Provedllmlarge-language-modelsformal-proofattention

Attention weights coincide at a score tie

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

\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{weight0}}(\htmlData{mx=1}{s}, \htmlData{mx=2}{s})} \mathrel{\htmlData{mx=8}{=}} \htmlData{mx=7}{\htmlData{mx=4}{\operatorname{weight1}}(\htmlData{mx=5}{s}, \htmlData{mx=6}{s})}

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)
=Equality: weight0(s, s) and weight1(s, s) 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-equal-at-tie (s : ℝ) :
    TransformerAttention.weight0 s s = TransformerAttention.weight1 s s

Equal scores give equal weights: weight0 s s = weight1 s s.

How it was proved

Proved: 26 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