ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-13

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.

The union bound can be strict for overlapping events

Example

On the uniform four-point space, let $A = {1, 2}$ , $B = {2, 3}$ , and $C = {2, 4}$ . Then the three events cover the space, but the sum of their probabilities is $3 / 2$ .

Facts & Assumptions

Given: The uniform space and events in the Example.

[L1]

Uniform event probabilities are cardinality ratios (The uniform probability space on a nonempty finite set).

[L2]

Two-event inclusion-exclusion subtracts the probability of the overlap (Two-event inclusion-exclusion: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ ).

[L3]

The finite union bound compares union probability with the sum of marginal probabilities (The finite union bound).

Verification

technique · direct

1.1

Each event has probability $1 / 2$ , while $A \cup B \cup C = Ω$ has probability $1$ .

2.1

Thus [L3] reads $1 \leq 3 / 2$ , and the inequality is strict.

step 1.1L3algebra

3.1

The common outcome $2$ is counted in all three marginal probabilities; [L2] displays this overlap correction already for each pair.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 6.2.1 (standard reference, not scraped)