Provedllmlarge-language-modelsformal-proofattention

The attention logit is symmetric in query and key

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

\htmlData{mx=4}{\htmlData{mx=0}{\operatorname{logit}}(\htmlData{mx=1}{d}, \htmlData{mx=2}{q}, \htmlData{mx=3}{k})} \mathrel{\htmlData{mx=10}{=}} \htmlData{mx=9}{\htmlData{mx=5}{\operatorname{logit}}(\htmlData{mx=6}{d}, \htmlData{mx=7}{k}, \htmlData{mx=8}{q})}

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 d. Lean writes f x; papers write f(x). (Fin d)
⟹Implies: If Fin(d), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin d → ℝ)
x : TVariables: q, k range over Fin d → ℝ: the statement must hold for every choice. ((q k : Fin d → ℝ))
logit: LLMAttention.logit: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMAttention.logit)
=Equality: logit(d, q, k) and logit(d, k, q) are the same. (LLMAttention.logit d q k = LLMAttention…)
RReal numbers: The real numbers: every point on the number line. (ℝ)

theorem attn-logit-sym (d : ℕ) (q k : Fin d → ℝ) :
    LLMAttention.logit d q k = LLMAttention.logit d k q

logit d q k = logit d k q: scaled dot-product scores treat query and key symmetrically.

How it was proved

Proved: 26 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: 16 lines of Lean

The attention logit is symmetric in query and key

How it was proved

More proved statements