we-solve (beta)

Math-2724 · mathematics · Logic

Global weak normalization implies strong normalization in pure type systems

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every pure type system with arbitrary sorts, axioms and rules, if every legal expression in every valid context has some terminating $\beta$-reduction sequence, must every $\beta$-reduction sequence of each such expression terminate? Reduction includes type annotations and the specification need not be functional.

Short English statement

For every pure type system with arbitrary sorts, axioms and rules, if every legal expression in every valid context has some terminating $\beta$-reduction sequence, must every $\beta$-reduction sequence of each such expression terminate? Reduction includes type annotations and the specification need not be functional.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2724-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2724

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 245; 2026-09-25).

Global weak normalization implies strong normalization in pure type systems

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 245; 2026-09-25), the asserted result is that weak beta normalization of an arbitrary pure type system implies strong beta normalization across all legal expressions and contexts, including reduction in annotations.

Mathematical statement of the claimed result

Manuscript 1.
A pure type system is specified by sorts $\mathcal S$, typing axioms $\mathcal A\subseteq\mathcal S^2$ and product rules $\mathcal R\subseteq\mathcal S^3$, with the usual dependent-product formation and $\beta$ reduction. The manuscript claims, for every such triple, \[ [\text{every legal expression in every valid context has a finite }\beta\text{ reduction to normal form}] \ \Longrightarrow\ [\text{every }\beta\text{ reduction sequence from every such expression is finite}]. \] Thus global weak $\beta$ normalization implies global strong $\beta$ normalization. Neither the axioms nor product rules are assumed functional.

Outline of the argument

For manuscript 1 (family 245; 2026-09-25), a finite counterexample reduces to finitely many sorts and rules. Sets of possible typing sorts replace unique types in nonfunctional systems; substitution-stable observation candidates and typed data channels turn an infinite reduction into an obstruction to any normal form.

Formal verification

For manuscript 1 (family 245; 2026-09-25), OpenAI lists Lean statements for this assertion: weak beta normalization of an arbitrary pure type system implies strong beta normalization across all legal expressions and contexts, including reduction in annotations.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are TypeSystemNormalization.lean.

Their named declarations include OAI.PureTypeSystem.weak_implies_strong.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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