 
Summary: Towards Normalization by Evaluation for the
Calculus of Constructions (Extended Version)
Andreas Abel
Project PI.R2, INRIA Rocquencourt and PPS, Paris
andreas.abel@ifi.lmu.de
Abstract. We consider the Calculus of Constructions with typed betaeta equal
ity and an algorithm which computes long normal forms. The normalization algo
rithm evaluates terms into a semantic domain, and reifies the values back to terms
in normal form. To show termination, we interpret types as partial equivalence
relations between values and type constructors as operators on PERs. This mod
els also yields consistency of the betaetaCalculus of Constructions. The model
construction can be carried out directly in impredicative type theory, enabling a
formalization in Coq.
Keywords: effective model of type theory, formalization, impredicativity, judgemental
equality, realizability.
1 Introduction
The proof assistant Coq [INR08] based on intensional type theory is used for large ver
ification projects in mathematics [Gon04] and computer science [Ler06]. However, to
this day no complete meta theory of its logical core, the Calculus of Inductive Construc
tions (CIC) exists. The CIC is a dependent type theory with at least one impredicative
