An induction principle for nested datatypes in intensional type theory

Matthes, Ralph

doi:10.1017/s095679680900731x

Cited by 15 publications

(34 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Matthes [11] gives induction rules for nested types (including truly nested ones) in an intensional type theory. He handles only rank-2 functors that underlie nested types (while we handle any functor of any rank with an initial algebra), but his insights may help guide choices of fibrations for truly nested types.…”

Section: Discussionmentioning

confidence: 99%

Fibrational Induction Rules for Initial Algebras

Ghani

Johann

Fumex

2010

Computer Science Logic

View full text Add to dashboard Cite

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, an 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 particular syntactic forms. We establish the correctness of our generic induction rule by reducing induction to iteration. We show how our rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The former lies outside the scope of Hermida and Jacobs' work because it is not polynomial; as far as we are aware, no induction rules have been known to exist for the latter two 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.

show abstract

Section: Discussionmentioning

confidence: 99%

Fibrational Induction Rules for Initial Algebras

Ghani

Johann

Fumex

2010

Computer Science Logic

View full text Add to dashboard Cite

show abstract

“…However, no reasoning principles, in particular no induction principles, were studied there. Newer work by the author [8] integrates rank-2 Mendler iteration into the Calculus of Inductive Constructions [9][10][11] that underlies the Coq theorem prover [12] and also justifies an induction principle for them. This is embodied in the system LNMIt, the "logic for natural Mendler-style iteration", defined in Section 3.1.…”

Section: Introductionmentioning

confidence: 99%

“…The articles [6,8] only concern plain iteration. While an extension of primitive Mendler-style recursion [13] to nested datatypes has been described earlier [14], we will present here an extension LNMRec of system LNMIt by an enriched Mendler-style recursor where the step term additionally has access to a map term for the unknown type transformation X that occurs there.…”

Section: Introductionmentioning

confidence: 99%

“…By way of examples, its merits will be studied. An overview of extensions to LNMRec of results established for LNMIt in [8] is given. However, the main theorem of [8] is not carried over to the present setting, i. e., we do not yet have a justification within the Calculus of Inductive Constructions.…”

Section: Introductionmentioning

confidence: 99%

“…An overview of extensions to LNMRec of results established for LNMIt in [8] is given. However, the main theorem of [8] is not carried over to the present setting, i. e., we do not yet have a justification within the Calculus of Inductive Constructions. Nevertheless, all the concepts and results have been formalised in the Coq system, using module functors with parameters of a module type that specifies our extension of LNMRec.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Recursion on Nested Datatypes in Dependent Type Theory

Matthes

Logic and Theory of Algorithms

Self Cite

View full text Add to dashboard Cite

Abstract. Nested datatypes are families of datatypes that are indexed over all types and where the datatype constructors relate different members of the family. This may be used to represent variable binding or to maintain certain invariants through typing. In dependent type theory, a major concern is the termination of all expressible programs, so that types that depend on object terms can still be type-checked mechanically. Therefore, we study iteration and recursion schemes that have this termination guarantee throughout. This is not based on syntactic criteria (recursive calls with "smaller" arguments) but just on types ("type-based termination"). An important concern are reasoning principles that are compatible with the ambient type theory, in our case induction principles. In previous work, the author has proposed an abstract description of nested datatypes together with a mapping operation (like map for lists) and an iterator on the term side and an induction principle on the logical side that could all be implemented within the Coq system (with impredicative Set that is just needed for the justification, not for the definition and the examples). For verification purposes, it is important to have naturality theorems for the obtained iterative functions. Although intensional type theory does not provide naturality in general, criteria for naturality could be established that are met in case studies on "bushes" and representations of lambda terms (also with explicit flattening). The new contribution is an extension of this abstract description to full primitive recursion and its illustration by way of examples that have been carried out in Coq. Unlike the iterative system, we do not yet have a justification within Coq.

show abstract

Deep Induction: Induction Rules for (Truly) Nested Types

Johann

Polonsky

2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

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.

show abstract

An induction principle for nested datatypes in intensional type theory

Cited by 15 publications

References 22 publications

Fibrational Induction Rules for Initial Algebras

Fibrational Induction Rules for Initial Algebras

Recursion on Nested Datatypes in Dependent Type Theory

Deep Induction: Induction Rules for (Truly) Nested Types

Contact Info

Product

Resources

About