Summary: Explicit Substitutions Calculi with Explicit Eta rules which
Preserve Subject Reduction #
Daniel Lima Ventura 1+ , Mauricio Ayala�Rinc� on 1# and Fairouz Kamareddine 2
1 Grupo de Teoria da Computac� �
ao, Departamento de Matem�atica,
Universidade de Bras��lia, Bras��lia D.F., Brasil
2 School of Mathematical and Computer Sciences,
Heriot�Watt University, Edinburgh, Scotland
{ventura,ayala}@mat.unb.br fairouz@macs.hw.ac.uk
Abstract. Subject reduction (for short SR) is an essential property of any type system.
This property guarantees that all terms of the system preserve their types during any
possible computation. It is well�known that the classic simply typed #�calculus has
this property, which means that any well�typed #�term preserves its type under #� and
#�contractions. It has been argued in the past decade that the notion of substitution
in the #�calculus needs to be made explicit. In this paper, we show that SR poses
computational difficulties when the #�calculus is extended with explicit substitutions.
In particular, we show that two important calculi of explicit substitutions enlarged
with Eta rules, when no explicit type and normalisation considerations are given, do
not preserve subject reduction. However, we show also that if Eta reduction was made
``explicit'' then SR will hold for these calculi. More specifically, our results can be

Source: Ayala-Rinc�n, Mauricio - Departamento de Matem�tica, Universidade de Bras�lia

Collections: Mathematics