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.

Naturality of a connecting map under a map of coefficient sequences

Example

Compare the cone sequences of $\times 2$ and $\times 4$ on $Z [0]$ . There is a morphism of short exact sequences $\begin{array}{ccccccccc} 0 & \to & Z [0] & \to & Cone (\times 2) & \to & Z [1] & \to & 0 \\ ↓ \times 2 & ↓ θ & ↓ 1 \\ 0 & \to & Z [0] & \to & Cone (\times 4) & \to & Z [1] & \to & 0, \end{array}$ where $θ$ is the identity in degree $1$ and multiplication by $2$ in degree $0$ . The connecting square commutes because the top connecting map is $\times 2$ and the bottom one is $\times 4$ .

Facts & Assumptions

Given: The morphism of cone sequences displayed in the example.

[L1]

The homology connecting morphism is natural under morphisms of short exact sequences (Naturality of the homology connecting morphism).

[L2]

In module categories, the connecting morphism is computed by the lift-boundary formula (Elementwise formula for the connecting map in module categories).

Verification

technique · direct

1.1

The map $θ$ is a chain map because the top differential is multiplication by $2$ and the bottom one is multiplication by $4$ , so $4 \cdot 1 = 2 \cdot 2$ on degree $1$ . Thus the displayed diagram is a morphism of short exact sequences.

L1givenconstruct

2.1

By [L2], the top connecting morphism is multiplication by $2$ and the bottom one is multiplication by $4$ . Therefore $(\times 2) \circ \partial_{t o p} = \partial_{b o t t o m} \circ 1$ on $H_{1} (Z [1])$ . This is exactly the commuting square asserted by [L1].

L1L2step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)