in predicate logic. The type theory is sound and complete for a categorical model in virtual equipments, which model both internal and enriched category theory. While the proofs in our type theory look like standard set-based arguments, the syntactic discipline ensure that all proofs and constructions carry over to enriched and internal settings as well.

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.

The implementation and semantics of dependent type theories can be studied in a syntax-independent way: the objective metatheory of dependent type theories exploits the universal properties of their syntactic categories to endow them with computational content, mathematical meaning, and practical implementation (normalization, type checking, elaboration). The semantic methods of the objective metatheory inform the design and implementation of correct-by-construction elaboration algorithms, promising a principled interface between real proof assistants and ideal mathematics. In this dissertation, I add synthetic Tait computability to the arsenal of the objective metatheorist. Synthetic Tait computability is a mathematical machine to reduce difficult problems of type theory and programming languages to trivial theorems of topos theory. First employed by Sterling and Harper to reconstruct the theory of program modules and their phase separated parametricity, synthetic Tait computability is deployed here to resolve the last major open question in the syntactic metatheory of cubical type theory: normalization of open terms.

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.

Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types. This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of higher inductive types. These new primitives make function and propositional extensionality as well as quotient types directly definable with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. This extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.

We prove the conjecture that any Grothendieck $(\infty ,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning internally to $(\infty ,1)$-toposes, just as higher-order logic is used for 1-toposes. As part of the proof, we give a new, more explicit, characterization of the fibrations in injective model structures on presheaf categories. In particular, we show that they generalize the coflexible algebras of 2-monad theory.

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.