ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Splicing two short exact sequences

Example

Concatenate two displayed short exact sequences with common middle endpoint and identify the resulting two-fold extension and its derived Ext product.

Facts & Assumptions

Given: Short exact sequences $0 \to N \to E \overset{u}{\to} L \to 0$ and $0 \to L \overset{v}{\to} F \to M \to 0$ .

Verification

technique · direct

1.1givenconstruct

Concatenating the maps gives $0 \to N \to E \overset{v u}{\to} F \to M \to 0$ . Its exactness at $E$ follows from $v$ monic and $\ker u = N$ . At $F$ , $im (v u) = v (im u) = v (L)$ , which is exactly the kernel of $F \to M$ .

2.1step 1.1algebra∎

This exact five-term sequence is the two-fold Yoneda extension obtained by splicing. Under the comparison with derived Ext, its class is the composition of the classes of the two displayed short exact sequences in ${Ext}^{2} (M, N)$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Charles A. Weibel, An Introduction to Homological Algebra, Chapters 3–4 (standard reference, not scraped)