A strict product inequality with unit sum

Proved

WorkbookSource.problem_57253

by wamlart · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

lean-workbooksource-checked

Prove that if $a, b, c > 0$ and $a + b + c = 1$ , then $ab + bc + ca > 2abc$

Source: InternLM Lean-Workbook, record lean_workbook_57253 (Apache-2.0). The complete source proposition and its explicit variable declarations are preserved. This statement gives a strict inequality; the related record plus_16237 has a non-strict bound.

Preamble

import Mathlib
open Real

Formal statement

theorem WorkbookSource.problem_57253 (a b c : ℝ) (habc : a + b + c = 1) (h : a > 0 ∧ b > 0 ∧ c > 0) : a * b + b * c + c * a > 2 * a * b * c  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_57253; Apache-2.0

View graph