Provedllmlarge-language-modelsformal-proofoptimization

The gradient ignores additive constants in the position variable

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

\htmlData{mx=12}{\htmlData{mx=6}{\htmlData{mx=0}{\operatorname{grad}}(\htmlData{mx=1}{a}, \htmlData{mx=2}{m}, \htmlData{mx=5}{\htmlData{mx=3}{x} + \htmlData{mx=4}{k}})} - \htmlData{mx=11}{\htmlData{mx=7}{\operatorname{grad}}(\htmlData{mx=8}{a}, \htmlData{mx=9}{m}, \htmlData{mx=10}{x})}} \mathrel{\htmlData{mx=17}{=}} \htmlData{mx=16}{\htmlData{mx=14}{2\htmlData{mx=13}{a}} \cdot \htmlData{mx=15}{k}}

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

theorem opt-grad-add-const (a m x k : ℝ) :
    LLMOptim.grad a m (x + k) - LLMOptim.grad a m x = 2 * a * k

grad a m (x + k) − grad a m x = 2a·k.

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