lean_workbook_plus_31033

⚠️ 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 $a,b,c,d$ be positive real numbers. \n\n (a) If $(a+b+c+d)(\frac 1{a}+\frac 1{b}+\frac 1{c}+\frac 1{d}) \ge 10+4\sqrt{6}$ , then \n\n $\sqrt{ (a+b+c+d)(\frac 1{a}+\frac 1{b}+\frac 1{c}+\frac 1{d})}\ge 2+\sqrt[4]{(a^{2}+b^{2}+c^{2}+d^{2})(\frac 1{a^{2}}+\frac 1{b^{2}}+\frac 1{c^{2}}+\frac 1{d^{2}})}$ , \n\n with equality for $a=b=\sqrt{cd}$ or any symmetric permutation; \n\n (b) If $(a+b+c+d)(\frac 1{a}+\frac 1{b}+\frac 1{c}+\frac 1{d}) \le 10+4\sqrt{6}$ , then \n\n $(a+b+c+d)(\frac 1{a}+\frac 1{b}+\frac 1{c}+\frac 1{d})-10 \ge \sqrt{3(a^{2}+b^{2}+c^{2}+d^{2})(\frac 1{a^{2}}+\frac 1{b^{2}}+\frac 1{c^{2}}+\frac 1{d^{2}})-12}$ , \n\n with equality for $a=b=c$ or any symmetric permutation; \n\n (c) If $(a+b+c+d)(\frac 1{a}+\frac 1{b}+\frac 1{c}+\frac 1{d})=10+4\sqrt{6}$ , then \n $(a^{2}+b^{2}+c^{2}+d^{2})(\frac 1{a^{2}}+\frac 1{b^{2}}+\frac 1{c^{2}}+\frac 1{d^{2}}) \le 36$ , \n\n with equality for $a=b=\sqrt{cd}$ or any symmetric permutation, and again for $a=b=c$ or any symmetric permutation.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_31033 (a b c d : ℝ) : (a + b + c + d) * (1 / a + 1 / b + 1 / c + 1 / d) ≥ 10 + 4 * Real.sqrt 6 → Real.sqrt ((a + b + c + d) * (1 / a + 1 / b + 1 / c + 1 / d)) ≥ 2 + (a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2) ^ (1 / 4) * (1 / a ^ 2 + 1 / b ^ 2 + 1 / c ^ 2 + 1 / d ^ 2) ^ (1 / 4)   :=  by sorry

The target loses both fourth-root factors and asks only sqrt(P)≥3.

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 exponent carries no type ascription, so it elaborates at type ℕ where division truncates (^ (1 / 4) → ^0).

Proposed corrected statement

Restore a,b,c,d>0 and use the genuine real fourth root of the product of the two square sums.

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.