The last 15 years have seen an explosion in work on explicit
substitution, most of which
is done in the style of the λσ-calculus. In Kamareddine and
Ríos (1995a), we extended
the λ-calculus with explicit substitutions by turning de
Bruijn's meta-operators into object-operators offering a style of
explicit substitution that differs from that of λσ. The resulting
calculus, λs, remains as close as possible to the
λ-calculus from an intuitive point of view
and, while preserving strong normalisation (Kamareddine and Ríos,
1995a), is extended in this paper to a confluent calculus on open terms: the
λse-caculus. Since the establishment of
these results, another calculus, λζ, came into being in
Muñoz Hurtado (1996) which preserves
strong normalisation and is itself confluent on open terms. However, we
believe that λse still
deserves attention because, while offering a new style to work with
explicit substitutions, it
is able to simulate one step of classical β-reduction, whereas
λζ is not. To prove confluence
we introduce a generalisation of the interpretation method (cf. Hardin,
1989; Curien et al.,
1992) to a technique which uses weak normal forms (instead of strong
ones). We consider
that this extended method is a useful tool to obtain confluence when
strong normalisation
of the subcalculus of substitutions is not available. In our case,
strong normalisation of the
corresponding subcalculus of substitutions se,
is still a challenging open problem to the rewrite
community, but its weak normalisation is established here via an effective strategy.
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