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Parallel reductions in \(\lambda\)-calculus. (English) Zbl 0661.03008

The notion of parallel reduction is extracted from the Tait-Martin-Löf proof of the Church-Rosser theorem (for \(\beta\)-reduction). We define parallel \(\beta\)-, \(\eta\)- and \(\beta\) \(\eta\)-reduction by induction, and use them to give simple proofs of some fundamental theorems in \(\lambda\)- calculus; the normal reduction theorem for \(\beta\)-reduction, that for \(\beta\) \(\eta\)-reduction, the postponement theorem of \(\eta\)-reduction (in \(\beta\) \(\eta\)-reduction), and some others.

MSC:

03B40 Combinatory logic and lambda calculus
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References:

[1] Barendregt, H. P., (The Lambda Calculus (1984), North-Holland: North-Holland Amsterdam) · Zbl 0549.03012
[2] Klop, J. W., (Combinatory Reduction Systems (1980), Mathematisch Centrum: Mathematisch Centrum Amsterdam)
[3] Levy, J. J., An algebraic interpretation of the λ-β-K calculus and a labelled \(λ\)-calculus, Springer Lec. Notes Comp., 37, 147-165 (1975)
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