TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

A Lipschitz function after an absolutely continuous function is absolutely continuous

If $F \in A C [a, b]$ and $h$ is Lipschitz on $F ([a, b])$ , then $h \circ F \in A C [a, b]$ .

Given: $F \in A C [a, b]$ and a Lipschitz constant $L$ for $h$ on its image.

technique · direct

1.1givenchoose

For $ε > 0$ , use the AC modulus of $F$ from Absolute continuity on a compact interval with tolerance $ε / L$ when $L > 0$ .

2.1step 1.1algebra

Sum $∣ h (F (v_{j})) - h (F (u_{j})) ∣ \leq L ∣ F (v_{j}) - F (u_{j}) ∣$ over a short disjoint family.

3.1step 2.1∎

If $L = 0$ the composite is constant; otherwise step 2.1 is the AC condition.

Depends on