Provedcommons

Every softmax weight is positive

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 $\htmlData{mx=2}{\htmlData{mx=0}{\operatorname{Nonempty}}\,\htmlData{mx=1}{\iota}}$ ):

0 \mathrel{\htmlData{mx=4}{<}} \htmlData{mx=3}{\htmlData{mx=0}{\operatorname{softmax}}(\htmlData{mx=1}{x}, \htmlData{mx=2}{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)
<Less than: 0 is strictly smaller than softmax(x, i). (0 < Commons.softmax x i)
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)

theorem softmax-positive {ι : Type} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (i : ι) :
    0 < Commons.softmax x i

From our commons: Commons.softmax_pos 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

Every softmax weight is positive

How it was proved

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