Provedllmlarge-language-modelsformal-proofmoe

The MoE gate permutes with the experts

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{s : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ , $\htmlData{mx=7}{e : \htmlData{mx=6}{\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \simeq \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{Fin}}\,\htmlData{mx=4}{n}}}}$ and $\htmlData{mx=3}{i \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}}}$ :

\htmlData{mx=8}{\htmlData{mx=0}{\operatorname{gate}}(\htmlData{mx=6}{k \mapsto \htmlData{mx=5}{\htmlData{mx=1}{s}(\htmlData{mx=4}{\htmlData{mx=2}{e}(\htmlData{mx=3}{k})})}}, \htmlData{mx=7}{i})} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{\htmlData{mx=9}{\operatorname{gate}}(\htmlData{mx=10}{s}, \htmlData{mx=13}{\htmlData{mx=11}{e}(\htmlData{mx=12}{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)
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)
≃Equivalence: A one-to-one correspondence between Fin(n) and Fin(n). (Fin n ≃ Fin n)
gate: LLMMoE.gate: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMMoE.gate)
=Equality: gate(k ↦ s(e(k)), i) and gate(s, e(i)) are the same. (LLMMoE.gate (fun k => s (e k)) i = LLMM…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
x ↦ f(x)Function (λ-abstraction): The function sending k to s(e(k)). Lean writes fun x => …; papers write x ↦ …. (fun k => s (e k))

theorem moe-gate-permute {n : ℕ} (s : Fin n → ℝ) (e : Fin n ≃ Fin n) (i : Fin n) :
    LLMMoE.gate (fun k => s (e k)) i = LLMMoE.gate s (e i)

gate (s ∘ e) i = gate s (e i): relabeling experts relabels the gate.

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: 17 lines of Lean

The MoE gate permutes with the experts

How it was proved

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