The Barendregt-Geuvers-Klop conjecture for beta reduction: weak normalization implies strong normalization in every pure type system

Theorem 1.1: for every pure type system $(\mathcal S,\mathcal A,\mathcal R)$, with no functionality assumption on $\mathcal A$ or $\mathcal R$, if every legal expression in every valid context is weakly $\beta$-normalizing then every such expression is strongly $\beta$-normalizing; reduction acts inside annotations and contexts may be open. Both properties are system-wide: the theorem does not say that an individual term with a normal form is strongly normalizing. It makes no claim about $\beta\eta$-reduction, the other half of Geuvers's formulation. The proof uses Tait-Girard style candidates indexed by sort profiles and a realization of Hurkens's paradox to exclude a graph configuration.

Theorem 1.1: for every pure type system $(\mathcal S,\mathcal A,\mathcal R)$, with no functionality assumption on $\mathcal A$ or $\mathcal R$, if every legal expression in every valid context is weakly $\beta$-normalizing then every such expression is strongly $\beta$-normalizing; reduction acts inside annotations and contexts may be open. Both properties are system-wide: the theorem does not say that an individual term with a normal form is strongly normalizing. It makes no claim about $\beta\eta$-reduction, the other half of Geuvers's formulation. The proof uses Tait-Girard style candidates indexed by sort profiles and a realization of Hurkens's paradox to exclude a graph configuration.