Prove2Me

A cyclic product bound by quadratic forms

Proved
WorkbookSource.base_3618

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

prove that 32 (x2+y2)(x2+z2)+32 (y2+z2)(x2+y2)+32 (x2+z2)(y2+z2)≥(3 x+y)2(x+y)(z+2 x+y)+(3 y+z)2(y+z)(2 y+z+x)+(3 z+x)2(z+x)(2 z+x+y)32\, \left( {x}^{2}+{y}^{2} \right) \left( {x}^{2}+{z}^{2} \right) +32\, \left( {y}^{2}+{z}^{2} \right) \left( {x}^{2}+{y}^{2} \right) +32\, \left( {x}^{2}+{z}^{2} \right) \left( {y}^{2}+{z}^{2} \right) \geq \left( 3\,x+y \right) ^{2} \left( x+y \right) \left( z+2\,x+y \right) + \left( 3\,y+z \right) ^{2} \left( y+z \right) \left( 2\,y+z+x \right) + \left( 3\,z+x \right) ^{2} \left( z+x \right) \left( 2\,z+x+y \right) x,y,z>0

Source: InternLM Lean-Workbook, record lean_workbook_3618 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_3618 (x y z : ℝ) : 32 * (x^2 + y^2) * (x^2 + z^2) + 32 * (y^2 + z^2) * (x^2 + y^2) + 32 * (x^2 + z^2) * (y^2 + z^2) ≥ (3 * x + y)^2 * (x + y) * (z + 2 * x + y) + (3 * y + z)^2 * (y + z) * (2 * y + z + x) + (3 * z + x)^2 * (z + x) * (2 * z + x + y)  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_3618; Apache-2.0

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