Counterexamples to applicative simulation and extensionality in non-deterministic call-by-need lambda-calculi with letrec

This note provides an example that demonstrates that in non-deterministic call-by-need lambda-calculi extended with cyclic let, extensionality as well as applicative bisimulation in general may not be used as criteria for contextual equivalence w.r.t. may- and two different forms of must-convergence. We also outline how the counterexample can be adapted to other calculi.

► Non-deterministic call-by-need lambda-calculi with letrec are not extensional.
► Applicative bisimulation is unsound in such calculi.
► The counterexample can be adapted to other extended pure lambda-calculi.