Recursive coalgebras provide an elegant categorical tool for modelling recursive algorithms and analysing their termination and correctness. By considering coalgebras over categories of suitably indexed families, the correctness of the corresponding algorithms follows intrinsically just from the type of the computed maps. However, proving recursivity of the underlying coalgebras is non-trivial, and proofs are typically ad hoc. This layer of complexity impedes the formalization of coalgebraically defined recursive algorithms in proof assistants. We introduce a framework for constructing coalgebras which are intrinsically recursive in the sense that the type of the coalgebra guarantees recursivity from the outset. Our approach is based on the novel concept of a well-founded functor on a category of families indexed by a well-founded relation. We show as our main result that every coalgebra for a well-founded functor is recursive, and demonstrate that well-known techniques for proving recursivity and termination such as ranking functions are subsumed by this abstract setup. We present a number of case studies, including Quicksort, the Euclidian algorithm, and CYK parsing. Both the main theoretical result and selected case studies have been formalized in Cubical Agda.

The paper “Sorting with Bialgebras and Distributive Laws” by Hinze et al. uses the framework of bialgebraic semantics to define sorting algorithms. From distributive laws between functors they construct pairs of sorting algorithms using both folds and unfolds. Pairs of sorting algorithms arising this way include insertion/selection sort and quick/tree sort. We extend this work to define intrinsically correct variants in cubical Agda. Our key idea is to index our data types by multisets, which concisely captures that a sorting algorithm terminates with an ordered permutation of its input list. By lifting bialgebraic semantics to the indexed setting, we obtain the correctness of sorting algorithms purely from the distributive law.