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.

Two resolution data and their change isomorphism

Example

Assume the Axiom of Dependent Choice. Let $P, Q$ be two supplied projective resolution data and $I, J$ two supplied injective resolution data on the same class $D$ in an abelian category $A$ , and let $F : A \to B$ be an additive functor to an abelian category. For every object $A \in D$ and every degree $n$ , the page's change-of-data theorems produce isomorphisms $L_{n}^{P} F (A) \tilde{\to} L_{n}^{Q} F (A) and R_{I}^{n} F (A) \tilde{\to} R_{J}^{n} F (A),$ natural in $A$ .

Facts & Assumptions

Given: The Axiom of Dependent Choice, the supplied data $P, Q, I, J$ , the additive functor $F$ , an object $A \in D$ , and an integer $n$ .

[L1]

Two supplied projective resolution data define naturally isomorphic left derived functors (Two supplied projective resolution data define naturally isomorphic left derived functors).

[L2]

Two supplied injective resolution data define naturally isomorphic right derived functors (Two supplied injective resolution data define naturally isomorphic right derived functors).

Verification

technique · direct

1.1

Apply [L1] at the object $A$ . This gives the projective-side change isomorphism $L_{n}^{P} F (A) \tilde{\to} L_{n}^{Q} F (A)$ , natural in $A$ .

L1given

2.1

Apply [L2] at the same object $A$ . This gives the injective-side change isomorphism $R_{I}^{n} F (A) \tilde{\to} R_{J}^{n} F (A)$ , also natural in $A$ .

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)