Provedllmlarge-language-modelsformal-proofattention

First attention weight is nonnegative

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

0 \mathrel{\htmlData{mx=4}{\leq}} \htmlData{mx=3}{\htmlData{mx=0}{\operatorname{weight0}}(\htmlData{mx=1}{s_{0}}, \htmlData{mx=2}{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))
≤At most: 0 is less than or equal to weight0(s0, s1). (0 ≤ TransformerAttention.weight0 s0 s1)
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 s0, s1. Lean writes f x; papers write f(x). (TransformerAttention.weight0 s0 s1)

theorem attn2-weight0-nonneg (s0 s1 : ℝ) :
    0 ≤ TransformerAttention.weight0 s0 s1

Softmax weights are nonnegative.

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: 14 lines of Lean