#
A Normalizing Computation Rule for Propositional Extensionality in Higher-Order Minimal Logic
Paper in proceeding, 2018

There are three kinds of typing judgement in PHOML. There are terms which inhabit types, which are the simple types over Ω. There are proofs which inhabit propositions, which are the terms of type Ω. The canonical propositions are those constructed from ⊥ by implication ⊃. Thirdly, there are paths which inhabit equations M = A N , where M and N are terms of type A. There are two ways to prove an equality: reflexivity, and propositional extensionality — logically equivalent propositions are equal. This system allows for some definitional equalities that are not present in cubical type theory, namely that transport along the trivial path is identity.

We present a call-by-name reduction relation for this system, and prove that the system satisfies canonicity: every closed typable term head-reduces to a canonical form. This work has been formalised in Agda.

canonicity

type theory

univalence

## Author

### Marc Bezem

University of Bergen

### Robin Adams

Chalmers, Computer Science and Engineering (Chalmers), Computing Science (Chalmers)

### Thierry Coquand

University of Gothenburg

#### Leibniz International Proceedings in Informatics, LIPIcs

18688969 (ISSN)

Vol. 97 39783959770651 (ISBN)

Novi Sad, Serbia,

### Subject Categories

Algebra and Logic

Computer Science

### Areas of Advance

Information and Communication Technology

### Roots

Basic sciences

### DOI

10.4230/LIPIcs.TYPES.2016.3