TheoremStatement: 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.

Long exact sequence in group homology

Statement

Assume the Axiom of Dependent Choice and fix supplied projective resolution data on all left $G$ -modules. A short exact sequence $0 \to A \to B \to C \to 0$ of left $G$ -modules induces a natural long exact sequence $\dots \to H_{1} (G; C) \to H_{0} (G; A) \to H_{0} (G; B) \to H_{0} (G; C) \to 0.$

Proof

Given: The displayed short exact sequence.

1.1given

Coinvariants are right exact. Under the stated Choice and supplied-resolution hypotheses, Left derived functors form a homological delta functor makes their left derived functors a homological delta functor.

2.1step 1.1∎

Its connector has degree $H_{n} (G; C) \to H_{n - 1} (G; A)$ , giving the stated order after substituting the definition of $H_{n}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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, §6.1 (standard reference, not scraped)