Provedllmlarge-language-modelsformal-proofoptimization

The gradient at the minimizer (explicit zero)

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

\htmlData{mx=4}{\htmlData{mx=0}{\operatorname{grad}}(\htmlData{mx=1}{a}, \htmlData{mx=2}{m}, \htmlData{mx=3}{m})} \mathrel{\htmlData{mx=5}{=}} 0

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 range over real numbers: the statement must hold for every choice. ((a m : 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. Lean writes f x; papers write f(x). (LLMOptim.grad a m m)
=Equality: grad(a, m, m) and 0 are the same. (LLMOptim.grad a m m = 0)

theorem opt-grad-min (a m : ℝ) :
    LLMOptim.grad a m m = 0

LLMOptim.grad a m m = 0 by ring.

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