Provedllmlarge-language-modelsformal-proofattention

Cross-weight sum: weight0 s1 s0 + weight1 s1 s0 = 1

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

\htmlData{mx=8}{\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{weight0}}(\htmlData{mx=1}{s_{1}}, \htmlData{mx=2}{s_{0}})} + \htmlData{mx=7}{\htmlData{mx=4}{\operatorname{weight1}}(\htmlData{mx=5}{s_{1}}, \htmlData{mx=6}{s_{0}})}} \mathrel{\htmlData{mx=9}{=}} 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 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 s1, s0. Lean writes f x; papers write f(x). (TransformerAttention.weight0 s1 s0)
+Addition: weight0(s1, s0) plus weight1(s1, s0). (TransformerAttention.weight0 s1 s0 + Tr…)
=Equality: weight0(s1, s0) + weight1(s1, s0) and 1 are the same. (TransformerAttention.weight0 s1 s0 + Tr…)
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-weights-cross-sum (s0 s1 : ℝ) :
    TransformerAttention.weight0 s1 s0 + TransformerAttention.weight1 s1 s0 = 1

The partition of unity holds in swapped-argument coordinates too.

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