Gradual typing has long been advocated as a means to bridge the gap between static and dynamic typing disciplines, enabling a range of use cases such as the gradual migration of existing dynamically typed code to more statically typed code, as well as making advanced static typing disciplines more accessible. To assess whether a given gradual language can effectively support these use cases, several formal properties have been proposed, most notably the refined criteria set forth by Siek et al. One criterion asserts that the dynamic extreme of the spectrum should be expressible in the gradual language, formalized by the existence of an adequate embedding from the corresponding dynamic language.

We observe that the existing dynamic embedding criterion does not capture the desirable property of being able to ascribe embedded code to a static type that it semantically satisfies, and ensure reliable interactions with other components within the gradual language. Specifically, we introduce the notion of robustness for gradual terms, meaning that when interacting with any gradual context, runtime failures that may occur ought to be caused by the context, not by the robust term itself. We then formulate the robust dynamic embedding criterion: if a dynamic component semantically satisfies a given static type, then its embedding subsequently ascribed to that static type should be a robust term. We demonstrate that robust dynamic embedding is not implied by any existing metatheoretical property from the literature, and is not upheld by various existing gradual languages. We show that robust dynamic embedding is achievable with a gradualized simply-typed language. All the results are formalized in the Rocq proof assistant. This novel criterion complements the set of criteria for gradual languages and opens several venues for further exploration, in particular for typing disciplines that enforce rich semantic properties.