lean_workbook_plus_34374

⚠️ 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.

The roots of the equation are $b-t,b,b+t$ where $t$ is the common difference. Then we have $b(b-t)(b+t)=d\implies b^2-t^2=\frac{d}{b}\implies t^2=\frac{b^3-d}{b}.$ So the common difference is $\sqrt{\frac{b^3-d}{b}}.$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_34374  (b d : ℝ)
  (t : ℝ)
  (f : ℝ → ℝ)
  (h₀ : 0 < b ∧ 0 < d)
  (h₁ : t^2 = (b^3 - d) / b)
  (h₂ : ∀ x, f x = (x - b) * (x - (b + t)) * (x - (b - t))) :
  f b = d   :=  by sorry

Conclude f(b)=0 for the middle root; the constant-term relation is f(0)=−d. For a nonnegative common difference t and b≠0, the source relation is t=√((b³−d)/b).

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

Proposed corrected statement

Conclude f(b)=0 for the middle root; the constant-term relation is f(0)=−d. For a nonnegative common difference t and b≠0, the source relation is t=√((b³−d)/b).

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.