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Summary: Comparing Calculi of Explicit Substitutions with Eta-reduction
Mauricio Ayala-Rincon ?1 , Flavio L. C. de Moura ??1 , and Fairouz Kamareddine 2
1 Departamento de Matematica, Universidade de Braslia, Braslia D.F., Brasil ayala@mat.unb.br,
flavio@mat.unb.br
2 Computing and Electrical Engineering, Heriot-Watt University, Edinburgh, Scotland fairouz@cee.hw.ac.uk
Abstract. The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous
work has illustrated the usefulness of these calculi for practical notions like the implementation of
typed functional programming languages and higher order proof assistants. Three styles of explicit
substitutions are treated in this paper: the and the se which have proved useful for solving practical
problems like higher order uni cation, and the suspension calculus related to the implementation of the
language Prolog. We enlarge the suspension calculus with an adequate eta-reduction which we show
to preserve termination and con
uence of the associated substitution calculus and to correspond to the
eta-reductions of the other two calculi. Additionally, we prove that and se as well as and the
suspension calculus are non comparable while se is more adequate than the suspension calculus.
Keywords: Calculi of Explicit substitutions, lambda-calculi, Eta Reduction.
1 Introduction
Recent years have witnessed an explosion of work on expliciting substitutions [1, 7, 9, 14, 15, 17, 19] and on
establishing its usefulness to computation: e.g., to automated deduction and theorem proving [24, 25], to
proof theory [31], to programming languages [8, 20, 23, 26] and to higher order uni cation HOU [2, 13].
This paper concentrates on three di erent styles of substitutions:
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