Provedllmlarge-language-modelsformal-proofsoftmax

lse in named form

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

\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{lse}}(\htmlData{mx=1}{w})} \mathrel{\htmlData{mx=7}{=}} \htmlData{mx=6}{\log \htmlData{mx=5}{\htmlData{mx=3}{\operatorname{denom}}(\htmlData{mx=4}{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)
lse: LLMSoftmax.lse: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMSoftmax.lse)
=Equality: lse(w) and log(denom(w)) are the same. (LLMSoftmax.lse w = Real.log (LLMSoftmax…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
logNatural logarithm: The logarithm of denom(w) to base e: the power e must be raised to to give denom(w). (Real.log (LLMSoftmax.denom w))
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-lse-def (n : ℕ) (w : Fin n → ℝ) :
    LLMSoftmax.lse w = Real.log (LLMSoftmax.denom w)

LLMSoftmax.lse w = Real.log (LLMSoftmax.denom w).

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

lse in named form

How it was proved

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