ExampleConstruction: AI-adaptedVerification: 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.

A p-adic integer is encoded by a compatible sequence of residue digits

Example

For fixed digits $a_{0}, a_{1}, a_{2}, \dots \in {0, \dots, p - 1}$ , the sequence

$x_{n} : = a_{0} + a_{1} p + \dots + a_{n - 1} p^{n - 1} (m o d p^{n})$

defines an element of $Z_{p}$ .

Facts & Assumptions

Given: Digits $a_{i} \in {0, \dots, p - 1}$ for $i \geq 0$ .

[L1]

An element of $Z_{p}$ is a compatible residue-class tuple (The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod p^n).

Verification

technique · direct

1.1

For each $n \geq 1$ , reducing $x_{n + 1}$ modulo $p^{n}$ removes only the final term $a_{n} p^{n}$ , so $x_{n + 1} m o d p^{n} = x_{n}$ . Thus the tuple $(x_{n})_{n \geq 1}$ is compatible.

givenalgebra

2.1

By [L1], every compatible tuple defines an element of $Z_{p}$ . Hence the displayed digit data encodes a $p$ -adic integer.

L1step 1.1∎

Depends on

The p-adic integers are the compatible residue-class tuples in the inverse limit of Z mod p^n

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Jordan Bell, Explicit construction of the p-adic numbers (standard reference, not scraped)