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

A contravariant functor derived via the opposite category

Example

Take the contravariant functor $G = {Hom}_{Z} (-, Z) : A b \to A b .$ Fix supplied projective resolution data $P$ on a class $D$ of abelian groups. Regarded as a covariant functor on ${A b}^{o p}$ , $G$ has right derived objects, for $A \in D$ , $R_{P^{o p}}^{n} G (A) = H^{n} ⁣ (G (P (A)_{d e l})),$ using the opposite-category interpretation of the projective resolutions as injective resolutions in ${A b}^{o p}$ .

Facts & Assumptions

Given: The contravariant functor $G = {Hom}_{Z} (-, Z)$ , supplied projective data $P$ on $D$ , and an object $A \in D$ .

[L1]

Hom is left exact in each variable, so the displayed Hom functor is a standard contravariant example (Hom is left exact in each variable).

[L2]

Contravariant derived functors are derived on the opposite category (Contravariant derived functors are derived on the opposite category).

Verification

technique · direct

1.1

By [L1], $G$ is the sort of contravariant additive functor that later produces Ext-style right derived objects.

L1given

2.1

Apply [L2] to $G$ , $P$ , and $A$ . The projective resolution $P (A)$ in $A b$ is read in ${A b}^{o p}$ as an injective resolution, giving the displayed right derived object there. This makes the variance bookkeeping explicit.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed. (standard reference, not scraped)