lean_workbook_plus_14813

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Let $P_n$ denote the number of ways to form these subsets for $n$ elements. $P_1=2$ . Note that given an exhaustive list of all possibilities for a set of $n$ elements, we have three options to make them into a set of $n+1$ elements: we can add our $n+1$ th element to the first set, we can add the $n+1$ th element to the second set, or we can add it to both. However, if our two sets are exactly the same, then adding $n+1$ th element to the first set is the same thing as adding $n+1$ th element to the second set, so we overcount by $1$ this way. Therefore, $P_{n+1}=3P_n-1$ , and our answer is 365.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_14813  (p : ℕ → ℕ)
  (h₀ : p 1 = 2)
  (h₁ : ∀ n, p (n + 1) = 3 * p n - 1) :
  p 8 = 365   :=  by sorry

The formal statement changes or incorrectly supplies part of the source problem; the accepted note identifies a concrete witness or definitional mechanism.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For P₁=2 and P_{n+1}=3P_n−1, prove P_n=(3^n+1)/2 for n≥1 and P₈=3281. The value 365 is P₆, not P₈.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.