ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Hensel lifting a simple root of X squared minus 2 in Z_7

Example

The congruence $x^{2} \equiv 2 (m o d 7)$ lifts to a root of $X^{2} - 2$ in $Z_{7}$ .

Facts & Assumptions

Given: The polynomial $f (X) = X^{2} - 2$ over $Z_{7}$ .

[L1]

Simple roots lift uniquely, and Newton iteration computes the lifted root (Simple roots lift uniquely in Z_p, Newton's criterion in Q_p).

Verification

technique · direct

1.1

The residue class $3$ satisfies $3^{2} - 2 = 7 \equiv 0 (m o d 7)$ , and $f^{'} (3) = 6 \equiv̸ 0 (m o d 7)$ . By [L1], there is a unique root $a \in Z_{7}$ with $a \equiv 3 (m o d 7)$ .

L1givenalgebra

2.1

The Newton step from $a_{0} = 3$ is $a_{1} = 3 - \frac{7}{6},$ which is well defined in $Q_{7}$ and already lies in the same residue class modulo $7$ ; iterating stays in that class and converges to the lifted root from step 1.1.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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, Hensel's Lemma (standard reference, not scraped)