lean_workbook_plus_4472

⚠️ Retired — mistyped fractional exponent

The exponents ^ (1 / 3), ^ (2 / 3), ^ (4 / 3), ^ (5 / 3), ^ (10 / 3) 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 that $4\sum x^{4} + 4\sum x^{\frac {10}{3}}y^{\frac {1}{3}}z^{\frac {1}{3}} \ge 6x^{\frac {4}{3}}y^{\frac {4}{3}} z^{\frac {4}{3}} + 3\sum x^{4}y^{4}z + 3\sum x^{\frac {5}{3}}y^{\frac {5}{3}}z^{\frac {2}{3}}$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_4472 : ∀ x y z : ℝ, 4 * (x ^ 4 + y ^ 4 + z ^ 4) + 4 * (x ^ (10 / 3) * y ^ (1 / 3) * z ^ (1 / 3) + x ^ (1 / 3) * y ^ (10 / 3) * z ^ (1 / 3) + x ^ (1 / 3) * y ^ (1 / 3) * z ^ (10 / 3)) ≥ 6 * (x ^ (4 / 3) * y ^ (4 / 3) * z ^ (4 / 3)) + 3 * (x ^ 4 * y ^ 4 * z + x ^ 4 * y ^ 4 * z + x ^ 4 * y ^ 4 * z) + 3 * (x ^ (5 / 3) * y ^ (5 / 3) * z ^ (2 / 3) + x ^ (5 / 3) * y ^ (5 / 3) * z ^ (2 / 3) + x ^ (5 / 3) * y ^ (5 / 3) * z ^ (2 / 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
• ^ (2 / 3) → ^0, so the radical is 1 for every base
• ^ (4 / 3) → ^1, i.e. the 4/3 power is replaced by the 1th power
• ^ (5 / 3) → ^1, i.e. the 5/3 power is replaced by the 1th power
• ^ (10 / 3) → ^3, i.e. the 10/3 power is replaced by the 3th power

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

Use real fractional exponents and expand each sum with its intended distinct cyclic or symmetric terms. Recover the apparent degree9 term or normalization from the original source.

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.