The Discriminating Power of the Let-In Operator in the Lazy Call-by-Name Probabilistic lambda-Calculus

We consider the notion of probabilistic applicative bisimilarity (PAB), recently introduced as a behavioural equivalence over a probabilistic extension of the untyped lambda-calculus. Alberti, Dal Lago and Sangiorgi have shown that PAB is not fully abstract with respect to the context equivalence induced by the lazy call-by-name evaluation strategy. We prove that extending this calculus with a let-in operator allows for achieving the full abstraction. In particular, we recall Larsen and Skou\u27s testing language, which is known to correspond with PAB, and we prove that every test is representable by a context of our calculus