ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30‡ 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.

Euler-Poincare for a finite complex

Example

Consider the split exact three-term complex $0 \to Z \overset{i}{\to} Z \oplus Z \overset{p}{\to} Z \to 0,$ with $i (a) = (a, 0)$ and $p (a, b) = b$ . Then $χ (C) = 1 - 2 + 1 = 0 = \sum_{n} (- 1)^{n} rank (H_{n} (C)) .$

Facts & Assumptions

Given: The split exact complex just displayed.

[L1]

$A b$ is an abelian category (Abelian groups form an abelian category).

[L2]

The Euler-Poincare formula holds for bounded complexes of finite-rank free abelian groups with finite-rank free homology (Euler-Poincare formula for finite free complexes).

Verification

technique · direct

1.1

The complex is split exact, hence acyclic, so every homology group is zero. Its chain groups have ranks $1$ , $2$ , and $1$ . Therefore $χ (C) = 1 - 2 + 1 = 0,$ while the alternating sum of homology ranks is also $0$ .

L1givenalgebra

2.1

This agrees with the theorem [L2], so the example is a direct Euler-Poincare computation in a finite free complex.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

P. Hekmati, Homological Algebra, section 3.1 (standard reference, not scraped)