Provedllmlarge-language-modelsformal-proofattention

weight1 is shift-invariant

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

\htmlData{mx=7}{\htmlData{mx=0}{\operatorname{weight1}}(\htmlData{mx=3}{\htmlData{mx=1}{s_{0}} + \htmlData{mx=2}{c}}, \htmlData{mx=6}{\htmlData{mx=4}{s_{1}} + \htmlData{mx=5}{c}})} \mathrel{\htmlData{mx=12}{=}} \htmlData{mx=11}{\htmlData{mx=8}{\operatorname{weight1}}(\htmlData{mx=9}{s_{0}}, \htmlData{mx=10}{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, c range over real numbers: the statement must hold for every choice. ((s0 s1 c : ℝ))
weight1: TransformerAttention.weight1: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (TransformerAttention.weight1)
f(x)Function application: TransformerAttention.weight1 applied to s0 + c, s1 + c. Lean writes f x; papers write f(x). (TransformerAttention.weight1 (s0 + c) (…)
=Equality: weight1(s0 + c, s1 + c) and weight1(s0, s1) are the same. (TransformerAttention.weight1 (s0 + c) (…)
+Addition: s0 plus c. (s0 + c)

theorem attn2-weight1-shift (s0 s1 c : ℝ) :
    TransformerAttention.weight1 (s0 + c) (s1 + c) = TransformerAttention.weight1 s0 s1

weight1 (s0 + c) (s1 + c) = weight1 s0 s1.

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: 16 lines of Lean