Prove2Me

A cyclic linear-over-squared sum has a symmetric lower bound

Proved
WorkbookSource.base_4249

by wamlart · Sep 13, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let x,y,z>0,x,y,z>0, x(x+y)2+y(y+z)2+z(z+x)2≥1x+y+z+10xyz(x+y)(y+z)(z+x)(x+y+z)\frac{x}{(x+y)^2}+\frac{y}{(y+z)^2}+\frac{z}{(z+x)^2} \ge \frac{1}{x+y+z}+\frac{10xyz}{(x+y)(y+z)(z+x)(x+y+z)}

Source: InternLM Lean-Workbook, record lean_workbook_4249 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_4249 (x y z : ℝ) (hx : x > 0) (hy : y > 0) (hz : z > 0) : (x / (x + y) ^ 2 + y / (y + z) ^ 2 + z / (z + x) ^ 2) ≥ 1 / (x + y + z) + 10 * x * y * z / ((x + y) * (y + z) * (z + x) * (x + y + z))  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_4249; Apache-2.0

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