Positivity of pairwise products and reciprocals

Proved

WorkbookSource.problem_25211

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

lean-workbooksource-checked

Prove $ab+bc+ac>0$ and $\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}>0$ if $a, b, c$ have the same sign.

Source: InternLM Lean-Workbook, record lean_workbook_25211 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_25211 (a b c : ℝ) (hab : a * b > 0) (hbc : b * c > 0) (hca : c * a > 0) : a * b + b * c + c * a > 0 ∧ 1 / (a * b) + 1 / (b * c) + 1 / (c * a) > 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_25211; Apache-2.0

View graph