Provedllmlarge-language-modelsformal-proofsoftmax

The denominator dominates each exponential

For all $\htmlData{mx=1}{n \in \htmlData{mx=0}{\mathbb{N}}}$ , $\htmlData{mx=5}{w : \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}} \mathrel{\htmlData{mx=4}{\to}} \htmlData{mx=3}{\mathbb{R}}}$ and $\htmlData{mx=3}{i \in \htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fin}}\,\htmlData{mx=1}{n}}}$ :

\htmlData{mx=3}{e^{\htmlData{mx=2}{\htmlData{mx=0}{w}(\htmlData{mx=1}{i})}}} \mathrel{\htmlData{mx=7}{\leq}} \htmlData{mx=6}{\htmlData{mx=4}{\operatorname{denom}}(\htmlData{mx=5}{w})}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
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 n. Lean writes f x; papers write f(x). (Fin n)
⟹Implies: If Fin(n), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin n → ℝ)
eˣExponential function: e raised to the power w(i), where e ≈ 2.718. (Real.exp (w i))
≤At most: exp(w(i)) is less than or equal to denom(w). (Real.exp (w i) ≤ LLMSoftmax.denom w)
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)

theorem smx-denom-ge-exp (n : ℕ) (w : Fin n → ℝ) (i : Fin n) :
    Real.exp (w i) ≤ LLMSoftmax.denom w

exp (w i) ≤ denom w: each term is at most the whole sum.

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

The denominator dominates each exponential

How it was proved

More proved statements