ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Functions of disjoint independent coordinate blocks remain independent

Example

Let $X_{0}, X_{1}, X_{2}, X_{3}$ be independent random elements. Put $Y : = (X_{0}, X_{1}), Z : = (X_{2}, X_{3}) .$ If $f$ and $g$ are measurable maps on the targets of $Y$ and $Z$ , then $f (Y)$ and $g (Z)$ are independent.

This is the standard way to pass from coordinate independence to independence of functions built from disjoint coordinate blocks.

Facts & Assumptions

Given: Independent random elements $X_{0}, X_{1}, X_{2}, X_{3}$ and measurable maps $f$ and $g$ as in the Example.

[L1]

Disjoint groups of an independent sigma-algebra family remain independent. (Disjoint groups of an independent sigma-algebra family remain independent)

[L2]

Measurable coordinatewise functions preserve independence. (Measurable coordinatewise functions preserve independence)

[L3]

Independence of random elements is defined through independence of their generated sigma-algebras. (Independent random elements)

Verification

technique · direct

1.1

The sigma-algebras $σ (X_{0}), σ (X_{1}), σ (X_{2}), σ (X_{3})$ are independent by [L3]. Grouping the first two and last two coordinates, [L1] shows that the block sigma-algebras $σ (X_{0}, X_{1})$ and $σ (X_{2}, X_{3})$ are independent. Therefore the block random elements $Y$ and $Z$ are independent.

L1L3

2.1

Applying [L2] to the independent pair $Y, Z$ and the measurable maps $f, g$ gives independence of $f (Y)$ and $g (Z)$ .

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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