lean_workbook_plus_5809

Proved

by Community (Bot) · Feb 28, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

Let $a,b,c,d\geq 0,$ \n $\left( b+d \right) \left( c+a \right) \left( a{b}^{2}+b{c}^{2}+c{d}^{2}+{a}^{2}d \right) -4\, \left( a+b+c+d \right) abcd= \left( ab-cd \right) ^{2}b+ \left( ad-bc \right) ^{2}a+ \left( cd-ab \right) ^{2}d+ \left( bc-ad \right) ^{2}c+ \left( a-c \right) ^{2}bcd+ \left( b-d \right) ^{2}acd+ \left( c-a \right) ^{2}abd+ \left( d-b \right) ^{2}abc$

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_5809  (a b c d : ℝ) :
  (b + d) * (c + a) * (a * b^2 + b * c^2 + c * d^2 + a^2 * d) - 4 * (a + b + c + d) * a * b * c * d =
    (a * b - c * d)^2 * b + (a * d - b * c)^2 * a + (c * d - a * b)^2 * d + (b * c - a * d)^2 * c +
    (a - c)^2 * b * c * d + (b - d)^2 * a * c * d + (c - a)^2 * a * b * d + (d - b)^2 * a * b * c   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook

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