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-L'of-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.

Software development depends on the use of libraries whose public specifications inform client code and impose obligations on private implementations; it follows that verification at scale must also be modular, preserving such abstraction. Hoare’s influential methodology uses abstraction functions to demonstrate the coherence between such concrete implementations and their abstract specifications. However, the Hoare methodology relies on a conventional separation between implementation and specification, providing no linguistic support for ensuring that this convention is obeyed. This paper proposes a synthetic account of Hoare’s methodology within univalent dependent type theory by encoding the data of abstraction functions within types themselves. This is achieved via a phase distinction , which gives rise to a gluing construction that renders an abstraction function as a type and a pair of modalities that fracture a type into its concrete and abstract parts. A noninterference theorem governing the phase distinction characterizes the modularity guarantees provided by the theory. This approach scales to verification of cost, allowing the analysis of client cost relative to a cost-aware specification. A monadic sealing effect facilitates modularity of cost, permitting an implementation to be upper-bounded by its specification in cases where private details influence observable cost. The resulting theory supports modular development of programs and proofs in a manner that hides private details of no concern to clients while permitting precise specifications of both the cost and behavior of programs.

As is evident in the programming language literature, many practitioners favor specifying dynamic program behavior using big-step over small-step semantics. Unlike small-step semantics, which must dwell on every intermediate program state, big-step semantics conveniently jumps directly to the ever-important result of the computation. Big-step semantics also typically involves fewer inference rules than their small-step counterparts. However, in exchange for ergonomics, big-step semantics gives up power: Small-step semantics describes program behaviors that are outside the grasp of big-step semantics, notably divergence. This work presents a little-known extension of big-step semantics with inductive definitions that captures diverging computations without introducing error states. This big-stop semantics is illustrated for typed, untyped, and effectful variants of PCF. Big-stop semantics extends the standard big-step inference rules with a few additional rules to define an evaluation judgment that is equivalent to the reflexive-transitive closure of small-step transitions. This simple extension contrasts with other solutions in the literature that sacrifice ergonomics by introducing many additional inference rules, global state, and/or less-commonly-understood reasoning principles like coinduction. The ergonomics of big-stop semantics is exemplified via concise Agda proofs for some key results and compilation theorems.

Linear Temporal Logic (LTL) is arguably the most popular specification language for formal verification of safety-critical systems. However, LTL formulas can be unintuitive and error- prone for human practitioners to specify and validate. Meanwhile, drawing timelines remains one of the most popular methods for specifying and validating requirements for indus-trial system designs, such as in aerospace operational concepts. Therefore, we provide a new timeline tool for visualizing LTL specifications as timelines, providing provably-correct, intuitive equivalents between these two specification formats. Our tool generates timeline visualizations by translating LTL formulas to intermediate representations as Buchi automata and then regular expressions, and finally simplifying and visualizing the expressions. We provide an algorithm for this visualization, a theoretical soundness analysis” and an implementation.