Prove2Me

A normalized triple product bounded by symmetric quadratic forms

Proved
WorkbookSource.base_14234

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

lean-workbooksource-checked

Prove that for all positive real numbers x, y, and z, the following inequality holds: 8xyz(x+y)(y+z)(z+x)≤3(x2+y2+z2+3(xy+yz+zx))4(x+y+z)2\frac{8xyz}{(x+y)(y+z)(z+x)}\le \frac{3(x^{2}+y^{2}+z^{2}+3(xy+yz+zx))}{4(x+y+z)^{2}}

Source: InternLM Lean-Workbook, record lean_workbook_14234 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_14234 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (8 * x * y * z) / (x + y) / (y + z) / (z + x) ≤ (3 * (x ^ 2 + y ^ 2 + z ^ 2 + 3 * (x * y + y * z + z * x))) / (4 * (x + y + z) ^ 2)  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_14234; Apache-2.0

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