Relating System F and Lambda2: A Case Study in Coq, Abella and Beluga

• Main Authors: Kaiser, Jonas, Pientka, Brigitte, Smolka, Gert
• Format: Conference Object
• Language: English
• Published: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik 2017
• Subjects: Pure Type Systems; System F; de Bruijn Syntax; Higher-Order Abstract Syntax; Contextual Reasoning; Data processing Computer science; Beluga; Beluga*
• Online Access: https://doi.org/10.4230/LIPIcs.FSCD.2017.21; https://nbn-resolving.org/urn:nbn:de:0030-drops-77248; https://drops.dagstuhl.de/opus/volltexte/2017/7724/

We give three formalisations of a proof of the equivalence of the usual, two-sorted presentation of System F and its single-sorted pure type system (PTS) variant Lambda2. This is established by reducing the typability problem of F to Lambda2 and vice versa. A key challenge is the treatment of variable binding and contextual information. The formalisations all share the same high level proof structure using relations to connect the type systems. They do, however, differ significantly in their representation and manipulation of variables and contextual information. In Coq, we use pure de Bruijn indices and parallel substitutions. In Abella, we use higher-order abstract syntax (HOAS) and nominal constants of the ambient reasoning logic. In Beluga, we also use HOAS but within contextual modal type theory. Our contribution is twofold. First, we present and compare a collection of machine-checked solutions to a non-trivial theoretical result. Second, we propose our proof as a benchmark, complementing the POPLmark and ORBI challenges by testing how well a given proof assistant or framework handles complex contextual information involving multiple type systems.