A product-of-shifts bound at fixed pairwise sum

Proved

WorkbookSource.base_52401

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

lean-workbooksource-checked

Prove that for $a, b, c > 0$ satisfying $ab + bc + ca = 3$ , the inequality $a^2 + b^2 + c^2 + 2abc + 3 \geq (1 + a)(1 + b)(1 + c)$ holds.

Source: InternLM Lean-Workbook, record lean_workbook_52401 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_52401 (a b c : ℝ) (ha : 0 < a) (hb : 0 < b) (hc : 0 < c) (hab : a * b + b * c + c * a = 3) : a^2 + b^2 + c^2 + 2 * a * b * c + 3 ≥ (1 + a) * (1 + b) * (1 + c)  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_52401; Apache-2.0

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