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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 | import OAI.Geometry.CylinderCovering.Cartesian.CapsDefinitions
namespace OAI
namespace CylinderCovering.CartesianCaps
theorem sqrt_quarter_cubic_remainder {v : ℝ} (hv : -(1 / 4 : ℝ) ≤ v) :
|Real.sqrt (1 / 4 + v) - (1 / 2 + v - v ^ 2)| ≤ 12 * |v| ^ 3 := by
let z : ℝ := Real.sqrt (1 / 4 + v) - 1 / 2
have ht0 := Real.sqrt_nonneg (1 / 4 + v)
have ht2 := Real.sq_sqrt (show 0 ≤ 1 / 4 + v by linarith)
have hzid : v = z + z ^ 2 := by dsimp [z]; nlinarith only [ht2]
have hzlower : -(1 / 2 : ℝ) ≤ z := by dsimp [z]; linarith only [ht0]
have hzbound : |z| ≤ 2 * |v| := by
by_cases hz : 0 ≤ z
· rw [abs_of_nonneg hz]
have hv0 : 0 ≤ v := by nlinarith only [hzid, hz, sq_nonneg z]
rw [abs_of_nonneg hv0]
nlinarith only [hzid, hz, sq_nonneg z]
· have hz0 := le_of_not_ge hz
rw [abs_of_nonpos hz0]
have hv0 : v ≤ 0 := by nlinarith only [hzid, hz0, hzlower]
rw [abs_of_nonpos hv0]
nlinarith only [hzid, hz0, hzlower]
have hvz : |v + z| ≤ 3 * |v| := (abs_add_le _ _).trans (by linarith)
have hzsq : |z| ^ 2 ≤ (2 * |v|) ^ 2 :=
(sq_le_sq₀ (abs_nonneg _) (by positivity)).2 hzbound
calc
_ = |z ^ 2 * (v + z)| := by
congr 1
dsimp [z] at *
nlinarith only [ht2]
_ = |z| ^ 2 * |v + z| := by rw [abs_mul, abs_pow]
_ ≤ (2 * |v|) ^ 2 * (3 * |v|) :=
mul_le_mul hzsq hvz (abs_nonneg _) (by positivity)
_ = 12 * |v| ^ 3 := by ring
theorem sqrt_quarter_perturbed_remainder {τ M a B : ℝ}
(hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) (hM : 0 ≤ M)
(ha : |a| ≤ 1) (hB : |B| ≤ M) (hsmall : (M + 1) * τ ≤ 1 / 4) :
|Real.sqrt (1 / 4 + τ * a + τ ^ 2 * B) -
(1 / 2 + τ * a + τ ^ 2 * (B - a ^ 2))| ≤
(12 * (M + 1) ^ 3 + M * (M + 2)) * τ ^ 3 := by
let v : ℝ := τ * a + τ ^ 2 * B
have hτ2 : τ ^ 2 ≤ τ := by nlinarith
have haτ : |τ * a| ≤ τ := by
rw [abs_mul, abs_of_nonneg hτ0]
simpa using mul_le_mul_of_nonneg_left ha hτ0
have hBτ : |τ ^ 2 * B| ≤ τ ^ 2 * M := by
rw [abs_mul, abs_of_nonneg (sq_nonneg τ)]
exact mul_le_mul_of_nonneg_left hB (sq_nonneg τ)
have hv : |v| ≤ (M + 1) * τ := by
have hMτ := mul_le_mul_of_nonneg_left hτ2 hM
exact (abs_add_le _ _).trans (by nlinarith only [haτ, hBτ, hMτ])
have hvdom : -(1 / 4 : ℝ) ≤ v := by
linarith only [(abs_le.mp hv).1, hsmall]
have hvpow : |v| ^ 3 ≤ ((M + 1) * τ) ^ 3 := by gcongr
have htaylor : |Real.sqrt (1 / 4 + v) - (1 / 2 + v - v ^ 2)| ≤
12 * (M + 1) ^ 3 * τ ^ 3 := by
calc
_ ≤ 12 * |v| ^ 3 := sqrt_quarter_cubic_remainder hvdom
_ ≤ 12 * ((M + 1) * τ) ^ 3 := by gcongr
_ = 12 * (M + 1) ^ 3 * τ ^ 3 := by ring
have hplus : |v + τ * a| ≤ (M + 2) * τ :=
(abs_add_le _ _).trans (by nlinarith only [hv, haτ])
have hminus : |v - τ * a| ≤ τ ^ 2 * M := by
simpa [v] using hBτ
have hsq : |v ^ 2 - (τ * a) ^ 2| ≤ M * (M + 2) * τ ^ 3 := by
calc
_ = |v - τ * a| * |v + τ * a| := by
rw [← abs_mul]
congr 1
ring
_ ≤ (τ ^ 2 * M) * ((M + 2) * τ) :=
