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785c727 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 | import OAI.Geometry.CylinderCovering.Angular.CartesianFailure
namespace OAI
namespace AngularCartesianValidation
theorem sqrt_quarter_cubic_remainder {v : ℝ} (hv : |v| ≤ 1 / 8) :
|Real.sqrt (1 / 4 + v) - (1 / 2 + v - v ^ 2)| ≤ 6 * |v| ^ 3 := by
have hvl := (abs_le.mp hv).1
have hvr := (abs_le.mp hv).2
have hv2 : v ^ 2 ≤ 1 / 64 := by nlinarith only [hvl, hvr]
have hq : 0 ≤ 1 / 4 + v := by linarith
let r := Real.sqrt (1 / 4 + v)
let p := 1 / 2 + v - v ^ 2
have hr0 : 0 ≤ r := Real.sqrt_nonneg _
have hrsq : r ^ 2 = 1 / 4 + v := Real.sq_sqrt hq
have hr : 1 / 4 ≤ r := by nlinarith only [hr0, hrsq, hvl]
have hp : 1 / 4 ≤ p := by dsimp [p]; linarith
have hden : 1 / 2 ≤ r + p := by linarith
have hden0 : 0 ≤ r + p := by linarith
have hid : (r - p) * (r + p) = 2 * v ^ 3 - v ^ 4 := by
dsimp [p]
nlinarith only [hrsq]
have hv4 : |v| ^ 4 ≤ |v| ^ 3 := by
have hv1 : |v| ≤ 1 := hv.trans (by norm_num)
nlinarith only [mul_le_mul_of_nonneg_left hv1 (pow_nonneg (abs_nonneg v) 3)]
have hres : |2 * v ^ 3 - v ^ 4| ≤ 3 * |v| ^ 3 := by
calc
|2 * v ^ 3 - v ^ 4| ≤ |2 * v ^ 3| + |v ^ 4| := abs_sub _ _
_ = 2 * |v| ^ 3 + |v| ^ 4 := by rw [abs_mul, abs_pow, abs_pow]; norm_num
_ ≤ 3 * |v| ^ 3 := by linarith
have hprod : |r - p| * (r + p) ≤ 3 * |v| ^ 3 := by
calc
|r - p| * (r + p) = |(r - p) * (r + p)| := by
rw [abs_mul, abs_of_nonneg hden0]
_ = |2 * v ^ 3 - v ^ 4| := congrArg abs hid
_ ≤ 3 * |v| ^ 3 := hres
have hlo : |r - p| / 2 ≤ |r - p| * (r + p) := by
nlinarith only [mul_nonneg (abs_nonneg (r - p)) (sub_nonneg.mpr hden)]
change |r - p| ≤ 6 * |v| ^ 3
linarith only [hlo, hprod]
theorem linear_quadratic_perturbation_bound {τ z a M : ℝ}
(hτ : 0 ≤ τ) (hz : |z| ≤ 1) (ha : |a| ≤ M)
(hsmall : τ * M ≤ 1) :
|τ * z + τ ^ 2 * a| ≤ 2 * τ := by
calc
|τ * z + τ ^ 2 * a| ≤ |τ * z| + |τ ^ 2 * a| := abs_add_le _ _
_ = τ * |z| + τ ^ 2 * |a| := by
rw [abs_mul, abs_of_nonneg hτ, abs_mul, abs_of_nonneg (sq_nonneg τ)]
_ ≤ τ * 1 + τ ^ 2 * M := by gcongr
_ ≤ 2 * τ := by
nlinarith only [mul_le_mul_of_nonneg_left hsmall hτ]
theorem linear_quadratic_square_error {τ z a M : ℝ}
(hτ : 0 ≤ τ) (hz : |z| ≤ 1) (ha : |a| ≤ M)
(hsmall : τ * M ≤ 1) :
|(τ * z + τ ^ 2 * a) ^ 2 - (τ * z) ^ 2| ≤ 3 * M * τ ^ 3 := by
have hv := linear_quadratic_perturbation_bound hτ hz ha hsmall
have hzτ : |τ * z| ≤ τ := by
rw [abs_mul, abs_of_nonneg hτ]
nlinarith only [mul_le_mul_of_nonneg_left hz hτ]
