lean_workbook_plus_60407

⚠️ Retired — mistyped fractional exponent

The exponent ^(1/3) in the Lean statement below is natural-number division, not a real exponent, so it collapses to an integer power and the displayed radical is not what is being asserted. Do not import this node or use it as a dependency.

Find all positive real numbers $x, y$ , and $z$ that satisfy both of the following equations.\nxyz = 1\nx^2 + y^2 + z^2 = 4x\sqrt{yz}- 2yz

Why this node was retired

The posted statement is

theorem lean_workbook_plus_60407 (x y z : ℝ) (h₁ : x*y*z = 1) (h₂ : x^2 + y^2 + z^2 = 4*x*Real.sqrt (y*z) - 2*y*z) : x = y ∧ y = z ∧ z = (4:ℝ)^(1/3)   :=  by sorry

A numeric-literal exponent carrying no type ascription is elaborated at type ℕ (via HPow ℝ ℕ ℝ, Monoid.npow), and natural-number division truncates. Each occurrence therefore collapses:

• ^(1/3) → ^0, so the radical is 1 for every base

The recorded counterexample refutes the collapsed statement. It says nothing about the exercise displayed on this page, which is a different proposition.

What a faithful statement would require

Every fractional exponent must be given a real type, e.g. x ^ ((1 : ℝ) / 3) using Real.rpow, or be written with the intended root operation (Real.sqrt, or the signed real cube root where the radicand may be negative). Real powers also need their own domain hypotheses: rpow is only the intended root for a nonnegative base.

No corrected replacement node exists yet, and this target has no record in the public Prove2Me statement audit — the defect above was identified directly from the Lean text.