A shifted cyclic pair-product ratio upper bound at fixed sum three

Proved

WorkbookSource.base_39214

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

lean-workbooksource-checked

Let $x,y,z>0,x+y+z=3$ ,prove that: $\frac{1+xy}{2+x+xy}+\frac{1+yz}{2+y+yz}+\frac{1+xz}{2+z+xz} \le \frac{3}{2}$

Source: InternLM Lean-Workbook, record lean_workbook_39214 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_39214 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 3) : (1 + x * y) / (2 + x + x * y) + (1 + y * z) / (2 + y + y * z) + (1 + z * x) / (2 + z + z * x) ≤ 3 / 2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_39214; Apache-2.0

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