LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge 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.

Degenerate bar chains are contractible

Statement

The degenerate bar chains form a contractible subcomplex of the unnormalized bar complex.

Proof

Given: The span $D$ of homogeneous bars with two consecutive equal vertices.

1.1

A face of a tuple with $g_{i} = g_{i + 1}$ is again degenerate, except for the two faces deleting $g_{i}$ or $g_{i + 1}$ ; those give the same remaining tuple with opposite signs. Hence $d D \subseteq D$ .

given

2.1

Filter $D_{n}$ by the least index at which two consecutive vertices agree. On each successive quotient, the signed degeneracy that repeats the vertex at that index contracts the quotient; the simplicial identities give $s d + d s = 1$ . Combining these homotopies along the finite filtration $0 \leq i < n$ gives a contraction of $D$ .

step 1.1∎

Depends on

Used by

Normalized and unnormalized bars are homotopy equivalent Theorem

Dependency tree · two levels

3 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

Weibel, §6.5 (standard reference, not scraped)