Completeness of bisimilarity for contextual equivalence in linear theories

In this paper, we develop new variations of methods from operational semantics, and show how to apply these to a linear type theory which has a lazy operational semantics. In particular, we consider how one can establish contextual equivalences in a linear theory with function types and tensor types by instead establishing bisimulations. Thus bisimilarity is sound for contextual equivalence. Further, we show that bisimilarity is complete for contextual equivalence.; We shall show that the notion of a program context in the linear setting is non-trivial. In particular, we give a definition of linear context which is amenable to mechanization in a theorem prover, and explain why a more naive approach to dealing with linear contexts would not be so tractable.; The central contributions of the paper are: the formulation, in a linear setting, of a good notion of program context and the associated contextual equivalence; an adaptation of Howe's method; the notion of a linear precongruence; and a proof that bisimilarity is sound and complete for contextual equivalence.