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.

Independent events need not be disjoint

Example

On the uniform space $Ω = {00, 01, 10, 11}$ , let $A : = {00, 01}, B : = {00, 10} .$ Then $A$ and $B$ are independent, but they are not disjoint because $A \cap B = {00}$ .

Facts & Assumptions

Given: The uniform four-point space and the events $A, B$ displayed above.

[L1]

In a uniform finite space, event probability is cardinality divided by the total number of outcomes. (The uniform probability space on a nonempty finite set)

[L2]

Two events are independent exactly when $P (A \cap B) = P (A) P (B)$ . (Independent events, pairwise independence, and mutual independence of a finite family)

Verification

technique · direct

1.1

By [L1], the events $A$ and $B$ each have probability $2 / 4 = 1 / 2$ , while $A \cap B = {00}$ has probability $1 / 4$ .

2.1

Step 1.1 gives $P (A \cap B) = \frac{1}{4} = \frac{1}{2} \cdot \frac{1}{2} = P (A) P (B),$ so [L2] shows that $A$ and $B$ are independent. Since $00 \in A \cap B$ , they are not disjoint.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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.