Generic Fibrational Induction

TL;DR

This paper introduces a universal, semantic induction rule applicable to all inductive data types, including non-polynomial ones, within a fibrational framework, enabling proofs of properties for complex data structures.

Contribution

It derives a sound, generic induction rule for all inductive types, extending previous work to non-polynomial data types and broadening the scope of properties that can be proved.

Findings

Induction rule applies to rose trees, hereditary sets, and hyperfunctions.

The rule is sound and reduces induction to iteration.

It works for data types outside polynomial scope, like hyperfunctions.

Abstract

This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobs' elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, a sound induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of a particular syntactic form. We establish the soundness of our generic induction rule by reducing induction to iteration. We then show how our generic induction rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The first of these lies outside the scope of Hermida and Jacobs' work because it is not polynomial, and as far as we are aware, no induction rules have been known to exist for the second and third in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set.