A squared pairwise-product bound under lower bounds

Proved

WorkbookSource.plus_75875

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

lean-workbooksource-checked

Prove that for positive numbers a, b, and c such that a + b + c = 3 and a, b, c are greater than or equal to 2/3, the following inequality holds: $\sum_{cyc} (ab)^2 \ge \sum_{cyc} ab$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_75875 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_75875 {a b c : ℝ} (ha : 2/3 ≤ a) (hb : 2/3 ≤ b) (hc : 2/3 ≤ c) (hab : a + b + c = 3) : (a * b) ^ 2 + (b * c) ^ 2 + (c * a) ^ 2 ≥ a * b + b * c + c * a   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_75875; Apache-2.0

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