Alphabeta Math
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.

An absolutely continuous function after a monotone Lipschitz function is absolutely continuous

Statement

If h∈AC[c,d] and g:[a,b]→[c,d] is monotone and L-Lipschitz, then h∘g∈AC[a,b].

Facts & Assumptions

Given: h∈AC[c,d] and monotone L-Lipschitz g.

Proof

technique · direct
1.1

Choose the AC modulus η of h from Absolute continuity on a compact interval and set δ=η/L for L>0.

givenchoose
2.1

Images under monotone g of a disjoint ordered interval family have disjoint interiors, and their total length is at most L times the original total length.

step 1.1algebra
3.1

Apply the AC condition for h to these image intervals. If L=0, g and the composite are constant.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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