Provedllmlarge-language-modelsformal-proofoptimization

Zero iterations is the identity

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

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

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, x range over real numbers: the statement must hold for every choice. ((a m η x : ℝ))
gd: LLMOptim.gd: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMOptim.gd)
f(x)Function application: LLMOptim.gd applied to a, m, η, 0, x. Lean writes f x; papers write f(x). (LLMOptim.gd a m η 0 x)
=Equality: gd(a, m, η, 0, x) and x are the same. (LLMOptim.gd a m η 0 x = x)

theorem opt-gd-zero-iterate (a m η x : ℝ) :
    LLMOptim.gd a m η 0 x = x

gd a m η 0 x = x.

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: 12 lines of Lean