On the strength of dependent products in the type theory of Martin-L\"of

TL;DR

This paper investigates the relative strength of different formulations of dependent product types in Martin-Löf type theory, showing the superiority of introduction/elimination rules, reformulating the elimination rule with identity types, and proposing an improved principle of function extensionality.

Contribution

It demonstrates that introduction and elimination rules are strictly stronger than abstraction/application, reformulates the elimination rule with identity types, and proposes an improved function extensionality principle.

Findings

Introduction/elimination rules are strictly stronger than abstraction/application.

Reformulation of the elimination rule using identity types is possible.

An alternative formulation of function extensionality rectifies previous deficiencies.

Abstract

One may formulate the dependent product types of Martin-L\"of type theory either in terms of abstraction and application operators like those for the lambda-calculus; or in terms of introduction and elimination rules like those for the other constructors of type theory. It is known that the latter rules are at least as strong as the former: we show that they are in fact strictly stronger. We also show, in the presence of the identity types, that the elimination rule for dependent products--which is a "higher-order" inference rule in the sense of Schroeder-Heister--can be reformulated in a first-order manner. Finally, we consider the principle of function extensionality in type theory, which asserts that two elements of a dependent product type which are pointwise propositionally equal, are themselves propositionally equal. We demonstrate that the usual formulation of this principle fails to verify a number of very natural propositional equalities; and suggest an alternative formulation which rectifies this deficiency.