A pair-product sum with a reciprocal product correction

Proved

WorkbookSource.plus_7663

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

lean-workbooksource-checked

Let $a, b, c$ be positive real numbers satisfying $a + b + c = 3$ . Prove that: $ab + bc + ca + \frac {1}{abc} \ge 3 + abc$

Source: InternLM Lean-Workbook, record lean_workbook_plus_7663 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.plus_7663 (a b c : ℝ) (ha : a > 0) (hb : b > 0) (hc : c > 0) (habc : a + b + c = 3) : a * b + b * c + c * a + 1 / (a * b * c) ≥ 3 + a * b * c   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_7663; Apache-2.0

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