Prove2Me

A refined upper bound for the pairwise reciprocal sum

Proved
WorkbookSource.base_8376

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

lean-workbooksource-checked

If x,y,z>0x,y,z>0 , then 1x+y+1y+z+1z+x≤3(x+y+z)2(xy+yz+zx)−x2+y2+z2−xy−yz−zx4(xy+yz+zx)(x+y+z)\frac{1}{x+y}+\frac{1}{y+z}+\frac{1}{z+x}\le \frac{3\left( x+y+z \right)}{2\left( xy+yz+zx \right)}-\frac{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx}{4\left( xy+yz+zx \right)\left( x+y+z \right)}

Source: InternLM Lean-Workbook, record lean_workbook_8376 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_8376 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (1 / (x + y) + 1 / (y + z) + 1 / (z + x)) ≤ (3 * (x + y + z)) / (2 * (x * y + y * z + z * x)) - (x ^ 2 + y ^ 2 + z ^ 2 - x * y - y * z - z * x) / (4 * (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_8376; Apache-2.0

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