lean_workbook_plus_57597

⚠️ Retired — mistyped fractional exponent

The exponents ^(1/2), ^(1/5) in the Lean statement below are natural-number division, not a real exponent, so they collapse 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.

Prove the inequality: Let $a$, $b$, and $c$ be positive numbers. Prove that $\sqrt[5]{\frac{a^5 + b^5}{2}} + \sqrt[5]{\frac{b^5 + c^5}{2}} + \sqrt[5]{\frac{c^5 + a^5}{2}} \leq \sqrt{\frac{3(a^3 + b^3 + c^3)(a + b + c)}{ab + bc + ca}}$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_57597 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (a^5 + b^5)^(1/5) + (b^5 + c^5)^(1/5) + (c^5 + a^5)^(1/5) ≤ (3 * (a^3 + b^3 + c^3) * (a + b + c) / (a * b + b * c + c * a))^(1/2)   :=  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:

• ^(1/2) → ^0, so the radical is 1 for every base
• ^(1/5) → ^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.

Proposed corrected statement

Restore the factors 1/2 inside each fifth root and encode genuine real fifth roots, retaining the source right-hand square root and positivity assumptions.

From the public Prove2Me statement audit (wamlat/prove2me-errors). That proposal is natural-language mathematics and is not Lean-verified; it is a specification for a corrected node, not a drop-in replacement.