Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07
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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.

A translation-compatible natural isomorphism of exact functors respects triangles

Statement

If η:(F,ξ)⇒(G,ζ) is a natural isomorphism satisfying ζXηX[1]=ηX[1]ξX, then its components form an isomorphism between the image triangles of every distinguished triangle.

Facts & Assumptions

Given: Exact functors and a translation-compatible natural isomorphism η.

Proof

1.1

Naturality at the first two maps gives the first two commuting squares of the image triangles.

given
2.1

The translation-compatibility equation, followed by naturality at the final map, gives the square into F(X)[1]; componentwise invertibility makes the resulting morphism an isomorphism of triangles.

step 1.1given∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources