A pairwise-product bound under two quadratic constraints

Proved

WorkbookSource.plus_51825

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

lean-workbooksource-checked

Let $a,b,c$ be three numbers positive real such that: $a^2+ab+b^2 \leq18$ and $b^2 +bc+c^2 \leq 6$ . Prove that: $ab+bc+ca \leq 12$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_51825 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_51825 (a b c: ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a^2 + a * b + b^2 ≤ 18) (hbc : b^2 + b * c + c^2 ≤ 6) : a * b + b * c + c * a ≤ 12   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_51825; Apache-2.0

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