Non-Derivability Results in Polymorphic Dependent Type Theory

TL;DR

This paper investigates the limitations of polymorphic dependent type theory, specifically $\lambda$P2, in defining data types with induction and coinduction principles, showing certain types are not definable without additional features.

Contribution

It provides partial answers to whether quotient and coinductive types with induction principles are definable in $\lambda$P2, highlighting the necessity of function extensionality.

Findings

Parametric quotient types are not definable in $\lambda$P2.

The stream type does not have a coinduction principle in $\lambda$P2.

Function extensionality is essential for proving induction principles.

Abstract

In the pure Calculus of Constructions (CC) one can define data types and function over these, and there is a powerful higher order logic to reason over these functions and data types. This is due to the combination of impredicativity and dependent types, and most of these features can already be observed in polymorphic (second order) dependent type theory $\lambda$P2. The impredicative encoding of data types (in $\lambda$P2 or CC) is powerful but not fully satisfactory: for example, the induction principle is not provable. As a matter of fact, it can be shown that induction is not provable for whatever possible representation of data types. In a recent paper, Awodey, Frey and Speight show that in an extension of $\lambda$P2 with Sigma-types, identity types with uniqueness of identity proofs and function extensionality, it is possible to define data types for which the induction principle is provable. More recently it has been shown that in this extension of $\lambda$P2, also quotient types can be defined with the proper induction principle, and, using quotient types, coinductive types can be defined with the proper coinduction principle. This leaves various questions open: Are quotient types with induction principle not definable in the original $\lambda$P2? And how about coinductive types, is it impossible to get a strong coinduction principle in $\lambda$P2? Looking at it from the other side: which of the extensions used are really needed to make induction and coinduction work? In this paper, we contribute partial answers to these questions: parametric quotient types are not definable in $\lambda$P2 and the well-known definable stream type does not have a coinduction principle. For the latter question we show that, if we just extend $\lambda$P2 with Sigma-types and identity types with uniqueness of identity proofs, we still cannot prove an induction principle for the natural numbers. So function extensionality is crucial in making induction provable. We show these results by studying models of $\lambda$P2 where the types representing these principles are empty, so these models act as counter models to the derivability of the principles.