A shifted reciprocal sum and pair-product upper bound

Proved

WorkbookSource.plus_72054

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

lean-workbooksource-checked

Let $x,y,z$ are positive real numbers and satisfying $x+y+z=3$ . Prove: $\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}+\frac{xy+yz+zx}{2}\leq 3$

Source: InternLM Lean-Workbook, record lean_workbook_plus_72054 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_72054 (x y z : ℝ) (hx : 0 < x) (hy : 0 < y) (hz : 0 < z) (h : x + y + z = 3) : 1 / (x + 1) + 1 / (y + 1) + 1 / (z + 1) + (x * y + y * z + z * x) / 2 ≤ 3   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_72054; Apache-2.0

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