A supposedly fun thing i may have to do again : a HOAS encoding of Howe's method

We formally verify in Abella that similarity in the call-by-name lambda calculus is a pre-congruence, using Howe’s method. This turns out to be a very challenging task for HOAS-based systems, as it entails a demanding combination of inductive and coinductive reasoning on open terms, for which no other existing HOAS-based system is equipped for. We also o er a proof using a version of Abella supplemented with predicate quantification; this results in a more structured presentation that is largely independent of the operational semantics as well of the chosen notion of (bi)similarity. While the end result is significantly more succinct and elegant than previous attempts, the exercise highlights some limitations of the two-level approach in general and of Abella in particular.