We construct a realizability model of linear dependent type theory from a linear combinatory algebra. Our model motivates a number of additions to the type theory. In particular, we add a universe with two decoding operations: one takes codes to cartesian types and the other takes codes to linear types. The universe is impredicative in the sense that it is closed under both large cartesian dependent products and large linear dependent products. We also add a rule for injectivity of the modality turning linear terms into cartesian terms. With all of the additions, we are able to encode (linear) inductive types. As a case study, we consider the type of lists over a linear type, and demonstrate that our encoding has the relevant uniqueness principle. The construction of the realizability model is fully formalized in the proof assistant Rocq.

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.

Martin-Löf’s identity types provide a generic (albeit opaque) notion of identification or “equality” between any two elements of the same type, embodied in a canonical reflexive graph structure $\left(\underset{𝐴}{=},\text{𝐫𝐞𝐟𝐥}\right)$ on any type $𝐴$. The miracle of Voevodsky’s univalence principle is that it ensures, for essentially any naturally occurring structure in mathematics, that the resultant notion of identification is equivalent to the type of isomorphisms in the category of such structures. Characterisations of this kind are not automatic and must be established one-by-one; to this end, several authors have employed reflexive graphs and displayed reflexive graphs to organise the characterisation of identity types. We contribute reflexive graph lenses, a new family of intermediate abstractions lying between families of reflexive graphs and displayed reflexive graphs that simplifies the characterisation of identity types for complex structures. Every reflexive graph lens gives rise to a (more complicated) displayed reflexive graph, and our experience suggests that many naturally occurring displayed reflexive graphs arise in this way. Evidence for the utility of reflexive graph lenses is given by means of several case studies, including the theory of reflexive graphs itself as well as that of polynomial type operators. Finally, we exhibit an equivalence between the type of reflexive graph fibrations and the type of univalent reflexive graph lenses.

In impredicative type theory (System F, also known as λ2), it is possible to define inductive data types, such as natural numbers and lists. It is also possible to define coinductive data types such as streams. They work well in the sense that their (co)recursion principles obey the expected computation rules (the β-rules). Unfortunately, they do not yield a (co)induction principle [Herman Geuvers, 2001; Ivar Rummelhoff, 2004], because the necessary uniqueness principles are missing (the η-rules). Awodey, Frey, and Speight [Steve Awodey et al., 2018] used an extension of the Calculus of Constructions [Thierry Coquand and Gérard P. Huet, 1988] (λ C) with Σ-types, identity-types, and functional extensionality to define System F style inductive types with an induction principle, by encoding them as a well-chosen subtype, making them initial algebras. In this paper, we extend their results to coinductive data types, and we detail the example of the stream data type with the desired coinduction principle (also called bisimulation). To do that, we first define quotient types (with the desired η-rules) and we also need a stronger form of the definable existential types. We also show that we can use the original method by Awodey, Frey and Speight for general inductive types by defining W-types with an induction principle. The dual approach for streams can be extended to M-types, the generic notion of coinductive types, and the dual of W-types.

We introduce Displayed Type Theory (dTT) , a multi-modal homotopy type theory with discrete and simplicial modes. In the intended semantics, the discrete mode is interpreted by a model for an arbitrary $\infty$ -topos, while the simplicial mode is interpreted by Reedy fibrant augmented semi-simplicial diagrams in that model. This simplicial structure is represented inside the theory by a primitive notion of display or dependency , guarded by modalities, yielding a partially-internal form of unary parametricity. Using the display primitive, we then give a coinductive definition, at the simplicial mode, of a type of semi-simplicial types. Roughly speaking, a semi-simplicial type consists of a type together with, for each , a displayed semi-simplicial type over . This mimics how simplices can be generated geometrically through repeated cones, and is made possible by the display primitive at the simplicial mode. The discrete part of then yields the usual infinite indexed definition of semi-simplicial types, both semantically and syntactically. Thus, dTT enables working with semi-simplicial types in full semantic generality.