Alphabeta Math
PropositionStatement: Literature-sourcedProof: 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.

Chain homotopy is an equivalence relation

Statement

For fixed chain complexes C∙ and D∙ in an abelian category, the relation of being chain homotopic is an equivalence relation on the set of chain maps C∙→D∙.

Facts & Assumptions

Given: Chain maps f,g,h:C∙→D∙ between complexes in an abelian category.

[L1]

A chain homotopy s:f≃g is a degree-1 family with f−g=ds+sd componentwise (A chain homotopy).

Proof

technique · direct
1.1

Reflexivity holds because the zero degree-1 family satisfies f−f=0=d0+0d, so [L1] gives f≃f.

L1givenalgebra
2.1

If s:f≃g, then [L1] gives f−g=ds+sd, hence g−f=d(−s)+(−s)d, so g≃f. If s:f≃g and t:g≃h, then f−h=(f−g)+(g−h)=d(s+t)+(s+t)d, so f≃h. Therefore the relation is symmetric and transitive as well.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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