Provedllmlarge-language-modelsformal-proofattention

Attention at zero scores with one value

For all $\htmlData{mx=1}{v \in \htmlData{mx=0}{\mathbb{R}}}$ :

\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{attend2}}(0, 0, \htmlData{mx=1}{v}, 0)} \mathrel{\htmlData{mx=5}{=}} \htmlData{mx=4}{\frac{\htmlData{mx=3}{v}}{2}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
attend2: TransformerAttention.attend2: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (TransformerAttention.attend2)
f(x)Function application: TransformerAttention.attend2 applied to 0, 0, v, 0. Lean writes f x; papers write f(x). (TransformerAttention.attend2 0 0 v 0)
=Equality: attend2(0, 0, v, 0) and v/2 are the same. (TransformerAttention.attend2 0 0 v 0 = …)
a/bDivision: v divided by 2. In Lean, dividing by 0 gives 0. (v / 2)

theorem attn2-attend2-both-zero-left (v : ℝ) :
    TransformerAttention.attend2 0 0 v 0 = v / 2

attend2 0 0 v 0 = v / 2.

How it was proved

Proved: 28 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: 15 lines of Lean