A symmetric polynomial substitution identity

Proved

WorkbookSource.problem_837

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

lean-workbooksource-checked

Prove that $xz=p(q-r)+r^2$ where $p=a^2+b^2+c^2$ , $q=ab+bc+ca$ , $r=(b-a)(b-c)$ , $x=a^2+2bc$ , $y=b^2+2ca$ , and $z=c^2+2ab$ .

Source: InternLM Lean-Workbook, record lean_workbook_837 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.

Preamble

import Mathlib
open Real Nat

Formal statement

theorem WorkbookSource.problem_837 :∀ a b c x y z p q r : ℝ,  p = a^2 + b^2 + c^2 ∧ q = a * b + b * c + c * a ∧ r = (b - a) * (b - c) ∧ x = a^2 + 2 * b * c ∧ y = b^2 + 2 * c * a ∧ z = c^2 + 2 * a * b → x * z = p * (q - r) + r^2  :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, record lean_workbook_837; Apache-2.0

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