import OAI.Combinatorics.EditApproximation.Geometry.Parameters namespace OAI /-! Computed numerical slack for comparison-band coverage at non-bad nodes. -/ namespace EditApproximation theorem computed_branching_fourth_power_le_quarterRoot (N : ℕ) : ((integerParameters N).M : ℝ) ^ 4 ≤ heightQuarterRoot N := by have hexponent : 16 * (heightExponent N / 20) ≤ heightExponent N := by have h := Nat.div_mul_le_self (heightExponent N) 20 omega have hpower : ((integerParameters N).M : ℝ) ^ 16 ≤ inputHeight N := by simp only [integerParameters, Nat.cast_pow, Nat.cast_ofNat, inputHeight] rw [← pow_mul] exact pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2) (by omega) apply (pow_le_pow_iff_left₀ (by positivity : (0 : ℝ) ≤ ((integerParameters N).M : ℝ) ^ 4) (Real.sqrt_nonneg _) (by decide : 4 ≠ 0)).mp change (((integerParameters N).M : ℝ) ^ 4) ^ 4 ≤ heightQuarterRoot N ^ 4 rw [← pow_mul, heightQuarterRoot_fourth_power] exact hpower theorem band_root_slack {ell root M : ℝ} (hell : 2 ≤ ell) (hroot : 0 ≤ root) (hlarge : ell ^ 5000 ≤ root) (_hM : 0 ≤ M) (hMfour : M ^ 4 ≤ root) : 128 * ell ^ 90 * M ^ 2 ≤ root := by have hellone : 1 ≤ ell := by linarith have hcoef : (16384 : ℝ) ≤ ell ^ 14 := by have h := pow_le_pow_left₀ (by norm_num : (0 : ℝ) ≤ 2) hell 14 norm_num at h exact h have hbig : 16384 * ell ^ 180 ≤ root := by calc 16384 * ell ^ 180 ≤ ell ^ 14 * ell ^ 180 := mul_le_mul_of_nonneg_right hcoef (by positivity) _ = ell ^ 194 := by rw [← pow_add] _ ≤ ell ^ 5000 := pow_le_pow_right₀ hellone (by decide) _ ≤ root := hlarge have hproduct := mul_le_mul hMfour hbig (by positivity : (0 : ℝ) ≤ 16384 * ell ^ 180) hroot apply (sq_le_sq₀ (by positivity : 0 ≤ 128 * ell ^ 90 * M ^ 2) hroot).mp calc (128 * ell ^ 90 * M ^ 2) ^ 2 = M ^ 4 * (16384 * ell ^ 180) := by ring _ ≤ root * root := hproduct _ = root ^ 2 := by ring theorem nonbad_band_numerical_bound {ell root M b K G W : ℝ} (hell : 2 ≤ ell) (hM : 1 ≤ M) (hb : 0 ≤ b) (hroot : 128 * ell ^ 90 * M ^ 2 ≤ root) (_hK : 0 ≤ K) (_hG : 0 ≤ G) (hW : 0 ≤ W) (hKsmall : K ≤ b / root) (hGsmall : G ≤ b / root) (hWsmall : W ≤ 8 * ell * b) : Real.sqrt (K * W) + 2 * G ≤ (1 / ell ^ 4) * (1 / ell ^ 40) * b / M := by have hellpos : 0 < ell := by linarith have hellone : 1 ≤ ell := by linarith have hMpos : 0 < M := by linarith have hMtwo : M ≤ M ^ 2 := by nlinarith only [hM] have hrootpos : 0 < root := (by positivity : 0 < 128 * ell ^ 90 * M ^ 2).trans_le hroot have hscale : 32 * ell ^ 89 * M ^ 2 ≤ root := by calc 32 * ell ^ 89 * M ^ 2 ≤ 128 * ell ^ 90 * M ^ 2 := by have hp : ell ^ 89 ≤ ell ^ 90 := pow_le_pow_right₀ hellone (by decide : 89 ≤ 90) have hc : 32 * ell ^ 89 ≤ 128 * ell ^ 90 := mul_le_mul (by norm_num : (32 : ℝ) ≤ 128) hp (by positivity) (by norm_num) exact mul_le_mul_of_nonneg_right hc (sq_nonneg M) _ ≤ root := hroot have hscale' : 4 * ell ^ 44 * M ≤ root := by calc 4 * ell ^ 44 * M ≤ 128 * ell ^ 90 * M ^ 2 := by have hp : ell ^ 44 ≤ ell ^ 90 := pow_le_pow_right₀ hellone (by decide : 44 ≤ 90) have hc : 4 * ell ^ 44 ≤ 128 * ell ^ 90 := mul_le_mul (by norm_num : (4 : ℝ) ≤ 128) hp (by positivity) (by norm_num) exact mul_le_mul hc hMtwo hMpos.le (by positivity) _ ≤ root := hroot have hKW : K * W ≤ (8 * ell * b ^ 2) / root := by calc K * W ≤ (b / root) * (8 * ell * b) := mul_le_mul hKsmall hWsmall hW (div_nonneg hb hrootpos.le) _ = (8 * ell * b ^ 2) / root := by ring have hsquare : K * W ≤ (b / (2 * ell ^ 44 * M)) ^ 2 := by calc K * W ≤ (8 * ell * b ^ 2) / root := hKW _ ≤ b ^ 2 / (4 * ell ^ 88 * M ^ 2) := by apply (div_le_div_iff₀ hrootpos (by positivity : 0 < 4 * ell ^ 88 * M ^ 2)).2 have h := mul_le_mul_of_nonneg_right hscale (sq_nonneg b) nlinarith only [h, show (8 * ell * b ^ 2) * (4 * ell ^ 88 * M ^ 2) = (32 * ell ^ 89 * M ^ 2) * b ^ 2 by ring] _ = (b / (2 * ell ^ 44 * M)) ^ 2 := by field_simp; ring have hsqrt : Real.sqrt (K * W) ≤ b / (2 * ell ^ 44 * M) := (Real.sqrt_le_left (by positivity)).2 hsquare have htwice : 2 * G ≤ b / (2 * ell ^ 44 * M) := by calc 2 * G ≤ 2 * (b / root) := mul_le_mul_of_nonneg_left hGsmall (by norm_num) _ = (2 * b) / root := by ring _ ≤ b / (2 * ell ^ 44 * M) := by apply (div_le_div_iff₀ hrootpos (by positivity : 0 < 2 * ell ^ 44 * M)).2 have h := mul_le_mul_of_nonneg_right hscale' hb nlinarith only [h] have hid : b / (2 * ell ^ 44 * M) + b / (2 * ell ^ 44 * M) = (1 / ell ^ 4) * (1 / ell ^ 40) * b / M := by field_simp ring exact (add_le_add hsqrt htwice).trans_eq hid theorem computed_nonbad_band_numerical_bound (N : ℕ) (hN : 0 < N) (hell : 2 ^ 1000 ≤ inputSmallLog N) {b K G W : ℝ} (hb : 0 ≤ b) (hK : 0 ≤ K) (hG : 0 ≤ G) (hW : 0 ≤ W) (hKsmall : K ≤ b / heightQuarterRoot N) (hGsmall : G ≤ b / heightQuarterRoot N) (hWsmall : W ≤ 8 * inputSmallLog N * b) : Real.sqrt (K * W) + 2 * G ≤ ((rationalParameters (inputSmallLog N)).tau : ℝ) * ((rationalParameters (inputSmallLog N)).theta : ℝ) * b / (integerParameters N).M := by have helltwo : (2 : ℝ) ≤ inputSmallLog N := by have htwo : 2 ≤ (2 : ℕ) ^ 1000 := by simpa only [pow_one] using pow_le_pow_right₀ (by decide : 1 ≤ (2 : ℕ)) (by decide : 1 ≤ 1000) exact_mod_cast htwo.trans hell have hroot := band_root_slack helltwo (Real.sqrt_nonneg _) (fourth_root_polynomial_slack (by linarith) (inputHeight_polynomial_slack N hN hell) (Real.sqrt_nonneg _) (heightQuarterRoot_fourth_power N)) (Nat.cast_nonneg (integerParameters N).M) (computed_branching_fourth_power_le_quarterRoot N) have hM : (1 : ℝ) ≤ (integerParameters N).M := by exact_mod_cast (Nat.one_le_two_pow (n := heightExponent N / 20)) have h := nonbad_band_numerical_bound helltwo hM hb hroot hK hG hW hKsmall hGsmall hWsmall simpa only [rationalParameters, Rat.cast_div, Rat.cast_one, Rat.cast_pow, Rat.cast_natCast] using h end EditApproximation end OAI