Provedcommons

Adding a constant to every logit leaves softmax unchanged

For all $\htmlData{mx=1}{\iota \in \htmlData{mx=0}{\operatorname{Type}}}$ , $\htmlData{mx=3}{x : \htmlData{mx=0}{\iota} \mathrel{\htmlData{mx=2}{\to}} \htmlData{mx=1}{\mathbb{R}}}$ , $\htmlData{mx=1}{c \in \htmlData{mx=0}{\mathbb{R}}}$ 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=8}{\htmlData{mx=0}{\operatorname{softmax}}(\htmlData{mx=6}{j \mapsto \htmlData{mx=5}{\htmlData{mx=3}{\htmlData{mx=1}{x}(\htmlData{mx=2}{j})} + \htmlData{mx=4}{c}}}, \htmlData{mx=7}{i})} \mathrel{\htmlData{mx=13}{=}} \htmlData{mx=12}{\htmlData{mx=9}{\operatorname{softmax}}(\htmlData{mx=10}{x}, \htmlData{mx=11}{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)
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)
=Equality: softmax(j ↦ x(j) + c, i) and softmax(x, i) are the same. (Commons.softmax (fun j => x j + c) i = …)
RReal numbers: The real numbers: every point on the number line. (ℝ)
x ↦ f(x)Function (λ-abstraction): The function sending j to x(j) + c. Lean writes fun x => …; papers write x ↦ …. (fun j => x j + c)
+Addition: x(j) plus c. (x j + c)

theorem softmax-shift-invariant {ι : Type} [Fintype ι] [Nonempty ι] (x : ι → ℝ) (c : ℝ) (i : ι) :
    Commons.softmax (fun j => x j + c) i = Commons.softmax x i

From our commons: Commons.softmax_add_const 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

Adding a constant to every logit leaves softmax unchanged

How it was proved

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