Multivariate quotients of polynomial functors.

Basic definition of multivariate QPF. QPFs form a compositional framework for defining inductive and coinductive types, their quotients and nesting.

The idea is based on building ever larger functors. For instance, we can define a list using a shape functor:

inductive list_shape (a b : Type)
| nil : list_shape
| cons : a -> b -> list_shape

This shape can itself be decomposed as a sum of product which are themselves QPFs. It follows that the shape is a QPF and we can take its fixed point and create the list itself:

def list (a : Type) := fix list_shape a -- not the actual notation

We can continue and define the quotient on permutation of lists and create the multiset type:

def multiset (a : Type) := qpf.quot list.perm list a -- not the actual notion

And multiset is also a QPF. We can then create a novel data type (for Lean):

inductive tree (a : Type)
| node : a -> multiset tree -> tree

An unordered tree. This is currently not supported by Lean because it nests an inductive type inside of a quotient. We can go further and define unordered, possibly infinite trees:

coinductive tree' (a : Type)
| node : a -> multiset tree' -> tree'

by using the cofix construct. Those options can all be mixed and matched because they preserve the properties of QPF. The latter example, tree', combines fixed point, co-fixed point and quotients.

Related modules

• constructions
  • fix
  • cofix
  • quot
  • comp
  • sigma / pi
  • prj
  • const

each proves that some operations on functors preserves the QPF structure

reference

• [Jeremy Avigad, Mario M. Carneiro and Simon Hudon, Data Types as Quotients of Polynomial Functors][avigad-carneiro-hudon2019]

Show that every mvqpf is a lawful mvfunctor.