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7f37411 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Sphere.SphereMeasure
namespace OAI
/-! Quantitative estimates for the distance-set construction. -/
section
section
namespace CurrentFalconer.TreeND
abbrev CellPath (d : ℕ) := List (Fin d → Bool)
variable {d : ℕ}
def dyadicPaths {d : ℕ} : ℕ → Finset (CellPath d)
| 0 => {[]}
| n+1 => Finset.univ.biUnion fun a : Fin d → Bool =>
(dyadicPaths n).image (List.cons a)
@[simp] theorem mem_dyadicPaths (n : ℕ) (p : CellPath d) :
p ∈ dyadicPaths n ↔ p.length = n := by
induction n generalizing p with
| zero => simp [dyadicPaths]
| succ n ih =>
cases p with
| nil => simp [dyadicPaths]
| cons a p => simp [dyadicPaths, ih]
lemma dyadicPaths_card_le (N : ℕ) : (dyadicPaths (d := d) N).card ≤ (2^d)^N := by
classical
induction N with
| zero => simp [dyadicPaths]
| succ N ih =>
calc
_ ≤ ∑ a ∈ (Finset.univ : Finset (Fin d → Bool)),
((dyadicPaths N).image (List.cons a)).card := Finset.card_biUnion_le
_ ≤ ∑ _a ∈ (Finset.univ : Finset (Fin d → Bool)), (2^d)^N :=
Finset.sum_le_sum (fun _ _ => Finset.card_image_le.trans ih)
_ = _ := by simp [pow_succ,Nat.mul_comm]
end CurrentFalconer.TreeND
end
end
end OAI
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