Proving Strong Normalization of CC by Modifying Realizability Semantics

This presentation corresponds to a part of the author 's PhD thesis [Alt93a], a preliminary version has been presented in [Alt93b]. In my thesis I present a more general soundness result for a class of models for CC CC-structures from which the strong normalization argument can be derived as an instance. Here we shall restrict ourselves to the reasoning needed for the strong normalization proof. The proof tha t every term typable in the cMculus of constructions is strongly normalizing is known to be notoriously difficult. The original proof in Coquand's PhD thesis [Coq85] contained a bug which was fixed in [CG90] by using a Kripkestyle interpretation of contexts. Although this solves the original problem the proof remains quite intricate due to the use of typed terms and contexts. Another construction is due to Geuvers and Nederhof (see [Geu93], p. 168), who define a forgetful, reduction-preserving map from CC to F ~ . Thereby, they reduce the problem to strong normalization for F ~, which can be shown using the usual Girard-Tait method. The main problem with this construction is that it is not M1 clear, how this argument can be extended to a system with large eliminations (e.g. see [Wer92]), this is a system which allows the definition of a dependent type by primitive recursion. As an example consider the recursive definition of