| import OAI.Combinatorics.ContingencyTables.Dense.DenseMixing | |
| namespace OAI | |
| /-! # Error budget of the bounded dense rejection schedule -/ | |
| namespace ContingencyTables.GridBoundary | |
| theorem succ_le_two_pow (h : ℕ) : h + 1 ≤ 2 ^ h := by | |
| exact Nat.succ_le_of_lt Nat.lt_two_pow_self | |
| theorem square_le_two_pow (d : ℕ) (hd : 4 ≤ d) : d ^ 2 ≤ 2 ^ d := by | |
| induction d, hd using Nat.le_induction with | |
| | base => norm_num | |
| | succ d hd ih => | |
| calc | |
| (d + 1) ^ 2 ≤ 2 * d ^ 2 := by nlinarith | |
| _ ≤ 2 * 2 ^ d := Nat.mul_le_mul_left _ ih | |
| _ = 2 ^ (d + 1) := by rw [pow_succ, Nat.mul_comm] | |
| theorem dense_trial_error_budget (d h : ℕ) (hd : 14 ≤ d) : | |
| ((d ^ 4 * (h + 1) * (1 + d) : ℕ) : ℝ) * | |
| (2 : ℝ) ^ (-(4 * ((h : ℝ) + d))) ≤ (2 : ℝ) ^ (-(h : ℝ) - 2) := by | |
| have hd2 : (d : ℝ) ^ 2 ≤ (2 : ℝ) ^ d := by exact_mod_cast square_le_two_pow d (by omega) | |
| have hd4 : (d : ℝ) ^ 4 ≤ (2 : ℝ) ^ (2 * d) := by | |
| calc | |
| _ = ((d : ℝ) ^ 2) ^ 2 := by ring | |
| _ ≤ ((2 : ℝ) ^ d) ^ 2 := pow_le_pow_left₀ (by positivity) hd2 _ | |
| _ = _ := by rw [← pow_mul]; congr 1; omega | |
| have hh : (h : ℝ) + 1 ≤ (2 : ℝ) ^ h := by | |
| exact_mod_cast succ_le_two_pow h | |
| have hd1 : (1 : ℝ) + d ≤ (2 : ℝ) ^ d := by | |
| exact_mod_cast (show 1 + d ≤ 2 ^ d by simpa only [Nat.add_comm] using succ_le_two_pow d) | |
| have hprod : ((d ^ 4 * (h + 1) * (1 + d) : ℕ) : ℝ) ≤ | |
| (2 : ℝ) ^ ((h : ℝ) + 3 * d) := by | |
| push_cast | |
| calc | |
| _ ≤ (2 : ℝ) ^ (2 * d) * (2 : ℝ) ^ h * (2 : ℝ) ^ d := by gcongr | |
| _ = _ := by | |
| rw [← pow_add, ← pow_add, ← Real.rpow_natCast] | |
| congr 1 | |
| push_cast | |
| ring | |
| calc | |
| _ ≤ (2 : ℝ) ^ ((h : ℝ) + 3 * d) * (2 : ℝ) ^ (-(4 * ((h : ℝ) + d))) := | |
| mul_le_mul_of_nonneg_right hprod (by positivity) | |
| _ = (2 : ℝ) ^ (-(3 * (h : ℝ) + d)) := by | |
| rw [← Real.rpow_add (by norm_num)] | |
| congr 1 | |
| ring | |
| _ ≤ _ := Real.rpow_le_rpow_of_exponent_le (by norm_num) (by | |
| have hd' : (14 : ℝ) ≤ d := by exact_mod_cast hd | |
| have hh' : (0 : ℝ) ≤ h := Nat.cast_nonneg _ | |
| linarith) | |
| theorem dense_failure_error_budget (d h : ℕ) (hd : 14 ≤ d) : | |
| (63 / 64 : ℝ) ^ (d ^ 4 * (h + 1)) ≤ (2 : ℝ) ^ (-(h : ℝ) - 2) := by | |
| have hd4 : 128 ≤ d ^ 4 := by nlinarith [show 14 ^ 4 ≤ d ^ 4 from Nat.pow_le_pow_left hd _] | |
| have hs : (63 / 64 : ℝ) ≤ Real.exp (-(1 / 64 : ℝ)) := by | |
| linarith [Real.add_one_le_exp (-(1 / 64 : ℝ))] | |
| calc | |
| _ ≤ (Real.exp (-(1 / 64 : ℝ))) ^ (d ^ 4 * (h + 1)) := | |
| pow_le_pow_left₀ (by norm_num) hs _ | |
| _ = Real.exp (-(((d ^ 4 * (h + 1) : ℕ) : ℝ) / 64)) := by | |
| rw [← Real.exp_nat_mul] | |
| congr 1 | |
| ring | |
| _ ≤ Real.exp (-(2 * ((h : ℝ) + 1))) := by | |
| apply Real.exp_le_exp.mpr | |
| have h' : (128 : ℝ) ≤ (d : ℝ) ^ 4 := by exact_mod_cast hd4 | |
| push_cast | |
| nlinarith [show (0 : ℝ) ≤ h from Nat.cast_nonneg h] | |
| _ ≤ (2 : ℝ) ^ (-(2 * ((h : ℝ) + 1))) := by | |
| rw [Real.rpow_def_of_pos (by norm_num)] | |
| apply Real.exp_le_exp.mpr | |
| have hl : Real.log 2 ≤ 1 := by linarith [Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2)] | |
| have hh' : (0 : ℝ) ≤ h := Nat.cast_nonneg _ | |
| nlinarith | |
| _ ≤ _ := Real.rpow_le_rpow_of_exponent_le (by norm_num) (by | |
| have hh' : (0 : ℝ) ≤ h := Nat.cast_nonneg _ | |
| linarith) | |
| theorem dense_total_error_budget (d h : ℕ) (hd : 14 ≤ d) : | |
| (63 / 64 : ℝ) ^ (d ^ 4 * (h + 1)) + | |
| ((d ^ 4 * (h + 1) * (1 + d) : ℕ) : ℝ) * | |
| (2 : ℝ) ^ (-(4 * ((h : ℝ) + d))) ≤ (2 : ℝ) ^ (-(h : ℝ)) := by | |
| have hf := dense_failure_error_budget d h hd | |
| have ht := dense_trial_error_budget d h hd | |
| have heq : (2 : ℝ) ^ (-(h : ℝ) - 2) = (2 : ℝ) ^ (-(h : ℝ)) / 4 := by | |
| rw [Real.rpow_sub (by norm_num)] | |
| norm_num | |
| rw [heq] at hf ht | |
| linarith [Real.rpow_pos_of_pos (by norm_num : (0 : ℝ) < 2) (-(h : ℝ))] | |
| end ContingencyTables.GridBoundary | |
| end OAI | |