have hplus : |(τ * z + τ ^ 2 * a) + τ * z| ≤ 3 * τ :=
(abs_add_le _ _).trans (by linarith)
have hminus : |(τ * z + τ ^ 2 * a) - τ * z| ≤ τ ^ 2 * M := by
calc
|(τ * z + τ ^ 2 * a) - τ * z| = τ ^ 2 * |a| := by
rw [add_sub_cancel_left, abs_mul, abs_of_nonneg (sq_nonneg τ)]
_ ≤ τ ^ 2 * M := mul_le_mul_of_nonneg_left ha (sq_nonneg τ)
have hM : 0 ≤ M := (abs_nonneg a).trans ha
calc
|(τ * z + τ ^ 2 * a) ^ 2 - (τ * z) ^ 2| =
|(τ * z + τ ^ 2 * a) - τ * z| * |(τ * z + τ ^ 2 * a) + τ * z| := by
rw [← abs_mul]
congr 1
ring
_ ≤ (τ ^ 2 * M) * (3 * τ) :=
mul_le_mul hminus hplus (abs_nonneg _) (mul_nonneg (sq_nonneg τ) hM)
_ = 3 * M * τ ^ 3 := by ring
theorem paired_sqrt_cubic_remainder {τ z a b M : ℝ}
(hτ : 0 ≤ τ) (hz : |z| ≤ 1) (ha : |a| ≤ M) (hb : |b| ≤ M)
(hM : 1 ≤ M) (hsmall : τ * M ≤ 1 / 16) :
|Real.sqrt (1 / 4 + τ * z + τ ^ 2 * a) +
Real.sqrt (1 / 4 - τ * z + τ ^ 2 * b) -
(1 + τ ^ 2 * (a + b - 2 * z ^ 2))| ≤ 128 * M * τ ^ 3 := by
have hsmall1 : τ * M ≤ 1 := hsmall.trans (by norm_num)
have hτ16 : τ ≤ 1 / 16 := by
nlinarith only [mul_le_mul_of_nonneg_left hM hτ, hsmall]
have hnz : |-z| ≤ 1 := by simpa only [abs_neg] using hz
have hva := linear_quadratic_perturbation_bound hτ hz ha hsmall1
have hvb := linear_quadratic_perturbation_bound hτ hnz hb hsmall1
have hqa := sqrt_quarter_cubic_remainder (hva.trans (by linarith : 2 * τ ≤ 1 / 8))
have hqb := sqrt_quarter_cubic_remainder (hvb.trans (by linarith : 2 * τ ≤ 1 / 8))
have hpowA : |τ * z + τ ^ 2 * a| ^ 3 ≤ (2 * τ) ^ 3 :=
pow_le_pow_left₀ (abs_nonneg _) hva 3
have hpowB : |τ * -z + τ ^ 2 * b| ^ 3 ≤ (2 * τ) ^ 3 :=
pow_le_pow_left₀ (abs_nonneg _) hvb 3
have hqa' : |Real.sqrt (1 / 4 + (τ * z + τ ^ 2 * a)) -
(1 / 2 + (τ * z + τ ^ 2 * a) - (τ * z + τ ^ 2 * a) ^ 2)| ≤ 48 * τ ^ 3 := by
nlinarith only [hqa, hpowA]
have hqb' : |Real.sqrt (1 / 4 + (τ * -z + τ ^ 2 * b)) -
(1 / 2 + (τ * -z + τ ^ 2 * b) - (τ * -z + τ ^ 2 * b) ^ 2)| ≤ 48 * τ ^ 3 := by
nlinarith only [hqb, hpowB]
have hsa := linear_quadratic_square_error hτ hz ha hsmall1
have hsb := linear_quadratic_square_error hτ hnz hb hsmall1
have hMτ : τ ^ 3 ≤ M * τ ^ 3 := by
nlinarith only [mul_le_mul_of_nonneg_right hM (pow_nonneg hτ 3)]
rw [abs_le] at hqa' hqb' hsa hsb ⊢
have hrewA : 1 / 4 + (τ * z + τ ^ 2 * a) = 1 / 4 + τ * z + τ ^ 2 * a := by ring
have hrewB : 1 / 4 + (τ * -z + τ ^ 2 * b) = 1 / 4 - τ * z + τ ^ 2 * b := by ring
rw [hrewA] at hqa'
rw [hrewB] at hqb'
constructor <;> nlinarith only [hqa'.1, hqa'.2, hqb'.1, hqb'.2,
