Technical reports

# Towards a proof theory of rewriting: the simply-typed 2-λ calculus

**Barnaby P. Hilken**

May 1994, 28 pages

**DOI:** 10.48456/tr-336

## Abstract

This paper describes the simply typed 2-λ-calculus, a language with three levels, types, terms and rewrites. The types and terms are those of the simply typed λ-calculus, and the rewrites are expressions denoting sequences of β-reductions and η-expansions. An equational theory is imposed on the rewrites, based on 2-categorical justifications, and the word problem for this theory is solved by finding a canonical expression in each equivalence class.

The canonical form of rewrites allows us to prove several properties of the calculus, including a strong form of confluence and a classification of the long-β-η-normal forms in terms of their rewrites. Finally we use these properties as the basic definitions of a theory of categorical rewriting, and find that the expected relationships between confluence, strong normalisation and normal forms hold.

## Full text

PDF (1.7 MB)

## BibTeX record

@TechReport{UCAM-CL-TR-336, author = {Hilken, Barnaby P.}, title = {{Towards a proof theory of rewriting: the simply-typed 2-$\lambda$ calculus}}, year = 1994, month = may, url = {https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-336.pdf}, institution = {University of Cambridge, Computer Laboratory}, doi = {10.48456/tr-336}, number = {UCAM-CL-TR-336} }