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

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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.

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A complete ordered field has cofinal canonical integers

Statement

Let $F$ be a Dedekind-complete ordered field with $1_{F} > 0$ . For every $x \in F$ , some canonical integer $n \cdot 1_{F}$ satisfies $x < n \cdot 1_{F}$ .

Facts & Assumptions

Given: A Dedekind-complete ordered field $F$ and an element $x \in F$ .

[A1]

Every nonempty subset of $F$ that is bounded above has a least upper bound in $F$ .

[A2]

The ordered field has $1_{F} > 0$ , addition preserves order, and its canonical integers satisfy $(n + 1) \cdot 1_{F} = n \cdot 1_{F} + 1_{F}$ .

Proof

1.1givenA1A2assume-contrachoose

Suppose the positive canonical integers $N = {n \cdot 1_{F} : n \geq 1}$ were bounded above by some $y \in F$ . By [A1], let $s = \sup N$ . Since $1_{F} > 0$ , $s - 1_{F} < s$ , so $s - 1_{F}$ is not an upper bound; choose $m \geq 1$ with $m \cdot 1_{F} > s - 1_{F}$ .

2.1A2step 1.1contradiction

Adding $1_{F}$ gives $(m + 1) \cdot 1_{F} > s$ , while $(m + 1) \cdot 1_{F} \in N$ and $s$ is an upper bound of $N$ . This contradiction proves that $N$ is unbounded above.

3.1givenstep 2.1assume-contradischarge-contradiction∎

Suppose no canonical integer exceeds $x$ . Then every positive canonical integer is at most $x$ , making $N$ bounded above, contrary to step 2.1. Thus some canonical integer exceeds $x$ .

Used by

A dyadic supertriangle contains the sites and excludes every input circumcircle Lemma

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Truman, Bruckner, and Bruckner, Elementary Real Analysis, 2nd ed., Section 1.7, The Archimedean Property (standard reference, not scraped)