Provedllmlarge-language-modelsformal-proofattention

The weights sum to one in squared form

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

\begin{aligned} & \htmlData{mx=22}{\htmlData{mx=16}{\htmlData{mx=5}{\htmlData{mx=4}{\htmlData{mx=1}{\operatorname{weight0}}(\htmlData{mx=2}{s_{0}}, \htmlData{mx=3}{s_{1}})}^{2}} + \htmlData{mx=15}{\htmlData{mx=10}{2\htmlData{mx=9}{\htmlData{mx=6}{\operatorname{weight0}}(\htmlData{mx=7}{s_{0}}, \htmlData{mx=8}{s_{1}})}} \cdot \htmlData{mx=14}{\htmlData{mx=11}{\operatorname{weight1}}(\htmlData{mx=12}{s_{0}}, \htmlData{mx=13}{s_{1}})}}} + \htmlData{mx=21}{\htmlData{mx=20}{\htmlData{mx=17}{\operatorname{weight1}}(\htmlData{mx=18}{s_{0}}, \htmlData{mx=19}{s_{1}})}^{2}}} \\ & \quad \htmlData{mx=0}{=} 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 range over real numbers: the statement must hold for every choice. ((s0 s1 : ℝ))
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)
f(x)Function application: TransformerAttention.weight0 applied to s0, s1. Lean writes f x; papers write f(x). (TransformerAttention.weight0 s0 s1)
aᵇPower: weight0(s0, s1) raised to the power 2. (TransformerAttention.weight0 s0 s1 ^ 2)
+Addition: weight0(s0, s1)^2 plus 2 · weight0(s0, s1) · weight1(s0, s1). (TransformerAttention.weight0 s0 s1 ^ 2 …)
=Equality: weight0(s0, s1)^2 + 2 · weight0(s0, s1) · weight1(s0, s1) … and 1 are the same. (TransformerAttention.weight0 s0 s1 ^ 2 …)
·Multiplication: 2 times weight0(s0, s1). (2 * TransformerAttention.weight0 s0 s1)
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-weight0-plus-weight1-squared (s0 s1 : ℝ) :
    TransformerAttention.weight0 s0 s1 ^ 2 + 2 * TransformerAttention.weight0 s0 s1 * TransformerAttention.weight1 s0 s1 + TransformerAttention.weight1 s0 s1 ^ 2 = 1

weight0² + 2·weight0·weight1 + weight1² = 1: the simplex relation squared.

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

The weights sum to one in squared form

How it was proved

More proved statements