Provedllmlarge-language-modelsformal-proofattention

Attention is determined by its weights

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

\begin{aligned} & \htmlData{mx=6}{\htmlData{mx=1}{\operatorname{attend2}}(\htmlData{mx=2}{s_{0}}, \htmlData{mx=3}{s_{1}}, \htmlData{mx=4}{v_{0}}, \htmlData{mx=5}{v_{1}})} \\ & \quad \htmlData{mx=0}{=} \htmlData{mx=19}{\htmlData{mx=12}{\htmlData{mx=10}{\htmlData{mx=7}{\operatorname{weight0}}(\htmlData{mx=8}{s_{0}}, \htmlData{mx=9}{s_{1}})} \cdot \htmlData{mx=11}{v_{0}}} + \htmlData{mx=18}{\htmlData{mx=16}{\htmlData{mx=13}{\operatorname{weight1}}(\htmlData{mx=14}{s_{0}}, \htmlData{mx=15}{s_{1}})} \cdot \htmlData{mx=17}{v_{1}}}} \end{aligned}

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, v1 range over real numbers: the statement must hold for every choice. ((s0 s1 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 s0, s1, v0, v1. Lean writes f x; papers write f(x). (TransformerAttention.attend2 s0 s1 v0 v1)
=Equality: attend2(s0, s1, v0, v1) and weight0(s0, s1) · v0 + weight1(s0, s1) · v1 are the same. (TransformerAttention.attend2 s0 s1 v0 v…)
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)
+Addition: weight0(s0, s1) · v0 plus weight1(s0, s1) · v1. (TransformerAttention.weight0 s0 s1 * v0…)
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)

theorem attn2-of-weights (s0 s1 v0 v1 : ℝ) :
    TransformerAttention.attend2 s0 s1 v0 v1 = TransformerAttention.weight0 s0 s1 * v0 + TransformerAttention.weight1 s0 s1 * v1

attend2 s0 s1 v0 v1 = weight0·v0 + weight1·v1 is the definition, restated with named parts.

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: 13 lines of Lean

Attention is determined by its weights

How it was proved

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