ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-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.

Census and singleton SRSWOR boundaries

Example

For $U = {1, 2, 3}$ with values $(1, 2, 4)$ , the population mean is $7 / 3$ and $S_{U}^{2} = 7 / 3$ . At a census ( $n = 3$ ), the sample mean is fixed and its design variance is zero. At $n = 1$ , the sample mean takes values $1, 2, 4$ equally often, with variance $14 / 9$ , agreeing with the SRSWOR formula. The usual sample variance $s^{2} = (n - 1)^{- 1} \sum_{i \in S} (y_{i} - {\overset{ˉ}{y}}_{S})^{2}$ is undefined when $n = 1$ . A population of size $N = 1$ is outside the theorem's $S_{U}^{2}$ convention because its denominator $N - 1$ is zero.

Facts & Assumptions

Given: $U = {1, 2, 3}$ with fixed values $(1, 2, 4)$ .

[F1]

Simple random sampling without replacement: At each admissible sample size, every subset of that size is equally likely.

[F2]

SRSWOR inclusion covariance and sample-mean variance: For $N > 1$ and $1 \leq n \leq N$ , ${Var}_{d} ({\overset{ˉ}{y}}_{S}) = (1 - n / N) S_{U}^{2} / n$ .

Verification

1.1givenF1F2algebra

The population mean is $(1 + 2 + 4) / 3 = 7 / 3$ and $S_{U}^{2} = ((1 - 7 / 3)^{2} + (2 - 7 / 3)^{2} + (4 - 7 / 3)^{2}) / 2 = 7 / 3$ . At $n = 3$ , [F1] gives the single sample $U$ , so ${\overset{ˉ}{y}}_{S} = 7 / 3$ is constant with variance zero; [F2] gives $(1 - 3 / 3) (7 / 3) / 3 = 0$ .

2.1givenF1F2step 1.1algebra∎

At $n = 1$ , [F1] gives sample means $1, 2, 4$ , each with probability $1 / 3$ ; their expectation is $7 / 3$ and variance is $\frac{1}{3} ((1 - 7 / 3)^{2} + (2 - 7 / 3)^{2} + (4 - 7 / 3)^{2}) = 14 / 9$ . Formula [F2] gives $(1 - 1 / 3) (7 / 3) = 14 / 9$ . The denominator $n - 1$ in $s^{2}$ is zero, while $N = 1$ would make the population-variance denominator zero; neither is substituted into an undefined expression.

Depends on

Used by

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