Provedllmlarge-language-modelsformal-prooflog-likelihood

perplex in named form

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

\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{perplex}}(\htmlData{mx=1}{p}, \htmlData{mx=2}{q})} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{e^{\htmlData{mx=7}{\htmlData{mx=4}{\operatorname{crossEnt}}(\htmlData{mx=5}{p}, \htmlData{mx=6}{q})}}}

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)
x : TVariables: p, q range over Fin n → ℝ: the statement must hold for every choice. ((p q : Fin n → ℝ))
perplex: LLMLM.perplex: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMLM.perplex)
=Equality: perplex(p, q) and exp(crossEnt(p, q)) are the same. (LLMLM.perplex p q = Real.exp (LLMLM.cro…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
eˣExponential function: e raised to the power crossEnt(p, q), where e ≈ 2.718. (Real.exp (LLMLM.crossEnt p q))
crossEnt: LLMLM.crossEnt: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMLM.crossEnt)

theorem lm-perplex-def {n : ℕ} (p q : Fin n → ℝ) :
    LLMLM.perplex p q = Real.exp (LLMLM.crossEnt p q)

LLMLM.perplex p q = Real.exp (LLMLM.crossEnt p q).

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: 14 lines of Lean

perplex in named form

How it was proved

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