Prove2Me

A cyclic shifted pair-product ratio sum is at least three quarters at unit product

Proved
WorkbookSource.base_26343

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

lean-workbooksource-checked

If a>0,b>0,c>0a>0,b>0,c>0 and abc=1abc=1 ,then prove that

a(a+1)(b+1)+b(b+1)(c+1)+c(c+1)(a+1)≥34\frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+\frac{c}{(c+1)(a+1)}\geq \frac{3}{4}

Source: InternLM Lean-Workbook, record lean_workbook_26343 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.base_26343 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (habc : a * b * c = 1) : (a / (a + 1) / (b + 1) + b / (b + 1) / (c + 1) + c / (c + 1) / (a + 1)) ≥ 3 / 4  :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_26343; Apache-2.0

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