Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31
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.

Every chain map factors as a cofibration-like inclusion followed by a homotopy equivalence

Statement

Every chain map factors as a degreewise split inclusion followed by a chain-homotopy equivalence.

Facts & Assumptions

Given: A chain map f:C∙→D∙.

[L1]

The mapping-cylinder factorization writes f=p i with i degreewise split monic and p a chain-homotopy equivalence (The mapping cylinder factors a chain map).

Proof

technique · direct
1.1

Apply [L1] to the given map f. This produces a factorization C∙→iCyl⁡(f)∙→pD∙.

L1givenalgebra
2.1

The same theorem already proves that i is a degreewise split inclusion and that p is a chain-homotopy equivalence. That is exactly the asserted factorization.

L1step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

7 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