Provedllmlarge-language-modelsformal-proofoptimization

Gradient in explicit polynomial form

For all $\htmlData{mx=1}{a, m, x \in \htmlData{mx=0}{\mathbb{R}}}$ :

\htmlData{mx=4}{\htmlData{mx=0}{\operatorname{grad}}(\htmlData{mx=1}{a}, \htmlData{mx=2}{m}, \htmlData{mx=3}{x})} \mathrel{\htmlData{mx=14}{=}} \htmlData{mx=13}{\htmlData{mx=8}{\htmlData{mx=6}{2\htmlData{mx=5}{a}} \cdot \htmlData{mx=7}{x}} - \htmlData{mx=12}{\htmlData{mx=10}{2\htmlData{mx=9}{a}} \cdot \htmlData{mx=11}{m}}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: a, m, x range over real numbers: the statement must hold for every choice. ((a m x : ℝ))
grad: LLMOptim.grad: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMOptim.grad)
f(x)Function application: LLMOptim.grad applied to a, m, x. Lean writes f x; papers write f(x). (LLMOptim.grad a m x)
=Equality: grad(a, m, x) and 2 · a · x − 2 · a · m are the same. (LLMOptim.grad a m x = 2 * a * x - 2 * a…)
·Multiplication: 2 times a. (2 * a)
-Subtraction: 2 · a · x minus 2 · a · m. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (2 * a * x - 2 * a * m)

theorem opt-grad-explicit (a m x : ℝ) :
    LLMOptim.grad a m x = 2 * a * x - 2 * a * m

grad a m x = 2·a·x − 2·a·m.

How it was proved

Proved: 28 September 2026
By: Ajax, with the agent “ompmuse”
Checked by: Lean's kernel (Lean 4.33.1, with Mathlib); the proof is signed and replays offline
Axioms: Classical.choice, Quot.sound, propext
Proof: 13 lines of Lean