LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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 nonempty set of positive integers has a least member

Statement

Let $S \subseteq N_{\geq 1}$ be nonempty. Then there is a unique $n_{*} \in S$ such that $n_{*} \leq n$ for every $n \in S$ .

Facts & Assumptions

Given: A nonempty set $S \subseteq N_{\geq 1}$ .

[F1]

The positive integers are a subset of $N$ . The natural numbers $N$ (von Neumann)

[F2]

Every nonempty subset of $N$ has a least element. The well-ordering principle

Proof

1.1givenF1

By [F1], $S \subseteq N$ ; by the Given, $S$ is nonempty.

2.1step 1.1F2algebra∎

Apply [F2] to obtain a least member $n_{*} \in S$ . If $n_{*}^{'}$ were another least member, then $n_{*} \leq n_{*}^{'}$ and $n_{*}^{'} \leq n_{*}$ , hence $n_{*} = n_{*}^{'}$ by antisymmetry.

Depends on

Used by

Dependency tree · two levels

14 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

Gary W. Oehlert, A First Course in Design and Analysis of Experiments, Chapter 7, §7.1, printed pp. 149–150 (standard reference, not scraped)