False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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.

FALSE: every lambda-system is closed under finite intersections

Statement

Every lambda-system is closed under finite intersections.

Facts & Assumptions

Given: The set $X : = {1, 2, 3, 4}$ and the family $D$ consisting of $\emptyset$ , $X$ , and all two-element subsets of $X$ .

[L1]

A lambda-system contains $X$ , is closed under relative differences, and is closed under increasing countable unions (Lambda-systems, or Dynkin systems).

[L2]

A pi-system is closed under binary intersections (Pi-systems).

Refutation

technique · counterexample

1.1L1algebra

The family $D$ contains $X$ and is closed under complements. Apart from $\emptyset \subseteq X$ , its proper containments are $\emptyset \subseteq A \subseteq X$ for a two-element set $A$ , and every corresponding difference remains in $D$ ; every increasing sequence in the finite family stabilizes. Thus $D$ is a lambda-system by [L1].

2.1step 1.1L2∎

Both ${1, 2}$ and ${1, 3}$ lie in $D$ , but their intersection ${1}$ does not. Hence $D$ is not a pi-system by [L2], refuting the statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

2 results within one dependency step 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

A. Dembo, Probability Theory lecture notes, Proposition 1.1.37 (standard reference, not scraped)