Provedllmlarge-language-modelsformal-proofattention

attend2 at tied scores with different values

For all $\htmlData{mx=1}{s, v_{0}, v_{1} \in \htmlData{mx=0}{\mathbb{R}}}$ :

\htmlData{mx=5}{\htmlData{mx=0}{\operatorname{attend2}}(\htmlData{mx=1}{s}, \htmlData{mx=2}{s}, \htmlData{mx=3}{v_{0}}, \htmlData{mx=4}{v_{1}})} \mathrel{\htmlData{mx=10}{=}} \htmlData{mx=9}{\frac{\htmlData{mx=8}{\htmlData{mx=6}{v_{0}} + \htmlData{mx=7}{v_{1}}}}{2}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: s, v0, v1 range over real numbers: the statement must hold for every choice. ((s v0 v1 : 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 s, s, v0, v1. Lean writes f x; papers write f(x). (TransformerAttention.attend2 s s v0 v1)
=Equality: attend2(s, s, v0, v1) and (v0 + v1)/2 are the same. (TransformerAttention.attend2 s s v0 v1 …)
a/bDivision: v0 + v1 divided by 2. In Lean, dividing by 0 gives 0. ((v0 + v1) / 2)
+Addition: v0 plus v1. (v0 + v1)

theorem attn2-attend2-tied-scores-mixed (s v0 v1 : ℝ) :
    TransformerAttention.attend2 s s v0 v1 = (v0 + v1) / 2

attend2 s s v0 v1 = (v0 + v1)/2 (mixed-value tied-score form).

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: 15 lines of Lean