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.

The correctness of type-checking implementations for proof assistants based on dependent type theory relies on metatheoretical properties that ensure the decidability of typing, some of which require substantial logical strength. Recent mechanizations of such algorithms have highlighted the importance of separating the algorithmic components of the proof - often intricate but requiring relatively low logical strength - from the logical components, which depend on stronger metatheoretical properties, such as normalization or the injectivity of type constructors. In this work, we revisit the logical relations technique and show how it can be used to derive these metatheoretical properties in a direct and uniform way for a core dependent type theory featuring Π-types, N, ⊥ and a universe U. Our presentation yields a compact and conceptually simplified argument that isolates the logically strong reasoning from the algorithmic core. We argue that this approach scales smoothly to richer type theories, and demonstrate this by extending our construction to Exceptional Type Theory (ExcTT), obtaining the first mechanized canonicity proof for this theory.

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.

Parametricity is a property of the syntax of type theory implying, e.g., that there is only one function having the type of the polymorphic identity function. Parametricity is usually proven externally, and does not hold internally. Internalising it is difficult because once there is a term witnessing parametricity, it also has to be parametric itself and this results in the appearance of higher dimensional cubes. In previous theories with internal parametricity, either an explicit syntax for higher cubes is present or the theory is extended with a new sort for the interval. In this paper we present a type theory with internal parametricity which is a simple extension of Martin-Löf type theory: there are a few new type formers, term formers and equations. Geometry is not explicit in this syntax, but emergent: the new operations and equations only refer to objects up to dimension 3. We show that this theory is modelled by presheaves over the BCH cube category. Fibrancy conditions are not needed because we use span-based rather than relational parametricity. We define a gluing model for this theory implying that external parametricity and canonicity hold. The theory can be seen as a special case of a new kind of modal type theory, and it is the simplest setting in which the computational properties of higher observational type theory can be demonstrated.