This paper presents a new bisimulation theory for parametric polymorphism which enables straightforward co-inductive proofs of program equivalences involving existential types. The theory is an instance of typed normal form bisimulation and demonstrates the power of this recent framework for modeling typed lambda calculi as labelled transition systems.
We develop our theory for a continuation-passing style calculus, Jump-With-Argument, where normal form bisimulation takes a simple form. We equip the calculus with both existential and recursive types. An "ultimate pattern matching theorem" enables us to define bisimilarity and we show it to be a congruence. We apply our theory to proving program equivalences, type isomorphisms and genericity.