ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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.

Two-out-of-three for a diagram of finite complexes

Example

Compare the cone sequences of $\times 2$ and $\times (- 2)$ on $Z [0]$ : $0 \to Z [0] \to Cone (\times 2) \to Z [1] \to 0,$ $0 \to Z [0] \to Cone (\times (- 2)) \to Z [1] \to 0.$ Take the vertical maps to be $- 1_{Z [0]}$ on the left, $1_{Z [1]}$ on the right, and the middle map that is multiplication by $1$ in degree $1$ and by $- 1$ in degree $0$ . The outer maps are obvious quasi-isomorphisms, so the middle one is too.

Facts & Assumptions

Given: The morphism of short exact sequences described in the example.

[L1]

In a morphism of short exact sequences, any two quasi-isomorphisms force the third (Two-out-of-three for quasi-isomorphisms in a short exact sequence diagram).

Verification

technique · direct

1.1L1givenconstruct

The left and right vertical maps are isomorphisms of stalk complexes, hence quasi-isomorphisms. With the left map also equal to $- 1$ , the middle vertical map commutes with the canonical inclusion $j_{n} (y) = (y, 0)$ and the projection to $C [1]$ , so it is a morphism of short exact sequences; it is a chain map because it changes the sign in degree $0$ exactly as needed to compare the differentials $2$ and $- 2$ .

2.1L1step 1.1algebra∎

Both cone complexes have homology $Z / 2$ in degree $0$ and $0$ elsewhere, so the middle vertical map induces an isomorphism on homology. This agrees with the prediction of [L1]: once the outer two maps are quasi-isomorphisms, the third must be as well.

Depends on

Two-out-of-three for quasi-isomorphisms in a short exact sequence diagram

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)