ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 homotopic maps with different components

Example

On the two-term identity complex $0 \to Z \overset{1_{Z}}{\to} Z \to 0,$ the identity chain map and the zero chain map are chain homotopic, even though their degree- $0$ components are different.

Facts & Assumptions

Given: The two-term identity complex $C_{∙}$ .

[L1]

The previous example supplies a degree- $1$ map $s$ with $1_{C} = d s + s d$ (A contracting homotopy for the two-term identity complex).

[L2]

A chain homotopy satisfies $f - g = d s + s d$ (A chain homotopy).

[L3]

Homotopic maps induce the same map on homology (Chain-homotopic maps induce the same map on homology).

Verification

technique · direct

1.1L1L2givenalgebra

Let $f = 1_{C}$ and $g = 0$ . By [L1], the chosen $s$ satisfies $f - g = 1_{C} = d s + s d,$ so [L2] gives $f ≃ g$ .

2.1L3step 1.1algebra∎

Nevertheless $f_{0} = 1_{Z}$ and $g_{0} = 0$ , so the degree- $0$ components are different. By [L3], these distinct maps still induce the same homology map.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)