Provedllmlarge-language-modelsformal-proofoptimization

The gradient doubles with doubled distance

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

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

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, d range over real numbers: the statement must hold for every choice. ((a m d : 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, m + 2 · d. Lean writes f x; papers write f(x). (LLMOptim.grad a m (m + 2 * d))
=Equality: grad(a, m, m + 2 · d) and 4 · a · d are the same. (LLMOptim.grad a m (m + 2 * d) = 4 * a *…)
+Addition: m plus 2 · d. (m + 2 * d)
·Multiplication: 2 times d. (2 * d)

theorem opt-grad-double-dist (a m d : ℝ) :
    LLMOptim.grad a m (m + 2 * d) = 4 * a * d

grad a m (m + 2d) = 4a*d.

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