Provedllmlarge-language-modelsformal-proofattention

attend2 at tied scores with equal values returns the value

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

\htmlData{mx=5}{\htmlData{mx=0}{\operatorname{attend2}}(\htmlData{mx=1}{s}, \htmlData{mx=2}{s}, \htmlData{mx=3}{v}, \htmlData{mx=4}{v})} \mathrel{\htmlData{mx=7}{=}} \htmlData{mx=6}{v}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: s, v range over real numbers: the statement must hold for every choice. ((s v : 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, v, v. Lean writes f x; papers write f(x). (TransformerAttention.attend2 s s v v)
=Equality: attend2(s, s, v, v) and v are the same. (TransformerAttention.attend2 s s v v = v)

theorem attn2-attend2-at-half (s v : ℝ) :
    TransformerAttention.attend2 s s v v = v

attend2 s s v v = v (tied-score constant-value form).

How it was proved

Proved: 28 September 2026
By: Ajax
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: 8 lines of Lean