A cubic correction under unit pairwise sum

Proved

WorkbookSource.base_36367

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

lean-workbooksource-checked

Prove that $a^2+b^2+c^2+abc(a+b+c)\ge \frac{4}{3}$ given $a,b,c>0$ and $ab+bc+ca=1$ .

Source: InternLM Lean-Workbook, record lean_workbook_36367 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_36367 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hab : a * b + b * c + c * a = 1) : a ^ 2 + b ^ 2 + c ^ 2 + a * b * c * (a + b + c) ≥ 4 / 3  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_36367; Apache-2.0

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