Prove2Me

A shifted-product inequality with a sphere constraint

Proved
WorkbookSource.plus_5342

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

lean-workbooksource-checked

Given a,b,c,d>0a,b,c,d > 0 and a2+b2+c2+d2=1. a^2+b^2+c^2+d^2 = 1 . Prove that 4(k−a)(k−b)≥(c+d)2+2(k2−1)4(k-a)(k-b) \geq (c+d)^2+2(k^2-1) Where k∈R.k\in R.

Source: InternLM Lean-Workbook, record lean_workbook_plus_5342 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.plus_5342 (a b c d k : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hd : 0 < d) (h : a^2 + b^2 + c^2 + d^2 = 1) : 4 * (k - a) * (k - b) ≥ (c + d)^2 + 2 * (k^2 - 1)   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_5342; Apache-2.0

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