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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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | 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
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