A quadratic lower bound under linked products

Proved

WorkbookSource.plus_36393

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

lean-workbooksource-checked

Let $a,b,c,d$ be reals such that $ab+bc=12$ and $ac+bd=3 .$ Prove that

a^2+b^2+c^2+d^2 \geq \frac{154}{9}

Source: InternLM Lean-Workbook, record lean_workbook_plus_36393 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_36393 (a b c d : ℝ) (hab : a * b + b * c = 12) (hac : a * c + b * d = 3) : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 ≥ 154 / 9   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_36393; Apache-2.0

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