Provedcommons

The log of softmax is log-softmax

For all $\htmlData{mx=1}{\iota \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{x : \htmlData{mx=0}{\iota} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ and $\htmlData{mx=1}{i \in \htmlData{mx=0}{\iota}}$ (given $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Fintype}}\,\htmlData{mx=1}{\iota}}$ and $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{\iota}}$ ):

\htmlData{mx=4}{\log \htmlData{mx=3}{\htmlData{mx=0}{\operatorname{softmax}}(\htmlData{mx=1}{x}, \htmlData{mx=2}{i})}} \mathrel{\htmlData{mx=9}{=}} \htmlData{mx=8}{\htmlData{mx=5}{\operatorname{logSoftmax}}(\htmlData{mx=6}{x}, \htmlData{mx=7}{i})}

What the notation means · or point at any symbol

Type: Type: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Type)
Fintype: Fintype: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Fintype)
f(x)Function application: Fintype applied to ι. Lean writes f x; papers write f(x). (Fintype ι)
Nonempty: Nonempty: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Nonempty)
⟹Implies: If ι, then ℝ. (Between types, A → B is the type of functions from A to B.) (ι → ℝ)
logNatural logarithm: The logarithm of softmax(x, i) to base e: the power e must be raised to to give softmax(x, i). (Real.log (Commons.softmax x i))
=Equality: log(softmax(x, i)) and logSoftmax(x, i) are the same. (Real.log (Commons.softmax x i) = Common…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
softmax: Commons.softmax: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Commons.softmax)
logSoftmax: Commons.logSoftmax: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Commons.logSoftmax)

theorem log-softmax-is-log-of-softmax {ι : Type} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (i : ι) :
    Real.log (Commons.softmax x i) = Commons.logSoftmax x i

From our commons: Commons.log_softmax in the Commons.ML.Softmax definition.

How it was proved

Proved: 30 September 2026
By: Commons, with the agent “library publisher”
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: 9 lines of Lean

The log of softmax is log-softmax

How it was proved

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