Provedllmlarge-language-modelsformal-proofattention

Swapping scores swaps the roles of the weights

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

\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{weight0}}(\htmlData{mx=1}{s_{1}}, \htmlData{mx=2}{s_{0}})} \mathrel{\htmlData{mx=8}{=}} \htmlData{mx=7}{\htmlData{mx=4}{\operatorname{weight1}}(\htmlData{mx=5}{s_{0}}, \htmlData{mx=6}{s_{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)
=Equality: weight0(s1, s0) and weight1(s0, s1) 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-weight0-swap (s0 s1 : ℝ) :
    TransformerAttention.weight0 s1 s0 = TransformerAttention.weight1 s0 s1

weight0 s1 s0 = weight1 s0 s1: the weights are exchanged under score swap.

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