ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

ex-ext-one-as-a-derived-category-morphism.md

Example

For positive integers $m, n$ , ${Hom}_{D (A b)} (Z / m [0], Z / n [1]) ≅ Z / \gcd (m, n)$ .

Facts & Assumptions

Given: For positive integers $m, n$ , ${Hom}_{D (A b)} (Z / m [0], Z / n [1]) ≅ Z / \gcd (m, n)$ .

[F1]

Ext computed by a supplied projective resolution is derived Hom into the positive shift (Ext is hom in the derived category).

Verification

1.1

Resolve $Z / m$ by $P = (Z \overset{m}{\to} Z)$ in degrees $- 1, 0$ , with its quotient augmentation. Since $m > 0$ the first map is injective, so this is a projective resolution. Hom into $Z / n$ has terms $Z / n$ in degrees zero and one, and differential $- m$ with the cochain Hom convention.

F1algebra

2.1

Degree-one cohomology is $(Z / n) / m (Z / n) = Z / (m Z + n Z) = Z / \gcd (m, n)$ . The Ext comparison identifies it with the claimed Hom group. If $m = 1$ or $n = 1$ the quotient is zero, as required for a zero input module.

F1step 1.1algebra∎

Depends on

Ext is hom in the derived category

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

10.4.7 and 10.7.5, pp. 388, 400 (standard reference, not scraped)