A shifted polynomial bound under a sum and lower bounds

Proved

WorkbookSource.base_38972

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

lean-workbooksource-checked

Prove that if positive reals $x,y$ satisfy $x+y= 3$ , $x,y \ge 1$ then $9(x- 1)(y- 1) + (y^2 + y+ 1)(x + 1) + (x^2-x+ 1)(y- 1) \ge 9$

Source: InternLM Lean-Workbook, record lean_workbook_38972 (Apache-2.0). Complete source proposition preserved; proof developed independently.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.base_38972 (x y : ℝ) (hx : 1 ≤ x) (hy : 1 ≤ y) (hxy : x + y = 3) : 9 * (x - 1) * (y - 1) + (y^2 + y + 1) * (x + 1) + (x^2 - x + 1) * (y - 1) ≥ 9  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_38972; Apache-2.0

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