Provedalphabaguettebaguette-zsoftmax

Softmax weights are nonnegative and sum to 1

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

\begin{aligned} & \left(\htmlData{mx=12}{\forall i,\ 0 \mathrel{\htmlData{mx=11}{\leq}} \htmlData{mx=10}{\frac{\htmlData{mx=4}{e^{\htmlData{mx=3}{\htmlData{mx=1}{s}(\htmlData{mx=2}{i})}}}}{\htmlData{mx=9}{\sum_{j} \htmlData{mx=8}{e^{\htmlData{mx=7}{\htmlData{mx=5}{s}(\htmlData{mx=6}{j})}}}}}}}\right) \\ & \htmlData{mx=0}{\land} \htmlData{mx=23}{\sum_{i} \htmlData{mx=22}{\frac{\htmlData{mx=16}{e^{\htmlData{mx=15}{\htmlData{mx=13}{s}(\htmlData{mx=14}{i})}}}}{\htmlData{mx=21}{\sum_{j} \htmlData{mx=20}{e^{\htmlData{mx=19}{\htmlData{mx=17}{s}(\htmlData{mx=18}{j})}}}}}}} \mathrel{\htmlData{mx=24}{=}} 1 \end{aligned}

What the notation means · or point at any symbol

NNatural numbers: The natural numbers 0, 1, 2, …. In Lean they start at 0. (ℕ)
<Less than: 0 is strictly smaller than n. (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)
∧And: Both hold: ∀ i, 0 ≤ exp(s(i))/(∑ j, exp(s(j))), and (∑ i, exp(s(i))/(∑ j, exp(s(j)))) = 1. ((∀ i, 0 ≤ Real.exp (s i) / ∑ j, Real.ex…)
∀For all: The statement after the comma holds for every i. (∀ i, 0 ≤ Real.exp (s i) / ∑ j, Real.exp…)
≤At most: 0 is less than or equal to exp(s(i))/(∑ j, exp(s(j))). (0 ≤ Real.exp (s i) / ∑ j, Real.exp (s j))
RReal numbers: The real numbers: every point on the number line. (ℝ)
eˣExponential function: e raised to the power s(i), where e ≈ 2.718. (Real.exp (s i))
a/bDivision: exp(s(i)) divided by ∑ j, exp(s(j)). In Lean, dividing by 0 gives 0. (Real.exp (s i) / ∑ j, Real.exp (s j))
∑Summation: Add up exp(s(j)) for every j. (∑ j, Real.exp (s j))
=Equality: ∑ i, exp(s(i))/(∑ j, exp(s(j))) and 1 are the same. (∑ i, Real.exp (s i) / ∑ j, Real.exp (s …)

theorem bz-softmax-weights-convex {n : ℕ} (hn : 0 < n) (s : Fin n → ℝ) :
    (∀ i, 0 ≤ Real.exp (s i) / ∑ j, Real.exp (s j)) ∧ ∑ i, Real.exp (s i) / ∑ j, Real.exp (s j) = 1

The hypotheses of the convex-combination bound hold for every attention row (n > 0). 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: 17 lines of Lean

Softmax weights are nonnegative and sum to 1

How it was proved

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