Abstract.
Inductive characterizations of the sets of terms, the subset of strongly normalizing terms and normal forms are studied in order to reprove weak and strong normalization for the simply-typed λ-calculus and for an extension by sum types with permutative conversions. The analogous treatment of a new system with generalized applications inspired by generalized elimination rules in natural deduction, advocated by von Plato, shows the flexibility of the approach which does not use the strong computability/candidate style à la Tait and Girard. It is also shown that the extension of the system with permutative conversions by η-rules is still strongly normalizing, and likewise for an extension of the system of generalized applications by a rule of ``immediate simplification''. By introducing an infinitely branching inductive rule the method even extends to Gödel's T.
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Received 12 October 1998 Published online: 19 December 2002
Mathematics Subject Classification (2000): 03B40
Keywords or phrases: λ-calculus – Permutative/commuting conversions – Sum types – Generalized application – Gödel's Tq-ω-rule – Inductive characterization
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Joachimski, F., Matthes, R. Short proofs of normalization for the simply- typed λ-calculus, permutative conversions and Gödel's T. Arch. Math. Logic 42, 59–87 (2003). https://doi.org/10.1007/s00153-002-0156-9
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DOI: https://doi.org/10.1007/s00153-002-0156-9