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e63eb7f | 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 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 | import OAI.Combinatorics.EditApproximation.Geometry.BandRounding
import OAI.Combinatorics.EditApproximation.Geometry.Parameters
namespace OAI
/-! The input-computed dyadic parameters discharge all spacing and basic rounding premises. -/
namespace EditApproximation
theorem computed_theta_dyadic (N : ℕ) :
(rationalParameters (inputSmallLog N)).theta =
(2 : ℚ) ^ (-((40 * smallLogExponent N : ℕ) : ℤ)) := by
change 1 / (((2 ^ smallLogExponent N : ℕ) : ℚ) ^ 40) = _
rw [Nat.cast_pow, Nat.cast_ofNat, ← pow_mul]
rw [zpow_neg, zpow_natCast]
rw [Nat.mul_comm (smallLogExponent N) 40, one_div]
theorem computed_P_dyadic (N : ℕ) :
(integerParameters N).P =
2 ^ (heightExponent N * (coarseExponent (inputSmallLog N) + 20)) := by
change (2 ^ heightExponent N) ^ (coarseExponent (inputSmallLog N) + 20) = _
rw [pow_mul]
theorem basic_band_small_constants {ell : ℚ} (hell : 2 ≤ ell) :
0 < 1 / ell ^ 4 ∧ 0 < 1 / ell ^ 40 ∧ 1 / ell ^ 40 ≤ 1 / 2 ∧
32 * ell * (1 / ell ^ 40) ≤ 1 ∧ 64 * ell * (1 / ell ^ 40) ≤ 1 := by
have hellpos : 0 < ell := by linarith
have hellone : 1 ≤ ell := by linarith
have hcoef : (64 : ℚ) ≤ ell ^ 6 := by
have h := pow_le_pow_left₀ (by norm_num : (0 : ℚ) ≤ 2) hell 6
norm_num at h
exact h
have hpower : (64 : ℚ) ≤ ell ^ 39 :=
hcoef.trans (pow_le_pow_right₀ hellone (by decide : 6 ≤ 39))
have hscale : 64 * ell ≤ ell ^ 40 := by
calc
64 * ell ≤ ell ^ 39 * ell := mul_le_mul_of_nonneg_right hpower hellpos.le
_ = ell ^ 40 := (pow_succ ell 39).symm
have h64 : 64 * ell * (1 / ell ^ 40) ≤ 1 := by
rw [mul_one_div]
exact (div_le_one₀ (pow_pos hellpos 40)).2 hscale
have h32 : 32 * ell * (1 / ell ^ 40) ≤ 1 := by
have hnonneg : 0 ≤ ell * (1 / ell ^ 40) := by positivity
nlinarith only [h64, hnonneg]
have htheta : 1 / ell ^ 40 ≤ 1 / 2 := by
apply (div_le_div_iff₀ (pow_pos hellpos 40) (by norm_num : (0 : ℚ) < 2)).2
have h := pow_le_pow_right₀ hellone (by decide : 1 ≤ 40)
simp only [pow_one] at h
linarith only [hell, h]
exact ⟨by positivity, by positivity, htheta, h32, h64⟩
theorem computed_band_basic_constants (N : ℕ) (hell : 2 ≤ inputSmallLog N) :
0 < (rationalParameters (inputSmallLog N)).tau ∧
0 < (rationalParameters (inputSmallLog N)).theta ∧
(rationalParameters (inputSmallLog N)).theta ≤ 1 / 2 ∧
(32 * inputSmallLog N : ℚ) * (rationalParameters (inputSmallLog N)).theta ≤ 1 ∧
(64 * inputSmallLog N : ℚ) * (rationalParameters (inputSmallLog N)).theta ≤ 4 := by
have h := basic_band_small_constants (show (2 : ℚ) ≤ inputSmallLog N by exact_mod_cast hell)
exact ⟨h.1, h.2.1, h.2.2.1, h.2.2.2.1, h.2.2.2.2.trans (by norm_num)⟩
theorem computed_band_spacing_dominates (N : ℕ) (hN : 0 < N)
(hell : 2 ^ 1000 ≤ inputSmallLog N) :
((integerParameters N).M : ℚ) ≤
(rationalParameters (inputSmallLog N)).theta * (integerParameters N).P := by
have hellone : (1 : ℝ) ≤ inputSmallLog N := by
have h : 1 ≤ inputSmallLog N := Nat.one_le_two_pow
exact_mod_cast h
have hH40 : (inputSmallLog N : ℝ) ^ 40 ≤ inputHeight N :=
(pow_le_pow_right₀ hellone (by decide : 40 ≤ 20000)).trans (inputHeight_polynomial_slack N hN hell)
have hH40Q : (inputSmallLog N : ℚ) ^ 40 ≤ inputHeight N := by exact_mod_cast hH40
have hM : (integerParameters N).M ≤ inputHeight N :=
pow_le_pow_right₀ (by decide : 1 ≤ (2 : ℕ)) (Nat.div_le_self _ _)
have hMQ : ((integerParameters N).M : ℚ) ≤ inputHeight N := by exact_mod_cast hM
have hHone : 1 ≤ inputHeight N := Nat.one_le_two_pow
have hPtwo : inputHeight N ^ 2 ≤ (integerParameters N).P :=
pow_le_pow_right₀ hHone (by omega : 2 ≤ coarseExponent (inputSmallLog N) + 20)
have hPtwoQ : (inputHeight N : ℚ) ^ 2 ≤ (integerParameters N).P := by exact_mod_cast hPtwo
have hproduct : ((integerParameters N).M : ℚ) * (inputSmallLog N : ℚ) ^ 40 ≤
(integerParameters N).P := by
have h := mul_le_mul hMQ hH40Q (pow_nonneg (Nat.cast_nonneg (inputSmallLog N)) 40)
(Nat.cast_nonneg (inputHeight N))
exact h.trans (by simpa only [pow_two] using hPtwoQ)
change ((integerParameters N).M : ℚ) ≤ (1 / (inputSmallLog N : ℚ) ^ 40) * (integerParameters N).P
rw [one_div_mul_eq_div]
have hellpos : (0 : ℚ) < inputSmallLog N := by exact_mod_cast (Nat.two_pow_pos (smallLogExponent N))
apply (le_div_iff₀ (pow_pos hellpos 40)).2
exact hproduct
end EditApproximation
end OAI
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