Provedllmlarge-language-modelsformal-proofattention

Attention interpolates from the second value toward the first

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

\htmlData{mx=5}{\htmlData{mx=0}{\operatorname{attend2}}(\htmlData{mx=1}{s_{0}}, \htmlData{mx=2}{s_{1}}, \htmlData{mx=3}{v_{0}}, \htmlData{mx=4}{v_{1}})} \mathrel{\htmlData{mx=16}{=}} \htmlData{mx=15}{\htmlData{mx=6}{v_{1}} + \htmlData{mx=14}{\htmlData{mx=10}{\htmlData{mx=7}{\operatorname{weight0}}(\htmlData{mx=8}{s_{0}}, \htmlData{mx=9}{s_{1}})} \cdot (\htmlData{mx=13}{\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 v1)
=Equality: attend2(s0, s1, v0, v1) and v1 + weight0(s0, s1) · (v0 − v1) are the same. (TransformerAttention.attend2 s0 s1 v0 v…)
+Addition: v1 plus weight0(s0, s1) · (v0 − v1). (v1 + 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)
·Multiplication: weight0(s0, s1) times v0 − v1. (TransformerAttention.weight0 s0 s1 * (v…)
-Subtraction: v0 minus v1. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (v0 - v1)

theorem attn2-interp-left (s0 s1 v0 v1 : ℝ) :
    TransformerAttention.attend2 s0 s1 v0 v1 = v1 + TransformerAttention.weight0 s0 s1 * (v0 - v1)

attend2 = v1 + weight0 * (v0 - v1): the first weight is the interpolation parameter.

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