lean_workbook_plus_70871

⚠️ 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.

Let $a,b,c$ are positive reals which satisfy: $a+b+c=6$ and $$ $abc=4.$ Prove that \n

$$\sqrt[3]{a} + \sqrt{b} +\sqrt{c} \geq 2+\sqrt[3]{4}$$ $$\sqrt[3]{a} + k\sqrt{b} +k\sqrt{c} \geq 2+k\sqrt[3]{4}$$

Where $k\geq 1.$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_70871 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a * b * c = 4) (h : a + b + c = 6) : (a^(1/3) + b^(1/2) + c^(1/2) ≥ 2 + 4^(1/3)) ∧ (∀ k : ℝ, k >= 1 → a^(1/3) + k * b^(1/2) + k * c^(1/2) ≥ 2 + k * 4^(1/3))   :=  by sorry

The untyped fractional exponent is a natural quotient equal to zero; the posted powers become 1 instead of genuine roots.

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.

Confirmed directly from the Lean text: the numeric-literal exponents carry no type ascription, so they elaborate at type ℕ where division truncates (^(1/2) → ^0; ^(1/3) → ^0).

Proposed corrected statement

Restore all displayed roots, using nonnegative real roots for nonnegative radicands and signed real cube roots for potentially negative radicands. Type other fractional exponents explicitly in ℝ. Source task: Let $a,b,c$ are positive reals which satisfy: $a+b+c=6$ and $$ $abc=4.$ Prove that \n

$$\sqrt[3]{a} + \sqrt{b} +\sqrt{c} \geq 2+\sqrt[3]{4}$$ $$\sqrt[3]{a} + k\sqrt{b} +k\sqrt{c} \geq 2+k\sqrt[3]{4}$$

Where $k\geq 1.$

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.

import Mathlib.Analysis.Complex.Basic

theorem lean_workbook_plus_70871 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a * b * c = 4) (h : a + b + c = 6) : (a^(1/3) + b^(1/2) + c^(1/2) ≥ 2 + 4^(1/3)) ∧ (∀ k : ℝ, k >= 1 → a^(1/3) + k * b^(1/2) + k * c^(1/2) ≥ 2 + k * 4^(1/3))   :=  by sorry