Schematic Polymorphism in the Abella Proof Assistant

TL;DR

This paper introduces schematic polymorphism in the Abella proof assistant, enabling generic reasoning over parameterized types while maintaining logical soundness, thus reducing redundancy in formalizations.

Contribution

It develops a novel approach allowing type parameterization in Abella, overcoming limitations of simply typed logic and preserving proof validity across type instances.

Findings

Implemented schematic polymorphism in Abella

Ensured proof validity across type instantiations

Enhanced expressiveness of the proof system

Abstract

The Abella interactive theorem prover has proven to be an effective vehicle for reasoning about relational specifications. However, the system has a limitation that arises from the fact that it is based on a simply typed logic: formalizations that are identical except in the respect that they apply to different types have to be repeated at each type. We develop an approach that overcomes this limitation while preserving the logical underpinnings of the system. In this approach object constructors, formulas and other relevant logical notions are allowed to be parameterized by types, with the interpretation that they stand for the (infinite) collection of corresponding constructs that are obtained by instantiating the type parameters. The proof structures that we consider for formulas that are schematized in this fashion are limited to ones whose type instances are valid proofs in the simply typed logic. We develop schematic proof rules that ensure this property, a task that is complicated by the fact that type information influences the notion of unification that plays a key role in the logic. Our ideas, which have been implemented in an updated version of the system, accommodate schematic polymorphism both in the core logic of Abella and in the executable specification logic that it embeds.