Recursion on Nested Datatypes in Dependent Type Theory

Abstract

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.

The power of parameterization in coinductive proof

“…We first review the idea of Mendler-style recursion in the setting of complete lattices and show how it can be employed to overcome Coq's strict positivity restriction in the case of predicates. This observation is not novel, and has been made before by Matthes [11].…”

Foundational nonuniform (Co)datatypes for higher-order logic

“…In particular, map functions and relators are missing and can be difficult to add. c) Other Work: The pioneering work of Bird and his collaborators on nonuniform datatypes [7], [8], [9] has been extended into several directions, including structures for efficient functional programming [23], [24], [30], datatypes with references [16], as well as work directly relevant for DTT proof assistants: reduction to W-types and container types [1], typed term rewriting frameworks for total programming [2], [3], [31], intensional-DTT induction [32]. Our current contribution was concerned with bootstrapping nonuniform datatypes in HOL on a sound and compositional basis.…”

Nested Datatypes with Generalized Mendler Iteration: Map Fusion and the Example of the Representation of Untyped Lambda Calculus with Explicit Flattening

“…Clearly, not only (generalized) iteration should be available for programs on nested datatypes. The author experiments with primitive recursion in Mendler style, but does not yet have termination guarantees [31].…”