Provedllmlarge-language-modelsformal-prooflayernorm

Variance is nonnegative (instance restated)

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

0 \mathrel{\htmlData{mx=3}{\leq}} \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{variance}}(\htmlData{mx=1}{x})}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
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)
≤At most: 0 is less than or equal to variance(x). (0 ≤ LLMLayerNorm.variance x)
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)
RReal numbers: The real numbers: every point on the number line. (ℝ)

theorem ln-variance-nonneg-instance {n : ℕ} (x : Fin n → ℝ) :
    0 ≤ LLMLayerNorm.variance x

0 ≤ variance x for any vector.

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: 14 lines of Lean

Variance is nonnegative (instance restated)

How it was proved

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