Provedllmlarge-language-modelsformal-proofattention

Attention is translation-equivariant in the values

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

\htmlData{mx=9}{\htmlData{mx=0}{\operatorname{attend2}}(\htmlData{mx=1}{s_{0}}, \htmlData{mx=2}{s_{1}}, \htmlData{mx=5}{\htmlData{mx=3}{v_{0}} + \htmlData{mx=4}{c}}, \htmlData{mx=8}{\htmlData{mx=6}{v_{1}} + \htmlData{mx=7}{c}})} \mathrel{\htmlData{mx=18}{=}} \htmlData{mx=17}{\htmlData{mx=15}{\htmlData{mx=10}{\operatorname{attend2}}(\htmlData{mx=11}{s_{0}}, \htmlData{mx=12}{s_{1}}, \htmlData{mx=13}{v_{0}}, \htmlData{mx=14}{v_{1}})} + \htmlData{mx=16}{c}}

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, c, v0, v1 range over real numbers: the statement must hold for every choice. ((s0 s1 c 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 + c, v1 + c. Lean writes f x; papers write f(x). (TransformerAttention.attend2 s0 s1 (v0 …)
=Equality: attend2(s0, s1, v0 + c, v1 + c) and attend2(s0, s1, v0, v1) + c are the same. (TransformerAttention.attend2 s0 s1 (v0 …)
+Addition: v0 plus c. (v0 + c)

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

Adding a constant to both values adds that constant to the output: the weights sum to one.

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

Attention is translation-equivariant in the values

How it was proved

More proved statements