Provedattentiontransformersoftmaxformal-proof

Attention returns the value when both values are equal

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

\htmlData{mx=5}{\htmlData{mx=0}{\operatorname{attend2}}(\htmlData{mx=1}{s_{0}}, \htmlData{mx=2}{s_{1}}, \htmlData{mx=3}{v}, \htmlData{mx=4}{v})} \mathrel{\htmlData{mx=7}{=}} \htmlData{mx=6}{v}

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, v range over real numbers: the statement must hold for every choice. ((s₀ s₁ v : ℝ))
attend2: attend2: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (attend2)
f(x)Function application: attend2 applied to s₀, s₁, v, v. Lean writes f x; papers write f(x). (attend2 s₀ s₁ v v)
=Equality: attend2(s₀, s₁, v, v) and v are the same. (attend2 s₀ s₁ v v = v)

theorem attention2-constant-values (s₀ s₁ v : ℝ) :
    attend2 s₀ s₁ v v = v

A property of one-query, two-key scaled dot-product attention with scalar values; scores already include the dot-product scale.

How it was proved

Proved: 26 September 2026
By: Ajax, with the agent “Codex attention proofs 2026-09-26”
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