lean_workbook_plus_7170

Proved

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

Let $a,b,c>0$ Prove that

\frac{3}{\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}}\geq 1+\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}

\iff

\frac{1}{\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}}-\frac{1}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}\geq\frac{1}{3}

Preamble

import Mathlib.Analysis.Complex.Basic

Formal statement

theorem lean_workbook_plus_7170 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (3 / (1 / (a + 1) + 1 / (b + 1) + 1 / (c + 1))) ≥ 1 + (3 / (1 / a + 1 / b + 1 / c)) ↔ (1 / (1 / (a + 1) + 1 / (b + 1) + 1 / (c + 1))) - (1 / (1 / a + 1 / b + 1 / c)) ≥ 1 / 3   :=  by sorry

Source

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

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