Provedbridge

Temperature softmax is softmax of the scaled logits

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

\htmlData{mx=4}{\htmlData{mx=0}{\operatorname{smT}}(\htmlData{mx=1}{T}, \htmlData{mx=2}{w}, \htmlData{mx=3}{i})} \mathrel{\htmlData{mx=14}{=}} \htmlData{mx=13}{\htmlData{mx=5}{\operatorname{softmax}}\left(\htmlData{mx=11}{j \mapsto \htmlData{mx=10}{\frac{\htmlData{mx=8}{\htmlData{mx=6}{w}(\htmlData{mx=7}{j})}}{\htmlData{mx=9}{T}}}}, \htmlData{mx=12}{i}\right)}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
RReal numbers: The real numbers: every point on the number line. (ℝ)
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)
smT: LLMSoftmax.smT: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMSoftmax.smT)
=Equality: smT(T, w, i) and softmax(j ↦ w(j)/T, i) are the same. (LLMSoftmax.smT T w i = Commons.softmax …)
softmax: Commons.softmax: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Commons.softmax)
x ↦ f(x)Function (λ-abstraction): The function sending j to w(j)/T. Lean writes fun x => …; papers write x ↦ …. (fun j => w j / T)
a/bDivision: w(j) divided by T. In Lean, dividing by 0 gives 0. (w j / T)

theorem bridge-smt-commons-softmax {n : ℕ} (T : ℝ) (w : Fin n → ℝ) (i : Fin n) :
    LLMSoftmax.smT T w i = Commons.softmax (fun j => w j / T) i

A bridge from your definitions to shared vocabulary (Mathlib or our commons), so statements on either side link. Posed with our commons' launch.

How it was proved

Proved: 30 September 2026
By: Ajax, with the agent “Bridges to Mathlib and the commons (Claude, one-off)”
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: 10 lines of Lean

Temperature softmax is softmax of the scaled logits

How it was proved

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