OpenAI · 245-01 · 2026-09-25 · Mathematical logic
Weak and strong normalization in pure type systems
preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026
Family 245. The -Barendregt–Geuvers–Klop conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a -normal form, every -reduction sequence terminates. This resolves the -Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts.
We prove that every weakly β-normalizing pure type system is strongly β-normalizing. Both properties quantify over all legal expressions in all valid contexts, and reduction acts inside type annotations. No functionality hypothesis is required. This resolves the β-Barendregt–Geuvers–Klop conjecture.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 245: Weak normalization implies strong normalization in pure type systems. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.