Prove2Me

A pair-product sum bounds cyclic rational differences

Proved
WorkbookSource.base_46175

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

lean-workbooksource-checked

Let a,b,c a,b,c be positive real numbers. Prove that ab+bc+ca≥4a3(b+c−a)(b+c)2+4b3(c+a−b)(c+a)2+4c3(a+b−c)(a+b)2 ab+bc+ca \ge \frac{4a^3(b+c-a)}{(b+c)^2}+\frac{4b^3(c+a-b)}{(c+a)^2}+\frac{4c^3(a+b-c)}{(a+b)^2}

Source: InternLM Lean-Workbook, record lean_workbook_46175 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_46175 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : a * b + b * c + c * a ≥ 4 * a ^ 3 * (b + c - a) / (b + c) ^ 2 + 4 * b ^ 3 * (c + a - b) / (c + a) ^ 2 + 4 * c ^ 3 * (a + b - c) / (a + b) ^ 2  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_46175; Apache-2.0

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