ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

The Hom double complex in low bidegrees

Example

Display K^{0,0}, K^{1,0}, K^{0,1}, and K^{1,1}, label the two unsigned differentials, and compute the signed total differential in degrees zero and one.

Facts & Assumptions

Given: Differentials $d_{P} : P_{1} \to P_{0}$ and $d_{I} : I^{0} \to I^{1}$ of supplied resolutions, and $K^{p, q} = Hom (P_{p}, I^{q})$ .

Verification

technique · direct

1.1

The low square has $K^{0, 0} = Hom (P_{0}, I^{0})$ , $K^{1, 0} = Hom (P_{1}, I^{0})$ , $K^{0, 1} = Hom (P_{0}, I^{1})$ , and $K^{1, 1} = Hom (P_{1}, I^{1})$ . Its unsigned arrows are $d_{h} (f) = f d_{P}$ and $d_{v} (f) = d_{I} f$ , and $d_{h} d_{v} = d_{v} d_{h}$ on each $f$ .

givenalgebra

2.1

With $D ∣_{K^{p, q}} = d_{h} + (- 1)^{p} d_{v}$ , ${Tot}^{0} = K^{0, 0}$ and ${Tot}^{1} = K^{1, 0} \oplus K^{0, 1}$ , so $D (f) = (f d_{P}, d_{I} f)$ . For $(a, b) \in {Tot}^{1}$ , its $K^{1, 1}$ component is $- d_{I} a + b d_{P}$ , displaying the sign which makes $D^{2} = 0$ .

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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 (standard reference, not scraped)