Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30‡ rests on unproved material (inherited)
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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The nine lemma follows from the snake lemma

Statement

The nine lemma can be proved by applying the snake lemma to the standard quotient diagram attached to a commutative 3×3 diagram with short exact columns.

Facts & Assumptions

Given: A commutative 3×3 diagram with short exact columns and middle row short exact.

[L1]

The snake lemma supplies the exact six-term sequence for a morphism of short exact sequences (Snake lemma in an abelian category).

Proof

technique · direct
1.1

Collapse the first two rows of the 3×3 diagram to their quotient row. The short exact columns identify the needed kernels and cokernels of that quotient diagram with the two outer rows of the original 3×3 picture.

L1givenconstruct
2.1

Applying [L1] to that quotient diagram yields a snake sequence whose endpoint exactness is exactly the missing exactness of the remaining outer row. Running the same argument in the opposite direction gives the converse implication.

L1step 1.1
3.1

Therefore the nine lemma is a direct consequence of the snake lemma.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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