Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks

Abstract

Guarded recursion is a powerful modal approach to recursion that can be seen as an abstract form of step- indexing. It is currently used extensively in separation logic to model programming languages with advanced features by solving domain equations also with negative occurrences. In its multi-clocked version, guarded recursion can also be used to program with and reason about coinductive types, encoding the productivity condition required for recursive definitions in types. This paper presents the first type theory combining multi-clocked guarded recursion with the features of Cubical Type Theory, as well as a denotational semantics. Using the combination of Higher Inductive Types (HITs) and guarded recursion allows for simple programming and reasoning about coinductive types that are traditionally hard to represent in type theory, such as the type of finitely branching labelled transition systems. For example, our results imply that bisimilarity for these imply path equality, and so proofs can be transported along bisimilarity proofs. Among our technical contributions is a new principle of induction under clocks. This allows universal quantification over clocks to commute with HITs up to equivalence of types, and is crucial for the encoding of coinductive types. Such commutativity requirements have been formulated for inductive types as axioms in previous type theories with multi-clocked guarded recursion, but our present formulation as an induction principle allows for the formulation of general computation rules.