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 TR3

Statement

The distinguished cone triangles in $K (A)$ satisfy TR3.

Facts & Assumptions

Given: A commuting square of first maps between two standard cone triangles.

Proof

1.1given

Choose chain-map representatives $a : C \to C^{'}$ and $b : D \to D^{'}$ for the two vertical arrows. Commutativity in $K (A)$ means that there is a chain homotopy $H$ with $b f - f^{'} a = d H + H d .$ With the cone convention of Distinguished cone triangle in the homotopy category, the matrix $(\begin{array}{cc} b & H \\ 0 & a [1] \end{array}) : Cone (f) ⟶ Cone (f^{'})$ is a chain map.

2.1step 1.1given∎

Its composites with the cone inclusion and projection agree in $K (A)$ with the prescribed vertical arrows (the possible $H$ -terms are precisely homotopies). Thus its homotopy class is the required third component of a morphism of triangles; isomorphic replacements of standard cone triangles preserve the conclusion.

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Lemma 13.9.2