Provedllmlarge-language-modelsformal-proofoptimization

The gradient is odd in the distance

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

\htmlData{mx=6}{\htmlData{mx=0}{\operatorname{grad}}(\htmlData{mx=1}{a}, \htmlData{mx=2}{m}, \htmlData{mx=5}{\htmlData{mx=3}{m} - \htmlData{mx=4}{d}})} \mathrel{\htmlData{mx=12}{=}} \htmlData{mx=11}{-\htmlData{mx=10}{\htmlData{mx=8}{2\htmlData{mx=7}{a}} \cdot \htmlData{mx=9}{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 − d. Lean writes f x; papers write f(x). (LLMOptim.grad a m (m - d))
=Equality: grad(a, m, m − d) and −(2 · a · d) are the same. (LLMOptim.grad a m (m - d) = -(2 * a * d))
-Subtraction: m minus d. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (m - d)
−xNegation: The negative of 2 · a · d. (-(2 * a * d))
·Multiplication: 2 times a. (2 * a)

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

grad a m (m − d) = −2·a·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