ExampleConstruction: AI-adaptedVerification: 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.

Two-look Pocock-type symmetric spending

Example

Assume AC, take $K = 2$ , $t_{1} = 1 / 2$ , $t_{2} = 1$ , and $α = 1 / 20$ . Use the symmetric two-sided Pocock-type spending curve $A_{P} (0) = 0, A_{P} (t) = α \log (1 + (e - 1) t) (0 < t \leq 1),$ where $e = \exp (1)$ and $\log$ is the natural logarithm. Its first spend is $a_{1} = α \log ((1 + e) / 2)$ . Let $Φ (x) = P_{0} (Z_{1} \leq x)$ be the standard normal CDF and let $q (u)$ be the unique real number with $Φ (q (u)) = u$ . The first symmetric boundary is $c_{1} = q (1 - a_{1} / 2)$ . The second boundary is the unique finite $c_{2}$ satisfying $P_{0} (∣ Z_{1} ∣ \geq c_{1} or ∣ Z_{2} ∣ \geq c_{2}) = α,$ where the canonical covariance gives ${Corr}_{0} (Z_{1}, Z_{2}) = 1 / \sqrt{2}$ . The joint probability of continuing past look 1 and crossing at look 2 is therefore $α - a_{1}$ . Conditional on continuing, the second-look crossing probability is $(α - a_{1}) / (1 - a_{1})$ . The Pocock-type spending curve does not assert constant standardized boundaries; each boundary is calibrated under the joint law.

Facts & Assumptions

Given: AC, the two planned information fractions $t_{1} = 1 / 2, t_{2} = 1$ , $α = 1 / 20$ , the canonical Gaussian null law, and the stated Pocock-type spending curve. Verify the curve is admissible and identify the two two-sided boundaries exactly.

[F1]

Canonical group-sequential alpha-spending design: A design has nondecreasing $A : [0, 1] \to [0, α]$ with $A (0) = 0$ , $A (1) = α$ , and canonical covariance $Σ_{j k} = \sqrt{t_{j} / t_{k}}$ for $j \leq k$ .

[F2]

Recursive group-sequential boundary calibration: At a positive increment the symmetric two-sided boundary solves $P_{0} (R_{k - 1} \cup {∣ Z_{k} ∣ \geq c_{k}}) = A (t_{k})$ .

[F3]

Group-sequential boundaries exist uniquely and spend alpha exactly: Every positive increment has a unique finite boundary and $P_{0} (R_{k}) = A (t_{k})$ at each look.

[F4]

Standard normal and normal laws: The standard normal law has density $ϕ (x) = e^{- x^{2} / 2} / \sqrt{2 π}$ .

[F5]

The standard normal density has total mass one: The density $ϕ$ is positive and has integral one.

[F6]

The real exponential function and the number $e$ by a power series: The real exponential is defined by its power series and $e : = \exp (1)$ .

[F7]

The natural logarithm as the inverse of the exponential function: For $x > 0$ , $\log x$ is the unique real $y$ with $\exp (y) = x$ .

[F8]

Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm: $\log$ is continuous and strictly increasing on $(0, \infty)$ , and $\log 1 = 0$ .

[F9]

The exponential function is strictly increasing: The real exponential is strictly increasing on $R$ .

[F10]

Continuity from below for measures: The measure of the union of an increasing sequence of measurable sets is the supremum of their measures.

[F11]

Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ : A continuous real function on a closed bounded interval attains every value between its endpoint values.

[A1]

The Axiom of Choice: Assume the Axiom of Choice.

Verification

1.1givenF1F6F7F8F9algebra

Since $\exp (0) = 1$ by its power series and $\exp$ is strictly increasing by [F9], $e = \exp (1) > 1$ . Thus $1 + (e - 1) t$ increases from $1$ to $e$ as $t$ goes from $0$ to $1$ . By [F7]–[F8], $A_{P} (0) = 0$ , $A_{P} (1) = α \log e = α$ , and $A_{P}$ is nondecreasing with values in $[0, α]$ . At $t_{1} = 1 / 2$ this gives $a_{1} = α \log ((1 + e) / 2)$ and $0 < a_{1} < α$ .

1.2F4F5F10F11A1algebra

The standard normal CDF satisfies $Φ (y) - Φ (x) = \int_{x}^{y} ϕ (s) d s$ for $x < y$ . By [F4]–[F5], this difference is positive and at most $(y - x) / \sqrt{2 π}$ , so $Φ$ is strictly increasing and continuous. Its limits at $- \infty$ and $+ \infty$ are $0$ and $1$ : [F10] applied to $(- n, \infty)$ gives the lower-tail limit, and [F10] applied to $(- \infty, n]$ gives the upper-tail limit; the density has no point masses. Therefore [F11] gives a unique $q (u)$ for every $u \in (0, 1)$ . In particular $c_{1} = q (1 - a_{1} / 2)$ is finite, positive, and symmetry and atomlessness give $P_{0} (∣ Z_{1} ∣ \geq c_{1}) = a_{1}$ .

2.1givenF1F2F3step 1.1step 1.2algebra∎

The canonical covariance matrix is $Σ = (\begin{array}{cc} 1 & 1 / \sqrt{2} \\ 1 / \sqrt{2} & 1 \end{array})$ , so its determinant is $1 / 2 > 0$ and the correlation is $1 / \sqrt{2}$ . Applying [F2]–[F3] at the second look gives the unique finite $c_{2}$ satisfying the displayed cumulative two-sided crossing equation. The event ${∣ Z_{1} ∣ \geq c_{1}}$ and the continued-path event ${∣ Z_{1} ∣ < c_{1}, ∣ Z_{2} ∣ \geq c_{2}}$ are disjoint and their union is the final rejection event. Hence the continued-path event has joint probability $α - a_{1}$ , and since $P_{0} (∣ Z_{1} ∣ < c_{1}) = 1 - a_{1} > 0$ , its conditional probability given continuation is $(α - a_{1}) / (1 - a_{1})$ . The total rejection probability is $A_{P} (1) = α$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

65 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

David L. DeMets and K. K. Gordon Lan, Interim Analysis: The Alpha Spending Function Approach, Statistics in Medicine 13 (1994), PDF pp. 6–7 (standard reference, not scraped)
Michael J. Grayling, James M. S. Wason, and Adrian P. Mander, Group Sequential Clinical Trial Designs for Normally Distributed Outcome Variables, The Stata Journal 18(2) (2018), PDF pp. 2–4 (standard reference, not scraped)
Giorgos Minas, Multivariate Global Testing and Adaptive Designs, PhD thesis, Chapter 5 §§5.3–5.3.1, printed pp. 72–73 (standard reference, not scraped)