A product comparison involving shifted pairwise sums

Proved

WorkbookSource.plus_28964

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

lean-workbooksource-checked

Let $a,$ $b$ and $c$ are non-negative numbers. Prove that

$(a + b)(b + c)(c + a)(1 + a)(1 + b)(1 + c)\geq abc(2 + a + b)(2 + b + c)(2 + c + a).$

Source: InternLM Lean-Workbook, record lean_workbook_plus_28964 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_28964 (a b c : ℝ) (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) : (a + b) * (b + c) * (c + a) * (1 + a) * (1 + b) * (1 + c) ≥ a * b * c * (2 + a + b) * (2 + b + c) * (2 + c + a)   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_28964; Apache-2.0

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