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New Notions of Reduction and NonSemantic Proofs of Strong βNormalization in Typed λCalculi
 PROCEEDINGS OF LOGIC IN COMPUTER SCIENCE
, 1995
"... Two notions of reduction for terms of the λcalculus are introduced and the question of whether a λterm is βstrongly normalizing is reduced to the question of whether a λterm is merely normalizing under one of the notions of reduction. This gives a method to prove strong βnormalization for typ ..."
Abstract

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Two notions of reduction for terms of the λcalculus are introduced and the question of whether a λterm is βstrongly normalizing is reduced to the question of whether a λterm is merely normalizing under one of the notions of reduction. This gives a method to prove strong βnormalization for typed λcalculi. Instead of the usual semantic proof style based on Tait's realizability or Girard's "candidats de réductibilité", termination can be proved using a decreasing metric over a wellfounded ordering. This proof method is applied to the simplytyped λcalculus and the system of intersection types, giving the first nonsemantic proof for a polymorphic extension of the λcalculus.