Provedllmlarge-language-modelsformal-proofsoftmax

Denominator of a constant single-token

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

\htmlData{mx=5}{\htmlData{mx=0}{\operatorname{denom}}(\htmlData{mx=4}{\cdot \mapsto \htmlData{mx=1}{x}})} \mathrel{\htmlData{mx=8}{=}} \htmlData{mx=7}{e^{\htmlData{mx=6}{x}}}

What the notation means · or point at any symbol

RReal numbers: The real numbers: every point on the number line. (ℝ)
denom: LLMSoftmax.denom: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMSoftmax.denom)
=Equality: denom(_ ↦ x) and exp(x) are the same. (LLMSoftmax.denom (fun _ : Fin 1 => x) =…)
x ↦ f(x)Function (λ-abstraction): The function sending _ to x. Lean writes fun x => …; papers write x ↦ …. (fun _ : Fin 1 => x)
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 1. Lean writes f x; papers write f(x). (Fin 1)
eˣExponential function: e raised to the power x, where e ≈ 2.718. (Real.exp x)

theorem smx-denom-one-of-const (x : ℝ) :
    LLMSoftmax.denom (fun _ : Fin 1 => x) = Real.exp x

denom (fun _ => x) = exp x over Fin 1.

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