Provedllmlarge-language-modelsformal-proofattention

weight0 of doubled scores

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

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

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 2 · s, 2 · s. Lean writes f x; papers write f(x). (TransformerAttention.weight0 (2 * s) (2…)
=Equality: weight0(2 · s, 2 · s) and 1/2 are the same. (TransformerAttention.weight0 (2 * s) (2…)
·Multiplication: 2 times s. (2 * s)
a/bDivision: 1 divided by 2. In Lean, dividing by 0 gives 0. (1 / 2)

theorem attn2-weight0-double (s : ℝ) :
    TransformerAttention.weight0 (2 * s) (2 * s) = 1 / 2

weight0 (2s) (2s) = 1/2 (scaling invariance of ties).

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