Provedllmlarge-language-modelsformal-proofattention

One-dimensional logit symmetry (explicit)

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

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

What the notation means · or point at any symbol

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 1. Lean writes f x; papers write f(x). (Fin 1)
⟹Implies: If Fin(1), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin 1 → ℝ)
x : TVariables: q, k range over Fin 1 → ℝ: the statement must hold for every choice. ((q k : Fin 1 → 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(1, q, k) and logit(1, k, q) are the same. (LLMAttention.logit 1 q k = LLMAttention…)
RReal numbers: The real numbers: every point on the number line. (ℝ)

theorem attn-logit-sym-d1 (q k : Fin 1 → ℝ) :
    LLMAttention.logit 1 q k = LLMAttention.logit 1 k q

logit 1 q k = logit 1 k q computed via Fin.sum_univ_one.

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: 16 lines of Lean

One-dimensional logit symmetry (explicit)

How it was proved

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