Provedllmlarge-language-modelsformal-proofattention

Scaling both scores preserves the tie

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

\htmlData{mx=7}{\htmlData{mx=0}{\operatorname{weight0}}(\htmlData{mx=3}{\htmlData{mx=1}{c} \cdot \htmlData{mx=2}{s}}, \htmlData{mx=6}{\htmlData{mx=4}{c} \cdot \htmlData{mx=5}{s}})} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{\frac{1}{2}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
x ∈ TVariables: c, s range over real numbers: the statement must hold for every choice. ((c s : 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 c · s, c · s. Lean writes f x; papers write f(x). (TransformerAttention.weight0 (c * s) (c…)
=Equality: weight0(c · s, c · s) and 1/2 are the same. (TransformerAttention.weight0 (c * s) (c…)
·Multiplication: c times s. (c * s)
a/bDivision: 1 divided by 2. In Lean, dividing by 0 gives 0. (1 / 2)

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

weight0 (c·s) (c·s) = 1/2 for any c.

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: 15 lines of Lean