ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05‡ 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.

Homology as a homological delta functor

Example

Fix an abelian category $A$ . For chain complexes in $A$ that vanish in negative degrees, the family $H_{n} : {Ch}_{\geq 0} (A) \to A, n \geq 0,$ together with the usual connecting morphisms of homology, is a homological delta functor.

Facts & Assumptions

Given: An abelian category $A$ and its abelian category of nonnegatively graded chain complexes.

[L1]

This page's concrete homology family is the candidate homological delta functor (The homological delta-functor carried by homology of complexes).

[L2]

Homology of complexes satisfies the exactness and naturality axioms (Homology of complexes satisfies the delta-functor naturality and exactness laws).

[L3]

A homological delta functor is exactly such a family of additive functors with natural connecting maps (Homological delta functor).

Verification

technique · direct

1.1

Restrict the integer-indexed homology family of [L1] to complexes that vanish in negative degrees. This full subcategory is closed under kernels and cokernels degreewise, hence is abelian, and its short exact sequences have $H_{- 1} = 0$ .

L1given

2.1

The exactness and naturality axioms required in [L3] are supplied by [L2]; step 1.1 makes the integer-indexed long sequence terminate as $H_{0} (C) \to 0$ . Therefore the restricted family is a homological delta functor indexed by $n \geq 0$ .

L2L3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)
Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)