CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Relative consistency of BPI without Choice together with Halpern–Läuchli

Statement

Let HL be the finite-product dense-matrix scheme stated by Halpern–Läuchli dense-matrix dichotomy. Then

$Con (Z F) ⟹ Con (Z F + B P I + \neg A C + H L) .$

Thus, conditional on $Con (Z F)$ , BPI together with the choice-free Halpern--Läuchli scheme still does not imply AC.

Facts & Assumptions

Given: Assume $Con (Z F)$ .

[F1]

Relative consistency of BPI without Choice over ZF gives $Con (Z F + B P I + \neg A C)$ by a finite proof reduction.

[F2]

Halpern–Läuchli dense-matrix dichotomy is a ZF theorem uniform in every positive finite dimension and every finite tree family.

[F3]

The standard certified provability predicate fixes the reading of consistency as absence of a standard finite refutation.

Proof

technique · external conservative extension by a theorem of the base theory

1.1

Suppose the displayed target were inconsistent and fix a standard finite refutation. It uses only finitely many displayed HL instances, or one use of the uniformly quantified F2 theorem after its standard coding. Replace each such occurrence by the corresponding fixed finite ZF derivation from F2. This is an external transformation of the alleged finite refutation; no arithmetized uniform proof transformer is needed.

F2F3assume-contra

2.1

The result is a refutation of $Z F + B P I + \neg A C$ , contradicting F1 under the given consistency hypothesis. Hence the target is consistent whenever ZF is.

F1step 1.1discharge-contradiction

3.1

If BPI+HL implied AC over ZF, the target theory would prove both AC and its negation, contrary to step 2.1. This gives the stated conditional nonimplication without asserting any theory's consistency outright.

step 2.1assume-contra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. D. Halpern and H. Läuchli, A partition theorem, Theorem 1, pp.360-367 (standard reference, not scraped)
J. D. Halpern and A. Lévy, The Boolean prime ideal theorem does not imply the axiom of choice, pp.83-134 (standard reference, not scraped)