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.

Repeated fixed-level tests do not give an all-times bound

Example

Let $U_{1}, U_{2}, \dots$ be independent random variables, each uniform on $[0, 1]$ , let $F_{t} = σ (U_{1}, \dots, U_{t})$ , and at every time $t \geq 1$ test the same null by rejecting when $U_{t} \leq α$ , for a fixed $α \in (0, 1)$ . Each single-time test has rejection probability $α$ , but for every integer $n \geq 2$ , the probability of at least one rejection by time $n$ is $1 - (1 - α)^{n} > α$ . Marginal level $α$ at each time therefore does not give the same $α$ bound for the event of ever rejecting.

Facts & Assumptions

Given: The probability space carries the independent $[0, 1]$ -uniform random variables in the Example.

[A1]

The variables $U_{1}, U_{2}, \dots$ are mutually independent and each satisfies $P (U_{t} \leq x) = x$ for $0 \leq x \leq 1$ .

Verification

1.1givenA1

For every $t \geq 1$ , [A1] gives $P (U_{t} \leq α) = α$ , so the test at that fixed time has level $α$ .

1.2givenA1algebra

Independence in [A1] gives $P (U_{t} > α for all 1 \leq t \leq n) = \prod_{t = 1}^{n} P (U_{t} > α) = (1 - α)^{n}$ . Taking the complement, $P (\exists t \leq n : U_{t} \leq α) = 1 - (1 - α)^{n}$ .

2.1givenA1step 1.2algebra∎

For $n \geq 2$ , $1 - (1 - α)^{n} = α + (1 - α) (1 - (1 - α)^{n - 1}) > α$ , since both factors in the added term are positive. Thus the pointwise error bounds do not supply the same bound for all tests through time $n$ .

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

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