Provedllmlarge-language-modelsformal-proofoptimization

step at the optimum with explicit gradient zero

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

\htmlData{mx=5}{\htmlData{mx=0}{\operatorname{step}}(\htmlData{mx=1}{a}, \htmlData{mx=2}{m}, \htmlData{mx=3}{\eta}, \htmlData{mx=4}{m})} \mathrel{\htmlData{mx=7}{=}} \htmlData{mx=6}{m}

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, η range over real numbers: the statement must hold for every choice. ((a m η : ℝ))
step: LLMOptim.step: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMOptim.step)
f(x)Function application: LLMOptim.step applied to a, m, η, m. Lean writes f x; papers write f(x). (LLMOptim.step a m η m)
=Equality: step(a, m, η, m) and m are the same. (LLMOptim.step a m η m = m)

theorem opt-step-min-left (a m η : ℝ) :
    LLMOptim.step a m η m = m

LLMOptim.step a m η m = m via grad a m m = 0.

How it was proved

Proved: 27 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