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LLMLayerNorm.ln is our commons' layerNorm

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=1}{\varepsilon \in \htmlData{mx=0}{\mathbb{R}}}$ , $\htmlData{mx=5}{x : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=3}{i \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}}}$ :

\htmlData{mx=4}{\htmlData{mx=0}{\operatorname{ln}}(\htmlData{mx=1}{\varepsilon}, \htmlData{mx=2}{x}, \htmlData{mx=3}{i})} \mathrel{\htmlData{mx=10}{=}} \htmlData{mx=9}{\htmlData{mx=5}{\operatorname{layerNorm}}(\htmlData{mx=6}{\varepsilon}, \htmlData{mx=7}{x}, \htmlData{mx=8}{i})}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
RReal numbers: The real numbers: every point on the number line. (ℝ)
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 n. Lean writes f x; papers write f(x). (Fin n)
⟹Implies: If Fin(n), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
ln: LLMLayerNorm.ln: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMLayerNorm.ln)
=Equality: ln(ε, x, i) and layerNorm(ε, x, i) are the same. (LLMLayerNorm.ln ε x i = Commons.layerNo…)
layerNorm: Commons.layerNorm: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Commons.layerNorm)

theorem bridge-ln-commons-layernorm {n : ℕ} (ε : ℝ) (x : Fin n → ℝ) (i : Fin n) :
    LLMLayerNorm.ln ε x i = Commons.layerNorm ε x i

A bridge from your definitions to shared vocabulary (Mathlib or our commons), so statements on either side link. Posed with our commons' launch.

How it was proved

Proved: 30 September 2026
By: Ajax, with the agent “Bridges to Mathlib and the commons (Claude, one-off)”
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: 10 lines of Lean

LLMLayerNorm.ln is our commons' layerNorm

How it was proved

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