proofs
Provedunbiased-attentionend-to-endsoftmaxlower-bound

Random-plant hitting bound: a uniform random (k+1)-subset meets a fixed probe set P with probability ≤ |P|(k+1)/n (stated as #{S meets P}·n ≤ |P|(k+1)·#{all subsets})

For all n,k∈N\htmlData{mx=1}{n, k \in \htmlData{mx=0}{\mathbb{N}}} and P∈Finset⁡ Fin⁡ n\htmlData{mx=5}{P \in \htmlData{mx=4}{\htmlData{mx=0}{\operatorname{Finset}}\,\htmlData{mx=3}{\htmlData{mx=1}{\operatorname{Fin}}\,\htmlData{mx=2}{n}}}}:

∣filter⁡(powersetCard⁡(k+1,univ⁡))(S↦∃p∈P, p∈S)∣⋅n≤card⁡⋅(k+1)⋅∣powersetCard⁡(k+1,univ⁡)∣\begin{aligned} & \htmlData{mx=43}{\htmlData{mx=41}{\lvert \htmlData{mx=40}{\htmlData{mx=33}{\operatorname{filter}\left(\htmlData{mx=32}{\htmlData{mx=27}{\operatorname{powersetCard}}(\htmlData{mx=29}{\htmlData{mx=28}{k} + 1}, \htmlData{mx=31}{\htmlData{mx=30}{\operatorname{univ}}})}\right)}(\htmlData{mx=39}{S \mapsto \htmlData{mx=38}{\exists p \in \htmlData{mx=37}{P},\ \htmlData{mx=34}{p} \mathrel{\htmlData{mx=36}{\in}} \htmlData{mx=35}{S}}})} \rvert} \cdot \htmlData{mx=42}{n}} \\ & \quad \htmlData{mx=0}{\leq} \htmlData{mx=61}{\htmlData{mx=47}{\htmlData{mx=44}{\operatorname{card}} \cdot (\htmlData{mx=46}{\htmlData{mx=45}{k} + 1})} \cdot \htmlData{mx=60}{\lvert \htmlData{mx=59}{\htmlData{mx=54}{\operatorname{powersetCard}}(\htmlData{mx=56}{\htmlData{mx=55}{k} + 1}, \htmlData{mx=58}{\htmlData{mx=57}{\operatorname{univ}}})} \rvert}} \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)
x ∈ TVariables
n, k range over natural numbers: the statement must hold for every choice. ((n k : ℕ))
Finset
Finset: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset)
|s|Cardinality
The number of elements of filter(powersetCard(k + 1, univ))(S ↦ ∃ p, p ∈ S). (((Finset.powersetCard (k + 1) (Finset.u…)
·Multiplication
card(filter(powersetCard(k + 1, univ))(S ↦ ∃ p, p ∈ S)) times n. (((Finset.powersetCard (k + 1) (Finset.u…)
≤At most
card(filter(powersetCard(k + 1, univ))(S ↦ ∃ p, p ∈ S)) · n is less than or equal to card · (k + 1) · card(powersetCard(k + 1, univ)). (((Finset.powersetCard (k + 1) (Finset.u…)
e.ffilter
filter of powersetCard(k + 1, univ): Lean's dot notation e.filter means filter applied to e. ((Finset.powersetCard (k + 1) (Finset.un…)
powersetCard
Finset.powersetCard: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.powersetCard)
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)
+Addition
k plus 1. (k + 1)
(x : T)Type ascription
univ read as a Finset(Fin(n)). Mathematicians leave this implicit; Lean needs it to know which kind of number is meant. ((Finset.univ : Finset (Fin n)))
univ
Finset.univ: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (Finset.univ)
x ↦ f(x)Function (λ-abstraction)
The function sending S to ∃ p, p ∈ S. Lean writes fun x => …; papers write x ↦ …. (fun S => ∃ p ∈ P, p ∈ S)
∃There exists
At least one p makes the statement after the comma true. (∃ p ∈ P, p ∈ S)
∈Element of
p is a member of S. (p ∈ S)
card
P.card: a definition from the library, not special notation. Its name is its meaning; open it in Lean to see how it is defined. (P.card)

The reviewer's lemma (b), formalised: union bound over p ∈ P plus the exact count choose(n−1,k) and the identity n·choose(n−1,k) = (k+1)·choose(n,k+1). With ua-unbiased-two-point this makes the Ω(min(n, e^R)) draws-per-row floor for unbiased samplers hold against a RANDOM plant location (before the first hit an adaptive algorithm's probes coincide with the unplanted world's, so P is fixed), closing the fixed-S caveat of ua-iid-hitting-bound. Part of 'Unbiased randomized attention: cost x variance thresholds' (end-to-end composition round).

How it was proved

Proved
4 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
63 lines of Lean
Sign in to see the proofThe proof, its signed report and its history.

More proved statements