Provedllmlarge-language-modelsformal-proofattention

Second weight times the softmax denominator is the second exponential

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

\htmlData{mx=9}{\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{weight1}}(\htmlData{mx=1}{s_{0}}, \htmlData{mx=2}{s_{1}})} \cdot (\htmlData{mx=8}{\htmlData{mx=5}{e^{\htmlData{mx=4}{s_{0}}}} + \htmlData{mx=7}{e^{\htmlData{mx=6}{s_{1}}}}})} \mathrel{\htmlData{mx=12}{=}} \htmlData{mx=11}{e^{\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 range over real numbers: the statement must hold for every choice. ((s0 s1 : 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, s1. Lean writes f x; papers write f(x). (TransformerAttention.weight1 s0 s1)
·Multiplication: weight1(s0, s1) times exp(s0) + exp(s1). (TransformerAttention.weight1 s0 s1 * (R…)
=Equality: weight1(s0, s1) · (exp(s0) + exp(s1)) and exp(s1) are the same. (TransformerAttention.weight1 s0 s1 * (R…)
eˣExponential function: e raised to the power s0, where e ≈ 2.718. (Real.exp s0)
+Addition: exp(s0) plus exp(s1). (Real.exp s0 + Real.exp s1)

theorem attn2-weight1-times-denom (s0 s1 : ℝ) :
    TransformerAttention.weight1 s0 s1 * (Real.exp s0 + Real.exp s1) = Real.exp s1

weight1 * (exp s0 + exp s1) = exp s1, the unnormalized score mass.

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: 13 lines of Lean