openAI-math-dataset / lean /OAI /Combinatorics /LowSpaceSubsetSum /Separators /SeparatorSamplingBounds.lean
| 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 | |