File size: 2,688 Bytes
f89a930 | 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 69 70 | import OAI.NumberTheory.Jacobsthal.Estimates.FiniteSampleSpace
namespace OAI
namespace Erdos970
section
namespace ErdosInverseSampling
attribute [local instance] Classical.decEq
theorem sample_length_le (n h : ℕ) (beta : ℝ) (hb : 0 < beta) (hb1 : beta ≤ 1)
(hsize : 4*(h : ℝ)/beta ≤ (n : ℝ)) : h ≤ n := by
have hh := (div_le_iff₀ hb).mp hsize
have hbmul := mul_le_mul_of_nonneg_right hb1 (Nat.cast_nonneg n : (0 : ℝ) ≤ n)
have hr : (h : ℝ) ≤ n := by nlinarith [show (0 : ℝ) ≤ h from Nat.cast_nonneg h]
exact_mod_cast hr
theorem falling_ratio_lower (n m h : ℕ) (beta : ℝ) (hb : 0 < beta) (hb1 : beta ≤ 1)
(hgood : (beta/2)*(n : ℝ) ≤ (m : ℝ))
(hsize : 4*(h : ℝ)/beta ≤ (n : ℝ)) :
(beta/4)^h ≤ (m.descFactorial h : ℝ)/(n.descFactorial h : ℝ) := by
have hhn := sample_length_le n h beta hb hb1 hsize
have hpay := (div_le_iff₀ hb).mp hsize
have hhm : h ≤ m := by
have hh : (h : ℝ) ≤ m := by nlinarith [show (0 : ℝ) ≤ h from Nat.cast_nonneg h]
exact_mod_cast hh
have hnum : ((beta/4)*(n : ℝ))^h ≤ (m.descFactorial h : ℝ) := by
rw [Nat.descFactorial_eq_prod_range,Nat.cast_prod]
calc
_ = ∏ _i ∈ Finset.range h,(beta/4)*(n : ℝ) := by simp
_ ≤ ∏ i ∈ Finset.range h,((m-i : ℕ) : ℝ) := by
apply Finset.prod_le_prod₀ (fun _ _ => by positivity)
intro i hi
have hih : i < h := Finset.mem_range.mp hi
have him : i ≤ m := by omega
rw [Nat.cast_sub him]
have hiR : (i : ℝ) ≤ h := by exact_mod_cast hih.le
nlinarith
have hden : (n.descFactorial h : ℝ) ≤ (n : ℝ)^h := by
exact_mod_cast Nat.descFactorial_le_pow n h
have hdenpos : (0 : ℝ) < n.descFactorial h := by exact_mod_cast Nat.descFactorial_pos.mpr hhn
apply (le_div_iff₀ hdenpos).mpr
calc
_ ≤ (beta/4)^h*(n : ℝ)^h := mul_le_mul_of_nonneg_left hden (by positivity)
_ = ((beta/4)*(n : ℝ))^h := by rw [mul_pow]
_ ≤ _ := hnum
theorem sampleFrequency_lower {α : Type*} (U W : Finset α) (hWU : W ⊆ U) (h : ℕ)
(beta : ℝ) (hb : 0 < beta) (hb1 : beta ≤ 1)
(hgood : (beta/2)*(U.card : ℝ) ≤ (W.card : ℝ))
(hsize : 4*(h : ℝ)/beta ≤ (U.card : ℝ)) :
(beta/4)^h ≤ sampleFrequency U W h := by
rw [sampleFrequency_eq U W hWU h]
exact falling_ratio_lower U.card W.card h beta hb hb1 hgood hsize
theorem sample_nonempty_of_length {α : Type*} (U : Finset α) (h : ℕ) (hh : h ≤ U.card) :
Nonempty (Sample U h) := by
apply Fintype.card_pos_iff.mp
rw [sample_card]
exact Nat.descFactorial_pos.mpr hh
end ErdosInverseSampling
end
end Erdos970
end OAI
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