ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

$x^{5} - x - 1$ has a real root

Example

The polynomial $f (x) = x^{5} - x - 1$ has a real root.

Facts & Assumptions

Given: The polynomial $f (x) = x^{5} - x - 1$ .

[L1]

Every odd-degree real polynomial has a real root (Every odd-degree real polynomial has a real root).

Verification

technique · direct

1.1givenalgebra

Direct evaluation gives $f (1) = 1 - 1 - 1 = - 1 and f (2) = 32 - 2 - 1 = 29.$

2.1step 1.1L1

The sign change in step 1.1 is exactly the kind of endpoint behaviour used in the proof of [L1]. Since $f$ has odd degree, [L1] gives a real root.

3.1step 2.1∎

Thus $x^{5} - x - 1$ is a concrete odd-degree polynomial with a real zero.

Depends on

Every odd-degree real polynomial has a real root

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

Keith Conrad, Applications of Galois Theory, Theorem 2.1 (standard reference, not scraped)

x5−x−1x5−x−1 has a real root