ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-01

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.

An explicit comparison homotopy

Example

On two copies of the standard projective resolution $0 \to Z \overset{n}{\to} Z \to Z / n Z \to 0,$ the degreewise maps $f_{0} = f_{1} = 1$ and $g_{0} = g_{1} = 1 + n$ are two different comparison maps lifting the identity on $Z / n Z$ . They are joined by the explicit chain homotopy $s_{0} = 1 : Z \to Z$ .

Facts & Assumptions

Given: An integer $n \geq 1$ and two copies of the standard resolution.

[L1]

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

Verification

technique · direct

1.1

Because $1 + n \equiv 1 (m o d n)$ and $(1 + n) n = n (1 + n)$ , the degreewise maps $f_{i} = 1$ and $g_{i} = 1 + n$ are both augmentation-preserving chain maps lifting $1_{Z / n Z}$ . They are distinct as maps on $Z$ whenever $n \neq 0$ .

givenalgebra

2.1

Let $s_{0} : Z \to Z$ be the identity map. Then $g_{0} - f_{0} = n = d_{1} s_{0}, g_{1} - f_{1} = n = s_{0} d_{1},$ so $s_{0}$ is a chain homotopy from $f_{∙}$ to $g_{∙}$ . This is the explicit homotopy predicted by [L1].

L1step 1.1discharge-construct∎

Depends on

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

Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)