Provedllmlarge-language-modelsformal-proofsoftmax

Explicit two-token positivity

For all $\htmlData{mx=4}{w : \htmlData{mx=1}{\htmlData{mx=0}{\operatorname{Fin}}\,2} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ :

0 \mathrel{\htmlData{mx=3}{<}} \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{sm}}(\htmlData{mx=1}{w}, 1)}

What the notation means · or point at any symbol

Fin: Fin: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fin)
f(x)Function application: Fin applied to 2. Lean writes f x; papers write f(x). (Fin 2)
⟹Implies: If Fin(2), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin 2 → ℝ)
<Less than: 0 is strictly smaller than sm(w, 1). (0 < LLMSoftmax.sm w 1)
sm: LLMSoftmax.sm: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMSoftmax.sm)
RReal numbers: The real numbers: every point on the number line. (ℝ)

theorem smx-two-positive-explicit (w : Fin 2 → ℝ) :
    0 < LLMSoftmax.sm w 1

0 < LLMSoftmax.sm w 1 over Fin 2.

How it was proved

Proved: 27 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: 14 lines of Lean