Coincidence of contextual and interaction equivalence in effectful λ-calculi

Prove that, in λ-calculi with effects where normal form bisimilarities are known to be complete for contextual equivalence (e.g., the settings studied by Biernacki–Lenglet–Polesiuk and by Støvring–Lassen), contextual equivalence coincides with interaction equivalence.

Background

The paper highlights that interaction equivalence provides a coinductive/observational characterization matching Böhm-tree equality in the head case. In some languages with effects, normal form bisimilarities are complete for contextual equivalence.

The authors conjecture that in those effectful settings, interaction equivalence collapses to contextual equivalence, suggesting that the added effects reconcile extensionality and interaction counting.

References

Complete Normal Form Bisimilarities. In some 2-calculi with effects, normal form bisimilarities are complete for contextual equivalences [Biernacki et al. 2019; Støvring and Lassen 2009]. We conjecture that, therein, contextual equivalence and interaction equivalence coincide.

— Interaction Equivalence (2409.18709 - Accattoli et al., 27 Sep 2024) in Section 10, Future Work (Complete Normal Form Bisimilarities)