openAI-math-dataset / lean /OAI /NumberTheory /Jacobsthal /Probability /FallingFactorialProbability.lean
| 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 | |