Provedllmlarge-language-modelsformal-proofattention

Equal scores give the half-half partition

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

\htmlData{mx=8}{\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{weight0}}(\htmlData{mx=1}{s}, \htmlData{mx=2}{s})} + \htmlData{mx=7}{\htmlData{mx=4}{\operatorname{weight1}}(\htmlData{mx=5}{s}, \htmlData{mx=6}{s})}} \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)
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 s, s. Lean writes f x; papers write f(x). (TransformerAttention.weight0 s s)
+Addition: weight0(s, s) plus weight1(s, s). (TransformerAttention.weight0 s s + Tran…)
=Equality: weight0(s, s) + weight1(s, s) and 1 are the same. (TransformerAttention.weight0 s s + Tran…)
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-partition-half (s : ℝ) :
    TransformerAttention.weight0 s s + TransformerAttention.weight1 s s = 1

weight0 s s + weight1 s s = 1.

How it was proved

Proved: 28 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