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

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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.

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The product of two absolutely continuous functions is absolutely continuous

Statement

If $F, G \in A C [a, b]$ , then $F G \in A C [a, b]$ .

Facts & Assumptions

Given: Absolutely continuous $F, G : [a, b] \to R$ .

Proof

technique · direct

1.1givenchoose

Absolute continuity Absolute continuity on a compact interval implies continuity, hence $∣ F ∣ \leq M$ and $∣ G ∣ \leq N$ on the compact interval. Choose the two AC moduli with error $ε / (2 \max {M, 1})$ and $ε / (2 \max {N, 1})$ .

2.1step 1.1algebra

For every interval, $∣ F (v) G (v) - F (u) G (u) ∣ \leq M ∣ G (v) - G (u) ∣ + N ∣ F (v) - F (u) ∣$ . Summing over a sufficiently short disjoint family gives $< ε$ .

3.1step 2.1∎

This is the defining condition, also when a factor or the interval is degenerate.

The product of two absolutely continuous functions is absolutely continuous

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources