Prove2Me

A quartic bound from its coefficients

Proved
WorkbookSource.plus_34031

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

lean-workbooksource-checked

We have b≥a2+c24b \geq \frac{a^2+c^2}{4} so

x4+ax3+bx2+cx+1=(x+a2)2x2−a24x2+(c2x+1)2−c24x2+bx2x^4+ax^3+bx^2+cx+1 =\left(x+\frac{a}{2}\right)^2x^2-\frac{a^2}{4}x^2 + \left(\frac{c}{2}x+1\right)^2-\frac{c^2}{4}x^2 + bx^2 =(x+a2)2x2+(c2x+1)2+(b−a24−c24)x2≥0=\left(x+\frac{a}{2}\right)^2x^2 +\left(\frac{c}{2}x+1\right)^2+\left(b-\frac{a^2}{4}-\frac{c^2}{4}\right)x^2 \geq 0

Source: InternLM Lean-Workbook, record lean_workbook_plus_34031 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble
import Mathlib
open Real Nat
Formal statement
theorem WorkbookSource.plus_34031  (a b c x : ℝ)
  (h₀ : b ≥ (a^2 + c^2) / 4) :
  x^4 + a * x^3 + b * x^2 + c * x + 1 ≥ 0   :=  by sorry
Source
https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_plus_34031; Apache-2.0

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