Triangle inequalities force positive variables

Proved

WorkbookSource.problem_1404

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

lean-workbooksource-checked

Given that $a + b > c$ , $b + c > a$ , and $c + a > b$ , prove that $a > 0$ , $b > 0$ , and $c > 0$ .

Source: InternLM Lean-Workbook, record lean_workbook_1404 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_1404 (a b c : ℝ) (hab : a + b > c) (hbc : b + c > a) (hca : c + a > b) : a > 0 ∧ b > 0 ∧ c > 0  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_1404; Apache-2.0

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