Provedllmlarge-language-modelsformal-proofattention

Zero second value collapses attention to the first weight times value

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

\htmlData{mx=4}{\htmlData{mx=0}{\operatorname{attend2}}(\htmlData{mx=1}{s_{0}}, \htmlData{mx=2}{s_{1}}, \htmlData{mx=3}{v_{0}}, 0)} \mathrel{\htmlData{mx=11}{=}} \htmlData{mx=10}{\htmlData{mx=8}{\htmlData{mx=5}{\operatorname{weight0}}(\htmlData{mx=6}{s_{0}}, \htmlData{mx=7}{s_{1}})} \cdot \htmlData{mx=9}{v_{0}}}

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, v0 range over real numbers: the statement must hold for every choice. ((s0 s1 v0 : 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, v0, 0. Lean writes f x; papers write f(x). (TransformerAttention.attend2 s0 s1 v0 0)
=Equality: attend2(s0, s1, v0, 0) and weight0(s0, s1) · v0 are the same. (TransformerAttention.attend2 s0 s1 v0 0…)
weight0: TransformerAttention.weight0: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (TransformerAttention.weight0)
·Multiplication: weight0(s0, s1) times v0. (TransformerAttention.weight0 s0 s1 * v0)

theorem attn2-right-zero-value (s0 s1 v0 : ℝ) :
    TransformerAttention.attend2 s0 s1 v0 0 = TransformerAttention.weight0 s0 s1 * v0

If only the second value is zero, the output is weight0 * v0.

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