A pairwise-product bound when the sum equals the squared norm

Proved

WorkbookSource.base_4633

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

lean-workbooksource-checked

If $a,b,c$ are non-negative numbers such that $a^{2}+b^{2}+c^{2}=a+b+c$ , then $ab+bc+ca \geq a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2}$ .

Source: InternLM Lean-Workbook, record lean_workbook_4633 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_4633 {a b c : ℝ} (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) (hab : a + b + c = a^2 + b^2 + c^2) : a * b + b * c + c * a ≥ a^2 * b^2 + b^2 * c^2 + c^2 * a^2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_4633; Apache-2.0

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