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This paper extends the proof methods used by the Nuprl proof assistant to reason about the computational behavior of its untyped programs. We have implemented new methods to prove non-trivial bisimulations between programs and have successfully applied these methods to formally optimize distributed programs such as our synthesized and verified version of Paxos, a widely used protocol to achieve software based replication. We prove new results about the basic computational equality relation on terms, and we extend the theory of partial types as the basis for stating internal results about the computation system that were previously treated only in the meta theory of Nuprl. All the lemmas presented in this paper have been formally proved in Nuprl.