LemmaStatement: Literature-sourcedProof: 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.

Cone triangles satisfy TR1

Statement

The distinguished cone triangles in $K (A)$ satisfy TR1.

Facts & Assumptions

Given: An additive category $A$ and a chain map $f$ .

Proof

1.1

The standard cone triangle of $f$ completes $f$ , and the definition closes this class under isomorphism.

given

2.1

The cone of $1_{C}$ is contractible and hence isomorphic to zero in $K (A)$ , so its standard triangle gives the TR1 identity triangle.

step 1.1given∎

Depends on

Used by

The homotopy category of an abelian category is triangulated Theorem

Dependency tree · two levels

11 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

Amnon Yekutieli, A Course on Derived Categories, Theorem 9.2.2 (standard reference, not scraped)