ExampleConstruction: AI-generatedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Independence of two comparison lifts on homology

Example

Assume the Axiom of Dependent Choice. Let $P$ be a supplied projective resolution datum on a class $D$ in an abelian category $A$ , let $F : A \to B$ be an additive functor to an abelian category, let $u : A \to B$ be a morphism with $A, B \in D$ , and let ${\tilde{u}}_{∙}, {\hat{u}}_{∙} : P_{∙} (A) \to P_{∙} (B)$ be two comparison lifts between chosen projective resolutions. Then for every $n$ they induce the same map $L_{n}^{P} F (A) \to L_{n}^{P} F (B) .$ This is what makes the left derived map well defined.

Facts & Assumptions

Given: The supplied datum $P$ , additive functor $F$ , morphism $u : A \to B$ with $A, B \in D$ , and two comparison lifts ${\tilde{u}}_{∙}, {\hat{u}}_{∙}$ .

[L1]

Two comparison maps lifting the same morphism are chain-homotopic (Projective comparison maps are unique up to chain homotopy).

[L2]

The induced homology map is independent of the chosen comparison lift (The induced homology map is independent of the chosen comparison lift).

Verification

technique · direct

1.1

By [L1], the two displayed lifts are chain-homotopic.

L1given

2.1

Apply [L2] to those two lifts. It follows that they induce the same map on every left derived object $L_{n}^{P} F (A) \to L_{n}^{P} F (B)$ .

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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, An Introduction to Homological Algebra, Chapter 2 `Derived Functors` (standard reference, not scraped)