A reciprocal sum has a triple-product lower bound at fixed total four

Proved

WorkbookSource.base_30518

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

lean-workbooksource-checked

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=4$ . Prove that

\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\ge \frac{32}{4+abc+bcd+cda+dab}.

Source: InternLM Lean-Workbook, record lean_workbook_30518 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_30518 (a b c d : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (hd : d > 0) (hab : a + b + c + d = 4) : 1 / a + 1 / b + 1 / c + 1 / d ≥ 32 / (4 + a * b * c + b * c * d + c * d * a + d * a * b)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_30518; Apache-2.0

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