Provedllmlarge-language-modelsformal-proofsoftmax

Denominator over Fin 2 in explicit form

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

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)
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)
=Equality: denom(w) and exp(w(0)) + exp(w(1)) are the same. (LLMSoftmax.denom w = Real.exp (w 0) + R…)
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))
+Addition: exp(w(0)) plus exp(w(1)). (Real.exp (w 0) + Real.exp (w 1))

theorem smx-denom-two-explicit (w : Fin 2 → ℝ) :
    LLMSoftmax.denom w = Real.exp (w 0) + Real.exp (w 1)

denom w = exp(w 0) + exp(w 1) over Fin 2.

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