Small bisimulations for reasoning about higher-order imperative programs

We introduce a new notion of bisimulation for showing contextual equivalence of expressions in an untyped lambda-calculus with an explicit store, and in which all expressed values, including higher-order values, are storable. Our notion of bisimulation leads to smaller and more tractable relations than does the method of Sumii and Pierce [31]. In particular, our method allows one to write down a bisimulation relation directly in cases where [31] requires an inductive specification, and where the principle of local invariants [22] is inapplicable. Our method can also express examples with higher-order functions, in contrast with the most widely known previous methods [4, 22, 32] which are limited in their ability to deal with such examples. The bisimulation conditions are derived by manually extracting proof obligations from a hypothetical direct proof of contextual equivalence.