A cubic bound under three triangle-style affine constraints

Proved

WorkbookSource.base_29420

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

lean-workbooksource-checked

Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that $a^2+b^2+c^2\leq 2abc+1.$

Source: InternLM Lean-Workbook, record lean_workbook_29420 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_29420 (a b c : ℝ) (ha : a ≥ 0) (hb : b ≥ 0) (hc : c ≥ 0) (hab : a + b ≤ c + 1) (hbc : b + c ≤ a + 1) (hca : c + a ≤ b + 1) : a^2 + b^2 + c^2 ≤ 2 * a * b * c + 1  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_29420; Apache-2.0

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