Alphabeta Math
CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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.

Events with the same probability need not be independent

Statement refuted

If two events have the same probability, then they are independent.

Facts & Assumptions

Given: The uniform probability space on Ω={1,2,3}.

[L1]

Uniform event probability is event cardinality divided by ∣Ω∣ (The uniform probability space on a nonempty finite set).

[L2]

Independence requires P(A∩B)=P(A)P(B) (Independent events, pairwise independence, and mutual independence of a finite family).

Counterexample

technique · constructive
1.1

Let A={1,2} and B={2,3}. Then P(A)=P(B)=2/3.

L1construct
2.1

Their intersection is {2}, so P(A∩B)=1/3≠4/9=P(A)P(B).

step 1.1L1algebra
3.1

Thus equal event probabilities do not imply independence.

step 2.1L2discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 20 results over 8 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources