Provedalphabaguettebaguette-zsoftmax

Softmax is unchanged by subtracting any constant (the running maximum)

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

\htmlData{mx=13}{\frac{\htmlData{mx=5}{e^{\htmlData{mx=4}{\htmlData{mx=2}{\htmlData{mx=0}{s}(\htmlData{mx=1}{i})} - \htmlData{mx=3}{c}}}}}{\htmlData{mx=12}{\sum_{j} \htmlData{mx=11}{e^{\htmlData{mx=10}{\htmlData{mx=8}{\htmlData{mx=6}{s}(\htmlData{mx=7}{j})} - \htmlData{mx=9}{c}}}}}}} \mathrel{\htmlData{mx=24}{=}} \htmlData{mx=23}{\frac{\htmlData{mx=17}{e^{\htmlData{mx=16}{\htmlData{mx=14}{s}(\htmlData{mx=15}{i})}}}}{\htmlData{mx=22}{\sum_{j} \htmlData{mx=21}{e^{\htmlData{mx=20}{\htmlData{mx=18}{s}(\htmlData{mx=19}{j})}}}}}}

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)
eˣExponential function: e raised to the power s(i) − c, where e ≈ 2.718. (Real.exp (s i - c))
a/bDivision: exp(s(i) − c) divided by ∑ j, exp(s(j) − c). In Lean, dividing by 0 gives 0. (Real.exp (s i - c) / ∑ j, Real.exp (s j…)
=Equality: exp(s(i) − c)/(∑ j, exp(s(j) − c)) and exp(s(i))/(∑ j, exp(s(j))) are the same. (Real.exp (s i - c) / ∑ j, Real.exp (s j…)
RReal numbers: The real numbers: every point on the number line. (ℝ)
-Subtraction: s(i) minus c. On ℕ, Lean's subtraction stops at 0: 2 − 5 = 0. (s i - c)
∑Summation: Add up exp(s(j) − c) for every j. (∑ j, Real.exp (s j - c))

theorem bz-softmax-shift-invariance {n : ℕ} (s : Fin n → ℝ) (c : ℝ) (i : Fin n) :
    Real.exp (s i - c) / ∑ j, Real.exp (s j - c) = Real.exp (s i) / ∑ j, Real.exp (s j)

exp(sᵢ − c)/Σexp(sⱼ − c) = exp(sᵢ)/Σexp(sⱼ). The integer max-subtraction used to keep exp2 arguments ≤ 0 is exact. Part of the plan 'Baguette-Z: closed-loop certified integer-state training'.

How it was proved

Proved: 3 October 2026
By: Ajax, with the agent “Baguette math & proofs (Claude Code, 2026-10-03)”
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: 15 lines of Lean

Softmax is unchanged by subtracting any constant (the running maximum)

How it was proved

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