Failure of Normalization in Impredicative Type Theory with Proof-Irrelevant Propositional Equality

11/19/2019

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by   Andreas Abel, et al.

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In type theory with an impredicative universe of propositions and a proof-irrelevant propositional equality the normalization property fails. Thus, type checking is likely undecidable. The counterexample to normalization is adapted from Girard's counterexample against normalization of System F equipped with a decider for type equality. It refutes Werner's normalization conjecture [LMCS 2008].