Provedllmlarge-language-modelsformal-proofperplexity

One-step perplexity is the reciprocal probability

For all $\htmlData{mx=4}{p : \htmlData{mx=1}{\htmlData{mx=0}{\operatorname{Fin}}\,1} \mathrel{\htmlData{mx=3}{\to}} \htmlData{mx=2}{\mathbb{R}}}$ :

0 \mathrel{\htmlData{mx=2}{<}} \htmlData{mx=1}{\htmlData{mx=0}{p}(0)} \mathrel{\htmlData{mx=12}{\implies}} \htmlData{mx=7}{\htmlData{mx=3}{\operatorname{perplex}}(\htmlData{mx=5}{\cdot \mapsto \htmlData{mx=4}{1}}, \htmlData{mx=6}{p})} \mathrel{\htmlData{mx=11}{=}} \htmlData{mx=10}{\frac{1}{\htmlData{mx=9}{\htmlData{mx=8}{p}(0)}}}

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 1. Lean writes f x; papers write f(x). (Fin 1)
⟹Implies: If Fin(1), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin 1 → ℝ)
<Less than: 0 is strictly smaller than p(0). (0 < p 0)
RReal numbers: The real numbers: every point on the number line. (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(_ ↦ 1, p) and 1/p(0) are the same. (LLMLM.perplex (fun _ => (1:ℝ)) p = 1 / …)
x ↦ f(x)Function (λ-abstraction): The function sending _ to 1. Lean writes fun x => …; papers write x ↦ …. (fun _ => (1:ℝ))
(x : T)As real numbers: 1 read as an element of real numbers. Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((1:ℝ))
a/bDivision: 1 divided by p(0). In Lean, dividing by 0 gives 0. (1 / p 0)

theorem perplex-single (p : Fin 1 → ℝ) :
    0 < p 0 → LLMLM.perplex (fun _ => (1:ℝ)) p = 1 / p 0

perplex of a delta at the only index equals 1 / p 0 for 0 < p 0.

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: 17 lines of Lean