Internalizing representation independence with univalence

In their usual form, representation independence metatheorems provide an external guarantee that two implementations of abstract interface are interchangeable when they related by operation-preserving correspondence. If our programming language is dependently-typed, however, we would like to appeal such invariance results within the itself, in order obtain correctness theorems for complex transferring them from simpler, implementations. Recent work proof assistants has shown Voevodsky's univalence principle allows between isomorphic types, but many instances involve non-isomorphic representations. this paper, develop techniques establishing internal relational dependent type theory, using higher inductive types simultaneously quotient implementation a heterogeneous correspondence them. The becomes isomorphism quotiented thereby allowing us equality univalence. We illustrate considering applications matrices, queues, and finite multisets. Our all formalized Cubical Agda, recent extension Agda which supports computationally well-behaved way.