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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A separating set identifies an unshielded collider

Statement

Let $i - k - j$ be an unshielded triple in a finite DAG and let $S \subseteq V ∖ {i, j}$ d-separate $i$ and $j$ . Then the triple is a v-structure $i \to k \leftarrow j$ if and only if $k \notin S$ .

Statement references

Kalisch and Bühlmann, §2.3

Meek, §2.1

Facts & Assumptions

Given: A finite DAG, an unshielded triple $i - k - j$ , and a separating set $S$ for its endpoints.

[F1]

A path is active given $S$ when its internal noncolliders are outside $S$ and each collider has a descendant in $S$ . Paths, active paths, and d-separation in a finite DAG

[F2]

An unshielded triple and a v-structure have the meanings stated in the CPDAG interface. DAG Markov equivalence, patterns, and CPDAGs

Proof

1.1givenF1F2

If $i \to k \leftarrow j$ , then $k$ is a collider on the length-two path from $i$ to $j$ . If $k \in S$ , it is its own conditioned descendant, so this path is active by [F1], contradicting that $S$ separates the endpoints. Therefore $k \notin S$ .

1.2givenF1F2

If the unshielded triple is not a v-structure, then $k$ is a noncollider on its length-two path. If $k \notin S$ , that path has no conditioned internal noncollider and no collider, so it is active by [F1], again contradicting separation. Hence a noncollider requires $k \in S$ .

2.1step 1.1step 1.2algebra∎

The two implications prove the equivalence.

Depends on

Used by

PC distinguishes a chain from an unshielded collider Example
The PC algorithm recovers the oracle CPDAG Theorem

Dependency tree · one level

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Sources

Kalisch and Bühlmann, Estimating High-Dimensional Directed Acyclic Graphs with the PC-Algorithm, §2.3 (standard reference, not scraped)
Meek, Causal Inference and Causal Explanation with Background Knowledge, §2.1 (standard reference, not scraped)