ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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

1.1

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 $- 2$ .

L1givenconstruct

2.1

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.

L1step 1.1algebra∎

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)