lean_workbook_plus_36363

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

Let $a,b,c$ be nonnegative real numbers such that $ab + bc + ca = 3$ . Prove that \n\n $(x^2a^2 + 1)(x^2b^2 + 1)(x^2c^2 + 1) =[x^3abc-x(a+b+c)]^2+[x^2(ab+bc+ca)-1]^2 \n\n$ \n\n $\ge [x^2(ab+bc+ca)-1]^2$

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_36363 (a b c x : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (hab : a * b + b * c + c * a = 3) : (x ^ 2 * a ^ 2 + 1) * (x ^ 2 * b ^ 2 + 1) * (x ^ 2 * c ^ 2 + 1) = (x ^ 3 * a * b * c - x * (a + b + c)) ^ 2 + (x ^ 2 * (a * b + b * c + c * a) - 1) ^ 2 ∧ (x ^ 2 * (a * b + b * c + c * a) - 1) ^ 2 ≤ (x ^ 2 * a ^ 2 + 1) * (x ^ 2 * b ^ 2 + 1) * (x ^ 2 * c ^ 2 + 1)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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