proofs
Provedbridge

LLMLayerNorm.variance is our commons' popVariance

For all n∈N\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}} and x:Fin⁡ n→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}}}:

variance⁡(x)=popVariance⁡(x)\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{variance}}(\htmlData{mx=1}{x})} \mathrel{\htmlData{mx=6}{=}} \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{popVariance}}(\htmlData{mx=4}{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)
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)
=Equality
variance(x) and popVariance(x) are the same. (LLMLayerNorm.variance x = Commons.popVa…)
RReal numbers
The real numbers: every point on the number line. (ℝ)
popVariance
Commons.popVariance: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Commons.popVariance)

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
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