Provedllmlarge-language-modelsformal-prooflayernorm

The two-point mean is the midpoint

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

\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{mean}}(\htmlData{mx=1}{w})} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{\frac{\htmlData{mx=7}{\htmlData{mx=4}{\htmlData{mx=3}{w}(0)} + \htmlData{mx=6}{\htmlData{mx=5}{w}(1)}}}{2}}

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 2. Lean writes f x; papers write f(x). (Fin 2)
⟹Implies: If Fin(2), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin 2 → 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(w) and (w(0) + w(1))/2 are the same. (LLMLayerNorm.mean w = (w 0 + w 1) / 2)
RReal numbers: The real numbers: every point on the number line. (ℝ)
a/bDivision: w(0) + w(1) divided by 2. In Lean, dividing by 0 gives 0. ((w 0 + w 1) / 2)
+Addition: w(0) plus w(1). (w 0 + w 1)

theorem ln-mean-two (w : Fin 2 → ℝ) :
    LLMLayerNorm.mean w = (w 0 + w 1) / 2

For Fin 2, mean w = (w 0 + w 1)/2.

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