Alphabeta Math
LemmaStatement: 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.

The product of two absolutely continuous functions is absolutely continuous

Statement

If F,G∈AC[a,b], then FG∈AC[a,b].

Facts & Assumptions

Given: Absolutely continuous F,G:[a,b]→R.

Proof

technique · direct
1.1

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

givenchoose
2.1

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

step 1.1algebra
3.1

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

step 2.1∎

Depends on

Used by

Dependency tree · two levels

6 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