A two-term cubic ratio lower bound

Proved

WorkbookSource.base_55693

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

lean-workbooksource-checked

Let $a,\,b,\,c$ are positive real numbers, prove that $\frac{a^3}{(a+b)^3}+\frac{b^3}{(b+c)^3} \geqslant \frac{5a-c}{8(a+c)}$

Source: InternLM Lean-Workbook, record lean_workbook_55693 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_55693 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) : (a^3 / (a + b)^3 + b^3 / (b + c)^3) ≥ (5 * a - c) / (8 * (a + c))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_55693; Apache-2.0

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