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LLMLM.crossEnt is our commons' Commons.crossEntropy

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

\htmlData{mx=3}{\htmlData{mx=0}{\operatorname{crossEnt}}(\htmlData{mx=1}{p}, \htmlData{mx=2}{q})} \mathrel{\htmlData{mx=8}{=}} \htmlData{mx=7}{\htmlData{mx=4}{\operatorname{crossEntropy}}(\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)
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)
=Equality: crossEnt(p, q) and crossEntropy(p, q) are the same. (LLMLM.crossEnt p q = Commons.crossEntro…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
crossEntropy: Commons.crossEntropy: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Commons.crossEntropy)

theorem bridge-llmlm-crossent-commons-crossentropy {n : ℕ} (p : Fin n → ℝ) (q : Fin n → ℝ) :
    LLMLM.crossEnt p q = Commons.crossEntropy p q

Found by our verifier when llm-lm was published: LLMLM.crossEnt equals our commons' Commons.crossEntropy, proved by intros; rfl. Once proved here, statements about either name link to each other.

How it was proved

Proved: 30 September 2026
By: Ajax
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: 10 lines of Lean

LLMLM.crossEnt is our commons' Commons.crossEntropy

How it was proved

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