A supposedly fun thing i may have to do again

Abstract

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 offer 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.

A two-level logic perspective on (simultaneous) substitutions

“…been attempted in Abella before on a restricted PCF with no recursive types [21]. That paper lamented the cumbersomeness of lifting a closed relation to open terms by means of inductive definitions instead of using the pen-and-paper definition based on universal quantification over substitutions.…”

“…That paper lamented the cumbersomeness of lifting a closed relation to open terms by means of inductive definitions instead of using the pen-and-paper definition based on universal quantification over substitutions. The purpose and peculiarities of the Howe relation are explained in much more detail in [29,21,22], so we omit it here.…”

“…This illustration is part of a larger development that formalizes the fact that applicative similarity for PCF is a (pre-)congruence. Showing substitutivity has been claimed to be a challenge to formalize in Abella in the past because of the difficulty of quantifying over substitutions [21]. The full formalization may be browsed on-line; 3 and this example is also distributed in the examples/programming-languages/howe sub-directory of the master branch of Abella, available from Github.…”

The Next 700 Challenge Problems for Reasoning with Higher-Order Abstract Syntax Representations

“…In fact, the Isabelle/HOL version of Hybrid uses sets [16] and Abella may also offer multi-sets in the near future. 9 Abella does not support existentials goals, and, to overcome this, the user needs to resort to some indirect encoding, as exemplified in [29]. clause/2.…”

“…One approach is to use a form of "weak" HOAS that replaces the s all rule (see page 29) with one that explicitly adds variables (using the VAR constructor) to the context. A better approach, which would not require sacrificing the benefits of full HOAS, is to implement the ∇-quantifier in Hybrid.…”