Provedllmlarge-language-modelsformal-proofoptimization

One step with learning rate 1

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

\htmlData{mx=4}{\htmlData{mx=0}{\operatorname{step}}(\htmlData{mx=1}{a}, \htmlData{mx=2}{m}, 1, \htmlData{mx=3}{x})} \mathrel{\htmlData{mx=14}{=}} \htmlData{mx=13}{\htmlData{mx=5}{x} - \htmlData{mx=12}{1(\htmlData{mx=11}{\htmlData{mx=7}{2\htmlData{mx=6}{a}} \cdot (\htmlData{mx=10}{\htmlData{mx=8}{x} - \htmlData{mx=9}{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, x range over real numbers: the statement must hold for every choice. ((a m x : 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, 1, x. Lean writes f x; papers write f(x). (LLMOptim.step a m 1 x)
=Equality: step(a, m, 1, x) and x − 1 · (2 · a · (x − m)) are the same. (LLMOptim.step a m 1 x = x - 1 * (2 * a …)
·Multiplication: 2 times a. (2 * a)
-Subtraction: x minus m. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (x - m)

theorem opt-step-eta-one (a m x : ℝ) :
    LLMOptim.step a m 1 x = x - 1 * (2 * a * (x - m))

step a m 1 x = x − 1·(2a(x−m)).

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: 12 lines of Lean