A comparison of shifted reciprocal sums

Proved

WorkbookSource.base_42603

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

lean-workbooksource-checked

For $a,b,c>0$ , prove that $\frac{1}{2a+1}+\frac{1}{2b+1}+\frac{1}{2c+1}\geq\frac{1}{a+b+1}+\frac{1}{b+c+1}+\frac{1}{c+a+1}$ .

Source: InternLM Lean-Workbook, record lean_workbook_42603 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_42603 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) : (1 / (2 * a + 1) + 1 / (2 * b + 1) + 1 / (2 * c + 1)) ≥ (1 / (a + b + 1) + 1 / (b + c + 1) + 1 / (c + a + 1))  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_42603; Apache-2.0

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