LemmaStatement: Literature-sourcedProof: 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.

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A polyhedral event restricts the contrast to a computable interval

Statement

In the setup of Polyhedral selection event and contrast interval, for every $y \in R^{n}$ , $A y \leq b ⟺ G (Z (y)) and L (Z (y)) \leq T (y) \leq U (Z (y)) .$ The functions $L, U$ are Borel measurable as extended-real-valued functions, and the set of pairs $(t, z)$ satisfying the right-hand side is Borel.

Facts & Assumptions

Given: Finite $n, m$ , fixed $A, b, η \neq 0$ , and the notation of the definition.

[F1]

The contrast decomposition is $y = c_{η} T (y) + Z (y)$ with $c_{η} = η / ∥ η ∥^{2}$ . Polyhedral selection event and contrast interval

[F2]

Each row has coefficient $r_{j} = a_{j}^{T} c_{η}$ and offset $d_{j} (z) = b_{j} - a_{j}^{T} z$ ; the sign of $r_{j}$ determines its upper, lower, or zero-row feasibility condition in the definitions of $L, U, G$ . Polyhedral selection event and contrast interval

Proof

1.1F1algebra

Since $P_{η} y = c_{η} (η^{T} y) = c_{η} T (y)$ and $Z (y) = (I_{n} - P_{η}) y$ , we have $y = c_{η} T (y) + Z (y)$ .

2.1F1F2step 1.1casesalgebra

For each row $j$ , the inequality $a_{j}^{T} y \leq b_{j}$ becomes $r_{j} T (y) \leq d_{j} (Z (y))$ . If $r_{j} > 0$ , division gives $T (y) \leq d_{j} (Z (y)) / r_{j}$ ; if $r_{j} < 0$ , division reverses the inequality and gives $T (y) \geq d_{j} (Z (y)) / r_{j}$ ; if $r_{j} = 0$ , the row is feasible exactly when $d_{j} (Z (y)) \geq 0$ . Intersecting the finitely many row conditions is therefore equivalent to $G (Z (y))$ , $T (y) \geq L (Z (y))$ , and $T (y) \leq U (Z (y))$ , with the empty-index conventions in [F2].

3.1F2step 2.1algebra∎

Each $d_{j}$ is affine and hence continuous. The finite maximum defining $L$ and finite minimum defining $U$ are Borel, with the constant infinite values in the empty-index cases; $G$ is a finite intersection of closed half-spaces. Thus the fiber set is Borel. The rowwise equivalence in step 2.1 proves both implications, including zero coefficients and empty or degenerate intervals.

Depends on

Polyhedral selection event and contrast interval

Used by

The truncated normal selective pivot is conditionally uniform Theorem

Dependency tree · two levels

2 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

Jason D. Lee, Dennis L. Sun, Yuekai Sun, and Jonathan E. Taylor, Exact Post-Selection Inference, with Application to the Lasso (standard reference, not scraped)