Provedllmlarge-language-modelsformal-proofattention

Attention is odd under negating both values

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

\htmlData{mx=7}{\htmlData{mx=0}{\operatorname{attend2}}(\htmlData{mx=1}{s_{0}}, \htmlData{mx=2}{s_{1}}, \htmlData{mx=4}{-\htmlData{mx=3}{v_{0}}}, \htmlData{mx=6}{-\htmlData{mx=5}{v_{1}}})} \mathrel{\htmlData{mx=15}{=}} \htmlData{mx=14}{-\htmlData{mx=13}{\htmlData{mx=8}{\operatorname{attend2}}(\htmlData{mx=9}{s_{0}}, \htmlData{mx=10}{s_{1}}, \htmlData{mx=11}{v_{0}}, \htmlData{mx=12}{v_{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, v0, v1 range over real numbers: the statement must hold for every choice. ((s0 s1 v0 v1 : ℝ))
attend2: TransformerAttention.attend2: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (TransformerAttention.attend2)
f(x)Function application: TransformerAttention.attend2 applied to s0, s1, −v0, −v1. Lean writes f x; papers write f(x). (TransformerAttention.attend2 s0 s1 (-v0…)
=Equality: attend2(s0, s1, −v0, −v1) and −attend2(s0, s1, v0, v1) are the same. (TransformerAttention.attend2 s0 s1 (-v0…)
−xNegation: The negative of v0. (-v0)

theorem attn2-neg-values (s0 s1 v0 v1 : ℝ) :
    TransformerAttention.attend2 s0 s1 (-v0) (-v1) = -TransformerAttention.attend2 s0 s1 v0 v1

Negating both values negates the attention output.

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