LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge 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.

Every positive real lies below a dyadic scale less than twice as large

Statement

For every real $a > 0$ , there is an integer $i$ such that $a \leq 2^{i} < 2 a$ .

Facts & Assumptions

Given: A positive real number $a$ .

[F1]

The reals are Archimedean: a natural number can be chosen larger than any fixed positive real (Every complete ordered field is Archimedean).

[F2]

Induction gives $2^{n} \geq n + 1$ for $n \geq 0$ , and integer powers obey $2^{- n} = 1 / 2^{n}$ and $2^{i} = 2 \cdot 2^{i - 1}$ (The principle of mathematical induction, Integer powers $a^{m}$ , Laws of integer exponents).

Proof

technique · direct

Given: A real $a > 0$ .

1.1F1F2givenalgebra

By [F1], choose an integer $n \geq 1$ with $n + 1 > \max {a, 1 / a}$ . By induction from [F2], $2^{n} \geq n + 1 > a$ , while $2^{- n} = 1 / 2^{n} \leq 1 / (n + 1) < a$ .

2.1F2step 1.1algebrachoose∎

In the finite ordered list of integers $- n, - n + 1, \dots, n$ , take the least $i$ with $2^{i} \geq a$ ; it exists because exponent $n$ qualifies and exponent $- n$ does not. Thus $i > - n$ , so $i - 1$ is in the list and fails the defining inequality: $2^{i - 1} < a \leq 2^{i}$ . The power law in [F2] gives $2^{i} = 2 \cdot 2^{i - 1} < 2 a$ .

Depends on

Used by

Correct-scale slab bound for multivariate trimming Proposition

Dependency tree · two levels

21 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

Gábor Lugosi and Shahar Mendelson, Robust Multivariate Mean Estimation: The Optimality of Trimmed Mean, §3.1 (standard reference, not scraped)
Gábor Lugosi, Mean Estimation in High Dimension, §3.2 (standard reference, not scraped)