Provedllmlarge-language-modelsformal-proofattention

Zero first value collapses attention to the second weight times value

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

\htmlData{mx=4}{\htmlData{mx=0}{\operatorname{attend2}}(\htmlData{mx=1}{s_{0}}, \htmlData{mx=2}{s_{1}}, 0, \htmlData{mx=3}{v_{1}})} \mathrel{\htmlData{mx=11}{=}} \htmlData{mx=10}{\htmlData{mx=8}{\htmlData{mx=5}{\operatorname{weight1}}(\htmlData{mx=6}{s_{0}}, \htmlData{mx=7}{s_{1}})} \cdot \htmlData{mx=9}{v_{1}}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: s0, s1, v1 range over real numbers: the statement must hold for every choice. ((s0 s1 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 s0, s1, 0, v1. Lean writes f x; papers write f(x). (TransformerAttention.attend2 s0 s1 0 v1)
=Equality: attend2(s0, s1, 0, v1) and weight1(s0, s1) · v1 are the same. (TransformerAttention.attend2 s0 s1 0 v1…)
weight1: TransformerAttention.weight1: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (TransformerAttention.weight1)
·Multiplication: weight1(s0, s1) times v1. (TransformerAttention.weight1 s0 s1 * v1)

theorem attn2-left-zero-value (s0 s1 v1 : ℝ) :
    TransformerAttention.attend2 s0 s1 0 v1 = TransformerAttention.weight1 s0 s1 * v1

If only the first value is zero, the output is weight1 * v1.

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: 14 lines of Lean