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A confluent reduction for the extensional typed \(\lambda\)-calculus with pairs, sums, recursion and terminal object. (English) Zbl 1422.03022

Lingas, Andrzej (ed.) et al., Automata, languages and programming. 20th international colloquium, ICALP 93, Lund, Sweden, July 5–9, 1993. Proceedings. Berlin: Springer-Verlag. Lect. Notes Comput. Sci. 700, 645-656 (1993).
Summary: We add extensional equalities for the functional and product types to the typed \(\lambda\)-calculus with not only products and terminal object, but also sums and bounded recursion (a version of recursion that does not allow recursive calls of infinite length). We provide a confluent and strongly normalizing (thus decidable) rewriting system for the calculus, that stays confluent when allowing unbounded recursion. For that, we turn the extensional equalities into expansion rules, and not into contractions as is done traditionally. We first prove the calculus to be weakly confluent, which is a more complex and interesting task than for the usual \(\lambda\)-calculus. Then we provide an effective mechanism to simulate expansions without expansion rules, so that the strong normalization of the calculus can be derived from that of the underlying, traditional, non extensional system. These results give us the confluence of the full calculus, but we also show how to deduce confluence without the weak confluence property, using only our technique of simulating expansions.
For the entire collection see [Zbl 0814.00020].

MSC:

03B40 Combinatory logic and lambda calculus
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