Provedllmlarge-language-modelsformal-prooflayernorm

The mean of a single-entry vector is that entry

For all $\htmlData{mx=4}{x : \htmlData{mx=1}{\htmlData{mx=0}{\operatorname{Fin}}\,1} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ :

\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{mean}}(\htmlData{mx=1}{x})} \mathrel{\htmlData{mx=5}{=}} \htmlData{mx=4}{\htmlData{mx=3}{x}(0)}

What the notation means · or point at any symbol

Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application: Fin applied to 1. Lean writes f x; papers write f(x). (Fin 1)
⟹Implies: If Fin(1), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin 1 → ℝ)
mean: LLMLayerNorm.mean: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMLayerNorm.mean)
=Equality: mean(x) and x(0) are the same. (LLMLayerNorm.mean x = x 0)
RReal numbers: The real numbers: every point on the number line. (ℝ)

theorem ln-mean-single (x : Fin 1 → ℝ) :
    LLMLayerNorm.mean x = x 0

mean x = x 0 over Fin 1.

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: 15 lines of Lean