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e320e82 | 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 | import Mathlib.Basic.Real.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
namespace OAI
/-!
# Finite dyadic mass for the selector entropy estimate
The positive dyadic levels have total mass at most one, including when there
are no levels.
-/
open scoped BigOperators
namespace TalagrandThreshold
private lemma selector_dyadic_range_sum (s : ℕ) :
(∑ i ∈ Finset.range s, (1 / 2 : ℝ) ^ (i + 1)) = 1 - (1 / 2 : ℝ) ^ s := by
induction s with
| zero => simp
| succ s ih =>
rw [Finset.sum_range_succ, ih]
simp only [pow_succ]
ring
/-- The dyadic weights of all positive selector levels sum to at most one. -/
theorem selector_dyadic_sum_le_one (s : ℕ) :
(∑ i : Fin s, (1 / 2 : ℝ) ^ (i.val + 1)) ≤ 1 := by
calc
(∑ i : Fin s, (1 / 2 : ℝ) ^ (i.val + 1)) =
∑ i ∈ Finset.range s, (1 / 2 : ℝ) ^ (i + 1) :=
Fin.sum_univ_eq_sum_range (fun i : ℕ => (1 / 2 : ℝ) ^ (i + 1)) s
_ = 1 - (1 / 2 : ℝ) ^ s := selector_dyadic_range_sum s
_ ≤ 1 := sub_le_self _ (pow_nonneg (by norm_num) s)
end TalagrandThreshold
end OAI
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