hsa.1, hsa.2, hsb.1, hsb.2, hMτ, pow_nonneg hτ 3]
theorem cartesian_root_sum_expansion {x y τ η : ℝ}
(hx : |x| ≤ 1) (hy : |y| ≤ 1) (hτ : 0 ≤ τ) (hη : 0 ≤ η)
(hsmall : τ * (2 + η) ≤ 1 / 16) :
|Real.sqrt (1 / 4 + τ * x * y +
τ ^ 2 * ((y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4)) +
Real.sqrt (1 / 4 - τ * x * y +
τ ^ 2 * ((x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4)) -
(1 + τ ^ 2 * ((x ^ 2 - y ^ 2) ^ 2 + η))| ≤
128 * (2 + η) * τ ^ 3 := by
have hx2 : x ^ 2 ≤ 1 := by
nlinarith only [(abs_le.mp hx).1, (abs_le.mp hx).2]
have hy2 : y ^ 2 ≤ 1 := by
nlinarith only [(abs_le.mp hy).1, (abs_le.mp hy).2]
have hx4 : x ^ 4 ≤ 1 := by
nlinarith only [mul_nonneg (sq_nonneg x) (sub_nonneg.mpr hx2), hx2]
have hy4 : y ^ 4 ≤ 1 := by
nlinarith only [mul_nonneg (sq_nonneg y) (sub_nonneg.mpr hy2), hy2]
have hz : |x * y| ≤ 1 := by
rw [abs_mul]
nlinarith only [mul_le_mul hx hy (abs_nonneg y) (by norm_num : (0 : ℝ) ≤ 1)]
have ha : |(y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4| ≤ 2 + η := by
rw [abs_le]
constructor <;> nlinarith only [hx2, hy2, hx4, hy4, hη,
sq_nonneg x, sq_nonneg y, sq_nonneg (x ^ 2), sq_nonneg (y ^ 2)]
have hb : |(x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4| ≤ 2 + η := by
rw [abs_le]
constructor <;> nlinarith only [hx2, hy2, hx4, hy4, hη,
sq_nonneg x, sq_nonneg y, sq_nonneg (x ^ 2), sq_nonneg (y ^ 2)]
have h := paired_sqrt_cubic_remainder hτ hz ha hb (by linarith : 1 ≤ 2 + η) hsmall
have hid : ((y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4) +
((x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4) - 2 * (x * y) ^ 2 =
(x ^ 2 - y ^ 2) ^ 2 + η := by ring
rw [hid] at h
simpa only [mul_assoc] using h
theorem cartesian_root_sum_gt_one {x y τ η : ℝ}
(hx : |x| ≤ 1) (hy : |y| ≤ 1) (hτ : 0 < τ) (hη : 0 < η)
(hsmall : τ * (2 + η) ≤ 1 / 16)
(hmargin : τ * (256 * (2 + η)) ≤ η) :
1 < Real.sqrt (1 / 4 + τ * x * y +
τ ^ 2 * ((y ^ 2 / 4 + y ^ 4) + η / 2 - x ^ 2 / 4)) +
Real.sqrt (1 / 4 - τ * x * y +
τ ^ 2 * ((x ^ 2 / 4 + x ^ 4) + η / 2 - y ^ 2 / 4)) := by
have hexp := cartesian_root_sum_expansion hx hy hτ.le hη.le hsmall
have hbudget := mul_le_mul_of_nonneg_right hmargin (sq_nonneg τ)
have hpositive := mul_pos hη (sq_pos_of_pos hτ)
have hquartic := mul_nonneg (sq_nonneg τ) (sq_nonneg (x ^ 2 - y ^ 2))
nlinarith only [(abs_le.mp hexp).1, hbudget, hpositive, hquartic]
end AngularCartesianValidation
end OAI
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