Provedllmlarge-language-modelsformal-prooflog-likelihood

The empty sequence has zero log-probability

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

\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{seqLogProb}}(\htmlData{mx=1}{p})} \mathrel{\htmlData{mx=3}{=}} 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 0. Lean writes f x; papers write f(x). (Fin 0)
⟹Implies: If Fin(0), then ℝ. (Between types, A → B is the type of functions from A to B.) (Fin 0 → ℝ)
seqLogProb: LLMLM.seqLogProb: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (LLMLM.seqLogProb)
=Equality: seqLogProb(p) and 0 are the same. (LLMLM.seqLogProb p = 0)
RReal numbers: The real numbers: every point on the number line. (ℝ)

theorem seqlog-empty (p : Fin 0 → ℝ) :
    LLMLM.seqLogProb p = 0

seqLogProb over Fin 0 is 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: 15 lines of Lean