Deep Induction: Induction Rules for (Truly) Nested Types

Abstract

This paper introduces deep induction, and shows that it is the notion of induction most appropriate to nested types and other data types defined over, or mutually recursively with, (other) such types. Standard induction rules induct over only the top-level structure of data, leaving any data internal to the top-level structure untouched. By contrast, deep induction rules induct over all of the structured data present. We give a grammar generating a robust class of nested types (and thus ADTs), and develop a fundamental theory of deep induction for them using their recently defined semantics as fixed points of accessible functors on locally presentable categories. We then use our theory to derive deep induction rules for some common ADTs and nested types, and show how these rules specialize to give the standard structural induction rules for these types. We also show how deep induction specializes to solve the long-standing problem of deriving principled and practically useful structural induction rules for bushes and other truly nested types. Overall, deep induction opens the way to making induction principles appropriate to richly structured data types available in programming languages and proof assistants. Agda implementations of our development and examples, including two extended case studies, are available.

(Deep) induction rules for GADTs

“…A summary of the various classes of data types considered in this paper is given in Table 1. G in (12) Deep induction [13] is a generalization of structural induction that fits this bill exactly. Whereas structural induction rules induct over only the top-level structure of data, leaving any data internal to the top-level structure untouched, deep induction rules induct over all of the structured data present.…”

“…The inductive data types handled by Coq include (possibly mutually inductive) polynomial algebraic data types (ADTs), and the induction rules Coq generates for them are the expected ones for standard structural induction. However, as discussed in [13], it has long been understood that these rules are too weak to be genuinely useful for deep ADTs. 1 The following data type of rose trees, here coded in Agda and defined in terms of the standard type List of lists (see Section 2), is a deep ADT: Unfortunately, this is neither the induction rule we intuitively expect, nor is it expressive enough to prove even basic properties of rose trees that ought to be amenable to inductive proof.…”

“…Deep induction was shown in [13] to deliver induction rules appropriate to nested types, including ADTs. For the (deep) ADT of rose trees, for example, it gives the following genuinely useful induction rule:…”

“…Here, List ∧ (resp., Rose ∧ ) lifts its predicate argument P (resp., Q) on data of type Rose A (resp., A) to a predicate on data of type List (Rose A) (resp., Rose A) asserting that P (resp., Q) holds for every element of its list (resp., rose tree) argument. 3 Deep induction was also shown in [13] to deliver the firstever induction rules -structural or otherwise -for the Bush 3 Predicate liftings such as List ∧ and Rose ∧ can either be supplied as primitives or generated automatically from their associated data type definitions as described in Section 2. The predicate lifting for a container type like List A or Rose A simply traverses containers of that type and applies its predicate argument pointwise to the constituent data of type A.…”

“…Significantly, deep induction rules for GADTs cannot be derived by somehow extending the approach of [13] to syntaxonly GADTs. Indeed, the approach taken there makes crucial use of the functoriality of data types' interpretations from [12], and functoriality is precisely what interpreting GADTs as the interpretations of their Church encodings fails to deliver; see [9] for a discussion of why Seq, e.g., is not functorial.…”