lean_workbook_plus_20964

⚠️ Retired — specification defect

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

A sequence of functions $f(x)= \begin{cases}0&0<x\leq \frac{a-1}{2a}\\ ax+\frac{1-a}{2}&\frac{a-1}{2a}<x\leq \frac{a+1}{2a}\\ 1&\frac{a+1}{2a}<x<1\\ \end{cases}$ with assumption that $a>1$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_20964 (a : ℝ) (ha : a > 1) (x : ℝ) : ∃ f : ℝ → ℝ, f x = if 0 < x ∧ x ≤ (a - 1) / (2 * a) then 0 else if (a - 1) / (2 * a) < x ∧ x ≤ (a + 1) / (2 * a) then a * x + (1 - a) / (2 * a) else if (a + 1) / (2 * a) < x ∧ x < 1 then 1 else 0   :=  by sorry

The middle formula changes (1−a)/2 to (1−a)/(2a), and ∀x∃f specifies only one value at a time rather than a single family of functions. The source fragment does not state a convergence property.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

For each a>1, define one function f_a on(0,1), equal to0 below (a−1)/(2a), ax+(1−a)/2 on the middle interval, and1 above (a+1)/(2a). State f_a(x) by this formula for every x. Recover the intended property of the family from the full source instead of inventing it.

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.