File size: 2,937 Bytes
5d5048e | 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 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.Combinatorics.LowSpaceSubsetSum.Separators.SeparatorExpectation
import Mathlib.Data.Nat.Choose.Bounds
namespace OAI
/-! # Finite rounded family sizes and expected incidence bounds -/
noncomputable section
namespace LowSpaceSubsetSum.SeparatorExpectation
/-- Rounding the sampled family size costs at most one extra trial. -/
theorem rounded_family_mass {θ : ℝ} (hθ : 0 < θ) (hθ1 : θ ≤ 1) :
(⌈10 / θ⌉₊ : ℝ) * θ ≤ 11 := by
have h := (Nat.ceil_lt_add_one (show 0 ≤ 10 / θ by positivity)).le
have hm := mul_le_mul_of_nonneg_right h hθ.le
rw [add_mul, div_mul_cancel₀ _ hθ.ne', one_mul] at hm
linarith
theorem rounded_family_incidence (A D N : ℕ) (hD : 0 < D) (hN : 0 < N)
(hDN : D ≤ N) :
(⌈10 / ((D : ℝ) / N)⌉₊ : ℝ) * ((A : ℝ) / N) ≤ 11 * ((A : ℝ) / D) := by
have hDR : (0 : ℝ) < D := by exact_mod_cast hD
have hNR : (0 : ℝ) < N := by exact_mod_cast hN
have hθ1 : (D : ℝ) / N ≤ 1 := (div_le_one hNR).mpr (by exact_mod_cast hDN)
have hm := rounded_family_mass (div_pos hDR hNR) hθ1
have he : (⌈10 / ((D : ℝ) / N)⌉₊ : ℝ) * ((A : ℝ) / N) =
((⌈10 / ((D : ℝ) / N)⌉₊ : ℝ) * ((D : ℝ) / N)) * ((A : ℝ) / D) := by
field_simp
rw [he]
exact mul_le_mul_of_nonneg_right hm (by positivity)
/-- This bound is uniform over every admissible sampled family. -/
theorem rounded_family_size (g e c : ℕ) (he : e ≤ g) (hfeas : e - c ≤ g / 2)
(hDN : (g / 2).choose (e - c) ≤ g.choose e) :
⌈10 / ((Nat.choose (g / 2) (e - c) : ℝ) / Nat.choose g e)⌉₊ ≤ 11 * 2 ^ g := by
have hD : 0 < (g / 2).choose (e - c) := Nat.choose_pos hfeas
have hN : 0 < g.choose e := Nat.choose_pos he
have hm := rounded_family_incidence (g.choose e) ((g / 2).choose (e - c))
(g.choose e) hD hN hDN
have hNR : (0 : ℝ) < g.choose e := by exact_mod_cast hN
rw [div_self hNR.ne', mul_one] at hm
have hr : (g.choose e : ℝ) / (g / 2).choose (e - c) ≤ g.choose e := by
apply div_le_self (by positivity)
exact_mod_cast hD
have hb : (⌈10 / ((Nat.choose (g / 2) (e - c) : ℝ) / Nat.choose g e)⌉₊ : ℝ) ≤
11 * (2 : ℝ) ^ g := by
calc
_ ≤ 11 * ((g.choose e : ℝ) / (g / 2).choose (e - c)) := hm
_ ≤ 11 * (g.choose e : ℝ) := mul_le_mul_of_nonneg_left hr (by norm_num)
_ ≤ _ := mul_le_mul_of_nonneg_left (by exact_mod_cast Nat.choose_le_two_pow g e) (by norm_num)
exact_mod_cast hb
theorem rounded_family_depth (g e c : ℕ) (he : e ≤ g) (hfeas : e - c ≤ g / 2)
(hDN : (g / 2).choose (e - c) ≤ g.choose e) :
Nat.clog 2 (⌈10 / ((Nat.choose (g / 2) (e - c) : ℝ) / Nat.choose g e)⌉₊) ≤ g + 4 := by
apply Nat.clog_le_of_le_pow
have h := rounded_family_size g e c he hfeas hDN
calc
_ ≤ 11 * 2 ^ g := h
_ ≤ 2 ^ (g + 4) := by rw [pow_add]; norm_num; omega
end LowSpaceSubsetSum.SeparatorExpectation
end
end OAI
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