Provedllmlarge-language-modelsformal-proofoptimization

The gradient is additive in the position

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

\begin{aligned} & \htmlData{mx=7}{\htmlData{mx=1}{\operatorname{grad}}(\htmlData{mx=2}{a}, \htmlData{mx=3}{m}, \htmlData{mx=6}{\htmlData{mx=4}{x} + \htmlData{mx=5}{y}})} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=23}{\htmlData{mx=18}{\htmlData{mx=12}{\htmlData{mx=8}{\operatorname{grad}}(\htmlData{mx=9}{a}, \htmlData{mx=10}{m}, \htmlData{mx=11}{x})} + \htmlData{mx=17}{\htmlData{mx=13}{\operatorname{grad}}(\htmlData{mx=14}{a}, \htmlData{mx=15}{m}, \htmlData{mx=16}{y})}} - \htmlData{mx=22}{\htmlData{mx=19}{\operatorname{grad}}(\htmlData{mx=20}{a}, \htmlData{mx=21}{m}, 0)}} \end{aligned}

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, y range over real numbers: the statement must hold for every choice. ((a m x y : 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 + y. Lean writes f x; papers write f(x). (LLMOptim.grad a m (x + y))
=Equality: grad(a, m, x + y) and grad(a, m, x) + grad(a, m, y) − grad(a, m, 0) are the same. (LLMOptim.grad a m (x + y) = LLMOptim.gr…)
+Addition: x plus y. (x + y)
-Subtraction: grad(a, m, x) + grad(a, m, y) minus grad(a, m, 0). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (LLMOptim.grad a m x + LLMOptim.grad a m…)

theorem opt-grad-add (a m x y : ℝ) :
    LLMOptim.grad a m (x + y) = LLMOptim.grad a m x + LLMOptim.grad a m y - LLMOptim.grad a m 0

grad a m (x + y) = grad a m x + grad a m y − grad a m 0.

How it was proved

Proved: 27 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