Probabilistic lifting of the logical relation for Pólya urn equivalence

Develop a probabilistic lifting of Cambria’s step-indexed logical relation along a probability monad, replacing admissible-relation membership with coupling lifting and re-establishing anti-reduction and compositionality, in order to formally prove the representation equivalence between the state-based and direct implementations of Pólya’s urn.

Background

Cambria’s deterministic logical relation establishes abstraction over the parameter type but does not cover the probabilistic primitives used by the two Pólya urn handlers. One handler stores urn contents as a pair of integers in local state, while the other represents the urn by a single Beta-distributed bias and samples independently from it.

The paper identifies the missing technical development: the logical relation must be lifted to probabilistic computations using couplings, after which anti-reduction and compositionality must be reproved. This would formalize the de Finetti-style representation equivalence between the two handlers.