A quadratic and triple-product minimum at fixed sum

Proved

WorkbookCorrected.plus_76546

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

corrected-formalizationlean-workbooksource-checked

Let $a,b,c \geq 0$ and $a+b+c=3$ . Find the min of $a^2+b^2+c^2+2abc$ .

Source: InternLM Lean-Workbook, record lean_workbook_plus_76546 (Apache-2.0). Natural-language proposition preserved; the source bound is completed with an exact attainment witness. Proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookCorrected.plus_76546 : (∀ (a b c : ℝ) (ha : 0 ≤ a) (hb : 0 ≤ b) (hc : 0 ≤ c) (habc : a + b + c = 3), 9 / 2 ≤ a ^ 2 + b ^ 2 + c ^ 2 + 2 * a * b * c) ∧ (∃ a b c : ℝ, (0 ≤ a) ∧ (0 ≤ b) ∧ (0 ≤ c) ∧ (a + b + c = 3) ∧ ( 9 / 2  =  a ^ 2 + b ^ 2 + c ^ 2 + 2 * a * b * c   )) := by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_76546; Apache-2.0

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