Provedattentiontransformersoftmaxformal-proof

Both weights in two-key softmax attention are strictly positive

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

0 \mathrel{\htmlData{mx=4}{<}} \htmlData{mx=3}{\htmlData{mx=0}{\operatorname{weight0}}(\htmlData{mx=1}{s_{0}}, \htmlData{mx=2}{s_{1}})} \mathbin{\htmlData{mx=10}{\land}} 0 \mathrel{\htmlData{mx=9}{<}} \htmlData{mx=8}{\htmlData{mx=5}{\operatorname{weight1}}(\htmlData{mx=6}{s_{0}}, \htmlData{mx=7}{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))
<Less than: 0 is strictly smaller than weight0(s₀, s₁). (0 < weight0 s₀ s₁)
∧And: Both hold: 0 < weight0(s₀, s₁), and 0 < weight1(s₀, s₁). (0 < weight0 s₀ s₁ ∧ 0 < weight1 s₀ s₁)
weight0: weight0: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (weight0)
f(x)Function application: weight0 applied to s₀, s₁. Lean writes f x; papers write f(x). (weight0 s₀ s₁)
weight1: weight1: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (weight1)

theorem attention2-softmax-weights-positive (s₀ s₁ : ℝ) :
    0 < weight0 s₀ s₁ ∧ 0 < weight1 s₀ s₁

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: 15 lines of Lean