A product bounds shifted harmonic pairwise products

Proved

WorkbookSource.base_17952

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

lean-workbooksource-checked

Prove that for positive numbers $x$ , $y$ , and $z$ , the inequality $(1+x)(1+y)(1+z) \geq (1+\frac{2xy}{x+y})(1+\frac{2yz}{y+z})(1+\frac{2xz}{z+x})$ holds.

Source: InternLM Lean-Workbook, record lean_workbook_17952 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_17952 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) : (1 + x) * (1 + y) * (1 + z) ≥ (1 + 2 * x * y / (x + y)) * (1 + 2 * y * z / (y + z)) * (1 + 2 * z * x / (z + x))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_17952; Apache-2.0

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