An absolute-difference inequality with sharp constant four thirds

Proved

WorkbookSource.base_80

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Let $a,\ b,\ c$ be real numbers. Prove that $|a-b|+|b-c|+|c-a|+ab+bc+ca\leq a^{2}+b^{2}+c^{2}+\frac{4}{3}$ true too.

Source: InternLM Lean-Workbook, record lean_workbook_80 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_80 (a b c : ℝ) : |a - b| + |b - c| + |c - a| + a * b + b * c + c * a ≤ a ^ 2 + b ^ 2 + c ^ 2 + (4:ℝ) / 3  :=  by sorry

Source