LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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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.

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The inductive horseshoe step

Statement

Suppose $0 \to K_{n}^{'} \to K_{n} \to K_{n}^{''} \to 0$ is the short exact sequence of current kernels arising in the horseshoe construction. If the tails of projective resolutions of $K_{n}^{'}$ and $K_{n}^{''}$ are already chosen, then one more degree of the horseshoe construction produces a projective object $P_{n + 1} = P_{n + 1}^{'} \oplus P_{n + 1}^{''}$ surjecting onto $K_{n}$ and a new short exact sequence of next kernels.

Facts & Assumptions

Given: The current short exact kernel sequence and the next projective terms of the two side resolutions.

[L1]

The degree-zero horseshoe construction produces the next surjection from a direct sum of projectives (The degree-zero horseshoe lift).

[L2]

The kernel of that new surjection again sits in a short exact sequence with the two side kernels (The horseshoe kernel fits into a short exact sequence).

Proof

technique · direct

1.1L1givenconstruct

Apply [L1] to the short exact sequence $0 \to K_{n}^{'} \to K_{n} \to K_{n}^{''} \to 0$ and to the degree-zero tails of the already chosen projective resolutions of $K_{n}^{'}$ and $K_{n}^{''}$ . This produces a projective object $P_{n + 1} = P_{n + 1}^{'} \oplus P_{n + 1}^{''}$ together with an epimorphism $P_{n + 1} ↠ K_{n}$ .

2.1L2step 1.1∎

Applying [L2] to that new surjection gives the next short exact sequence of kernels. Therefore the horseshoe construction advances by one degree.

The inductive horseshoe step

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources