Provedllmlarge-language-modelsformal-prooflayernorm

ln in named form

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=19}{=}} \htmlData{mx=18}{\frac{\htmlData{mx=11}{\htmlData{mx=7}{\htmlData{mx=5}{x}(\htmlData{mx=6}{i})} - \htmlData{mx=10}{\htmlData{mx=8}{\operatorname{mean}}(\htmlData{mx=9}{x})}}}{\htmlData{mx=17}{\sqrt{\htmlData{mx=16}{\htmlData{mx=14}{\htmlData{mx=12}{\operatorname{variance}}(\htmlData{mx=13}{x})} + \htmlData{mx=15}{\varepsilon}}}}}}

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 (x(i) − mean(x))/√(variance(x) + ε) are the same. (LLMLayerNorm.ln ε x i = (x i - LLMLayer…)
a/bDivision: x(i) − mean(x) divided by √(variance(x) + ε). In Lean, dividing by 0 gives 0. ((x i - LLMLayerNorm.mean x) / Real.sqrt…)
-Subtraction: x(i) minus mean(x). On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (x i - LLMLayerNorm.mean x)
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)
√Square root: The non-negative number whose square is variance(x) + ε. (Real.sqrt (LLMLayerNorm.variance x + ε))
variance: LLMLayerNorm.variance: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMLayerNorm.variance)
+Addition: variance(x) plus ε. (LLMLayerNorm.variance x + ε)

theorem ln-ln-def {n : ℕ} (ε : ℝ) (x : Fin n → ℝ) (i : Fin n) :
    LLMLayerNorm.ln ε x i = (x i - LLMLayerNorm.mean x) / Real.sqrt (LLMLayerNorm.variance x + ε)

LLMLayerNorm.ln ε x i = (x i − mean x)/sqrt(variance x + ε).

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: 13 lines of Lean

ln in named form

How it was proved

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