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.

The bar differential is equivariant and squares to zero

Statement

For $n \geq 1$ , the maps $d_{n} : B_{n} (G) \to B_{n - 1} (G)$ are $Z [G]$ -linear. For $n \geq 2$ , $d_{n - 1} d_{n} = 0$ , and for $n = 1$ one has $ε d_{1} = 0$ .

Proof

Given: The alternating face maps of the homogeneous bar construction.

1.1

Deleting a coordinate commutes with diagonal left multiplication, so each face and hence $d_{n}$ is $Z [G]$ -linear.

given

2.1

For $n \geq 2$ , each double deletion of positions $i < j$ occurs twice, first as $d_{i} d_{j}$ and then as $d_{j - 1} d_{i}$ , with opposite signs. Pairing these terms proves $d_{n - 1} d_{n} = 0$ . For $n = 1$ , both faces have augmentation one, so $ε (d_{0} - d_{1}) = 0$ .

step 1.1∎

Depends on

The augmented unnormalized homogeneous bar complex

Used by

The first bar differentials Example
Degenerate bar chains are contractible Lemma
The augmented bar complex is exact Lemma
The bar complex is a free resolution of the trivial module Theorem

Dependency tree · two levels

2 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

Löh, Remark 1.2.2 (standard reference, not scraped)