This article presents Trocq , a new proof transfer framework for dependent type theory. Trocq is based on a novel formulation of type equivalence, used to generalize the univalent parametricity translation. This framework takes care of avoiding dependency on the axiom of univalence when possible, and may be used with more relations than just equivalences. We have implemented a corresponding plugin for the Rocq/Coq interactive theorem prover, in the Coq-Elpi meta-language.

Abstract This article presents Trocq, a new proof transfer framework for dependent type theory. Trocq is based on a novel formulation of type equivalence, used to generalize the univalent parametricity translation. This framework takes care of avoiding dependency on the axiom of univalence when possible, and may be used with more relations than just equivalences. We have implemented a corresponding plugin for the interactive theorem prover, in the meta-language.

Abstract Trocq [5] is both the name of a calculus, describing a parametricity framework, and of a plugin [6] that provides tactics for performing representation changes in goals, as well as vernacular commands for specifying the expected translations.

In the context of interactive theorem provers based on a dependent type theory, automation tactics (dedicated decision procedures, call of automated solvers, ...) are often limited to goals which are exactly in some expected logical fragment. This very often prevents users from applying these tactics in other contexts, even similar ones.