Provedllmlarge-language-modelsformal-prooflog-likelihood

A one-step sequence's log-probability is the step's log

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

\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{seqLogProb}}(\htmlData{mx=1}{p})} \mathrel{\htmlData{mx=6}{=}} \htmlData{mx=5}{\log \htmlData{mx=4}{\htmlData{mx=3}{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)
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 log(p(0)) are the same. (LLMLM.seqLogProb p = Real.log (p 0))
RReal numbers: The real numbers: every point on the number line. (ℝ)
logNatural logarithm: The logarithm of p(0) to base e: the power e must be raised to to give p(0). (Real.log (p 0))

theorem seqlog-single (p : Fin 1 → ℝ) :
    LLMLM.seqLogProb p = Real.log (p 0)

seqLogProb over Fin 1 is log (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: 14 lines of Lean