## Abstract

One may formulate the dependent product types of Martin-Löf 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.

Original language | English |
---|---|

Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Annals of Pure and Applied Logic |

Volume | 160 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 2009 |

Externally published | Yes |

## Keywords

- Dependent products
- Dependent type theory
- Function extensionality