Provedllmlarge-language-modelsformal-proofsoftmax

Softmax weight over a single token

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

\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{sm}}(\htmlData{mx=1}{w}, 0)} \mathrel{\htmlData{mx=10}{=}} \htmlData{mx=9}{\frac{\htmlData{mx=5}{e^{\htmlData{mx=4}{\htmlData{mx=3}{w}(0)}}}}{\htmlData{mx=8}{e^{\htmlData{mx=7}{\htmlData{mx=6}{w}(0)}}}}}

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 1. Lean writes f x; papers write f(x). (Fin 1)
⟹Implies: If Fin(1), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin 1 → ℝ)
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)
=Equality: sm(w, 0) and exp(w(0))/exp(w(0)) are the same. (LLMSoftmax.sm w 0 = Real.exp (w 0) / Re…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
eˣExponential function: e raised to the power w(0), where e ≈ 2.718. (Real.exp (w 0))
a/bDivision: exp(w(0)) divided by exp(w(0)). In Lean, dividing by 0 gives 0. (Real.exp (w 0) / Real.exp (w 0))

theorem smx-sm-one-explicit (w : Fin 1 → ℝ) :
    LLMSoftmax.sm w 0 = Real.exp (w 0) / Real.exp (w 0)

sm w 0 = exp(w 0)/exp(w 0) = 1 by division.

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: 13 lines of Lean