CorollaryStatement: Literature-sourcedProof: 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.

A contractible complex is acyclic

Statement

Every contractible chain complex is acyclic.

Facts & Assumptions

Given: A contractible chain complex $C_{∙}$ .

[L1]

A contractible complex is chain homotopy equivalent to the zero complex (A contractible complex).

[L2]

A chain homotopy equivalence is a quasi-isomorphism (A chain homotopy equivalence is a quasi-isomorphism).

Proof

technique · direct

1.1

By [L1], the unique map $C_{∙} \to 0$ is a chain homotopy equivalence.

L1given

2.1

Then [L2] makes $C_{∙} \to 0$ a quasi-isomorphism. Since the zero complex has zero homology in every degree, $H_{n} (C) = 0$ for all $n$ , so $C_{∙}$ is acyclic.

L2step 1.1algebra∎

Depends on

Used by

An acyclic noncontractible complex from a nonsplit extension Counterexample
FALSE: every acyclic complex is contractible False statement
Zero homology does not make an object zero in the homotopy category Proposition

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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed. (standard reference, not scraped)