Provedllmlarge-language-modelsformal-proofoptimization

Gradient with curvature 1

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

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

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: m, d range over real numbers: the statement must hold for every choice. ((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 1, m, m + d. Lean writes f x; papers write f(x). (LLMOptim.grad 1 m (m + d))
=Equality: grad(1, m, m + d) and 2 · 1 · d are the same. (LLMOptim.grad 1 m (m + d) = 2 * 1 * d)
+Addition: m plus d. (m + d)
·Multiplication: 2 times 1. (2 * 1)

theorem opt-grad-aunit (m d : ℝ) :
    LLMOptim.grad 1 m (m + d) = 2 * 1 * d

grad 1 m (m + d) = 1 scaled distance form.

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