Guarded Interaction Trees are a structure and a fully formalized framework for representing higher-order computations with higher-order effects in Rocq. We present an extension of Guarded Interaction Trees to support formal reasoning about context-dependent effects. That is, effects whose behaviors depend on the evaluation context, e.g., call/cc, shift and reset. Using and reasoning about such effects is challenging since certain compositionality principles no longer hold in the presence of such effects. For example, the so-called “bind rule” in modern program logics is no longer valid. The goal of our extension is to support representation and reasoning about context-dependent effects in the most painless way possible. To that end, our extension is conservative: the reasoning principles for context-independent effects remain the same. We use it to give direct-style denotational semantics for higher-order programming languages with call/cc and with delimited continuations. We extend the program logic for Guarded Interaction Trees to account for context-dependent effects, and we use the program logic to prove that the denotational semantics is adequate with respect to the operational semantics. Additionally, we retain the ability to combine multiple effects in a modular way, which we demonstrate by showing type soundness for safe interoperability of a programming language with delimited continuations and a programming language with higher-order store. Furthermore, as another contribution, in addition to context-dependent effects, we show how to extend Guarded Interaction Trees with preemptive concurrency. To support implementation and verification of concurrent data structures and algorithms in the presence of preemptive concurrency one requires atomic state modification operations, e.g., compare-and-exchange.

Guarded Interaction Trees are a structure and a fully formalized framework for representing higher-order computations with higher-order effects in Coq. We present an extension of Guarded Interaction Trees to support formal reasoning about context-dependent effects. That is, effects whose behaviors depend on the evaluation context, e.g., call/cc, shift, and reset. Using and reasoning about such effects is challenging since certain compositionality principles no longer hold in the presence of such effects. For example, the so-called “bind rule” in modern program logics (which allows one to reason modularly about a term inside a context) is no longer valid. The goal of our extension is to support representation and reasoning about context-dependent effects in the most painless way possible. To that end, our extension is conservative: the reasoning principles (and the Coq implementation) for context-independent effects remain the same. We show that our implementation of context-dependent effects is viable and powerful. We use it to give direct-style denotational semantics for higher-order programming languages with call/cc and with delimited continuations. We extend the program logic for Guarded Interaction Trees to account for context-dependent effects, and we use the program logic to prove that the denotational semantics is adequate with respect to the operational semantics. This is achieved by constructing logical relations between syntax and semantics inside the program logic. Additionally, we retain the ability to combine multiple effects in a modular way, which we demonstrate by showing type soundness for safe interoperability of a programming language with delimited continuations and a programming language with higher-order store.

Constructing solutions to recursive domain equations is a well-known, important problem in the study of programs and programming languages. Mathematically speaking, the problem is finding a fixed point (up to isomorphism) of a suitable functor over a suitable category. A particularly useful instance, inspired by the step-indexing technique, is where the functor is over (a subcategory of) the category of presheaves over the ordinal ω and the functors are locally-contractive, also known as guarded functors. This corresponds to step-indexing over natural numbers. However, for certain problems, e.g., when dealing with infinite non-determinism, one needs to employ trans-finite step-indexing, i.e., consider presheaf categories over higher ordinals. Prior work on trans-finite step-indexing either only considers a very narrow class of functors over a particularly restricted subcategory of presheaves over higher ordinals, or treats the problem very generally working with sheaves over an arbitrary complete Heyting algebra with a well-founded basis. In this paper we present a solution to the guarded domain equations problem over all guarded functors over the category of presheaves over ordinal numbers, as well as its mechanization in the Rocq Prover. As the categories of sheaves and presheaves over ordinals are equivalent, our main contribution is simplifying prior work from the setting of the category of sheaves to the setting of the category of presheaves and mechanizing it - presheaves are more amenable to mechanization in a proof assistant.

This paper considers direct encodings of generalized algebraic data types (GADTs) in a minimal suitable lambda-calculus. To this end, we develop an extension of System ${𝐹}_{𝜔}$ with recursive types and internalized type equalities with injective constant type constructors. We show how GADTs and associated pattern-matching constructs can be directly expressed in the calculus, thus showing that it may be treated as a highly idealized modern functional programming language. We prove that the internalized type equalities in conjunction with injectivity rules increase the expressive power of the calculus by establishing a non-macro-expressibility result in ${𝐹}_{𝜔}$, and prove the system type-sound via a syntactic argument. Finally, we build two relational models of our calculus: a simple, unary model that illustrates a novel, two-stage interpretation technique, necessary to account for the equational constraints; and a more sophisticated, binary model that relaxes the construction to allow, for the first time, formal reasoning about data-abstraction in a calculus equipped with GADTs.