Provedllmlarge-language-modelsformal-prooflayernorm
The mean is homogeneous
For all , and :
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) - 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(i ↦ c · x(i)) and c · mean(x) are the same. (
LLMLayerNorm.mean (fun i => c * x i) = …) - x ↦ f(x)Function (λ-abstraction)
- The function sending i to c · x(i). Lean writes fun x => …; papers write x ↦ …. (
fun i => c * x i) - ·Multiplication
- c times x(i). (
c * x i)
theorem ln-mean-smul {n : ℕ} (c : ℝ) (x : Fin n → ℝ) :
LLMLayerNorm.mean (fun i => c * x i) = c * LLMLayerNorm.mean xmean (c·x) = c · mean x.
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
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