Provedllmlarge-language-modelsformal-proofsoftmax

A two-token tie splits the mass evenly

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

\htmlData{mx=1}{\htmlData{mx=0}{w}(0)} \mathrel{\htmlData{mx=4}{=}} \htmlData{mx=3}{\htmlData{mx=2}{w}(1)} \mathrel{\htmlData{mx=16}{\implies}} \htmlData{mx=7}{\htmlData{mx=5}{\operatorname{sm}}(\htmlData{mx=6}{w}, 0)} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{\frac{1}{2}} \mathbin{\htmlData{mx=15}{\land}} \htmlData{mx=12}{\htmlData{mx=10}{\operatorname{sm}}(\htmlData{mx=11}{w}, 1)} \mathrel{\htmlData{mx=14}{=}} \htmlData{mx=13}{\frac{1}{2}}

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)
=Equality: w(0) and w(1) are the same. (w 0 = w 1)
RReal numbers: The real numbers: every point on the number line. (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)
∧And: Both hold: sm(w, 0) = 1/2, and sm(w, 1) = 1/2. (LLMSoftmax.sm w 0 = 1 / 2 ∧ LLMSoftmax.…)
a/bDivision: 1 divided by 2. In Lean, dividing by 0 gives 0. (1 / 2)

theorem smx-two-tie (w : Fin 2 → ℝ) :
    w 0 = w 1 → LLMSoftmax.sm w 0 = 1 / 2 ∧ LLMSoftmax.sm w 1 = 1 / 2

w 0 = w 1 implies both weights are 1/2.

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: 15 lines of Lean