mul_le_mul hminus hplus (abs_nonneg _) (by positivity)
_ = M * (M + 2) * τ ^ 3 := by ring
have hsplit : Real.sqrt (1 / 4 + τ * a + τ ^ 2 * B) -
(1 / 2 + τ * a + τ ^ 2 * (B - a ^ 2)) =
(Real.sqrt (1 / 4 + v) - (1 / 2 + v - v ^ 2)) -
(v ^ 2 - (τ * a) ^ 2) := by
dsimp [v]
rw [show 1 / 4 + (τ * a + τ ^ 2 * B) = 1 / 4 + τ * a + τ ^ 2 * B by ring]
ring
rw [hsplit]
exact (abs_sub _ _).trans (by nlinarith only [htaylor, hsq])
theorem sqrt_quarter_pair_remainder {τ M a B C : ℝ}
(hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) (hM : 0 ≤ M)
(ha : |a| ≤ 1) (hB : |B| ≤ M) (hC : |C| ≤ M)
(hsmall : (M + 1) * τ ≤ 1 / 4) :
|Real.sqrt (1 / 4 + τ * a + τ ^ 2 * B) +
Real.sqrt (1 / 4 - τ * a + τ ^ 2 * C) -
(1 + τ ^ 2 * (B + C - 2 * a ^ 2))| ≤
(24 * (M + 1) ^ 3 + 2 * M * (M + 2)) * τ ^ 3 := by
have hfirst := sqrt_quarter_perturbed_remainder hτ0 hτ1 hM ha hB hsmall
have hsecond := sqrt_quarter_perturbed_remainder hτ0 hτ1 hM
(a := -a) (by simpa only [abs_neg] using ha) hC hsmall
have hneg : 1 / 4 + τ * -a + τ ^ 2 * C = 1 / 4 - τ * a + τ ^ 2 * C := by ring
rw [hneg] at hsecond
have hsplit : Real.sqrt (1 / 4 + τ * a + τ ^ 2 * B) +
Real.sqrt (1 / 4 - τ * a + τ ^ 2 * C) -
(1 + τ ^ 2 * (B + C - 2 * a ^ 2)) =
(Real.sqrt (1 / 4 + τ * a + τ ^ 2 * B) -
(1 / 2 + τ * a + τ ^ 2 * (B - a ^ 2))) +
(Real.sqrt (1 / 4 - τ * a + τ ^ 2 * C) -
(1 / 2 + τ * -a + τ ^ 2 * (C - (-a) ^ 2))) := by ring
rw [hsplit]
exact (abs_add_le _ _).trans (by nlinarith only [hfirst, hsecond])
theorem cartesian_second_coefficient_bound {η x y : ℝ} (hη : 0 ≤ η)
(hx : |x| ≤ 1) (hy : |y| ≤ 1) :
|D y + η / 2 - x ^ 2 / 4| ≤ η + 2 := 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 hy4 : y ^ 4 ≤ 1 := by
nlinarith only [mul_nonneg (sq_nonneg y) (sub_nonneg.mpr hy2), hy2]
unfold D
rw [abs_le]
constructor <;> nlinarith only [hx2, hy2, hy4, sq_nonneg x, sq_nonneg y, sq_nonneg (y ^ 2), hη]
theorem cartesian_pair_expansion {τ η x y : ℝ}
(hτ0 : 0 ≤ τ) (hτ1 : τ ≤ 1) (hη : 0 ≤ η)
(hx : |x| ≤ 1) (hy : |y| ≤ 1) (hsmall : (η + 3) * τ ≤ 1 / 4) :
|Real.sqrt (1 / 4 + τ * (x * y) + τ ^ 2 * (D y + η / 2 - x ^ 2 / 4)) +
Real.sqrt (1 / 4 - τ * (x * y) + τ ^ 2 * (D x + η / 2 - y ^ 2 / 4)) -
(1 + τ ^ 2 * ((x ^ 2 - y ^ 2) ^ 2 + η))| ≤
(24 * (η + 3) ^ 3 + 2 * (η + 2) * (η + 4)) * τ ^ 3 := by
have hxy : |x * y| ≤ 1 := by
rw [abs_mul]
calc
|x| * |y| ≤ 1 * 1 := mul_le_mul hx hy (abs_nonneg _) (by norm_num)
_ = 1 := by norm_num
have h := sqrt_quarter_pair_remainder hτ0 hτ1 (M := η + 2) (by positivity)
hxy (cartesian_second_coefficient_bound hη hx hy)
(cartesian_second_coefficient_bound hη hy hx) (by linarith)
have hcoef : (D y + η / 2 - x ^ 2 / 4) + (D x + η / 2 - y ^ 2 / 4) -
2 * (x * y) ^ 2 = (x ^ 2 - y ^ 2) ^ 2 + η := by unfold D; ring
rw [hcoef] at h
convert h using 1
ring
theorem cartesian_pair_gt_one {τ η x y : ℝ}
(hτ : 0 < τ) (hτ1 : τ ≤ 1) (hη : 0 < η)
(hx : |x| ≤ 1) (hy : |y| ≤ 1) (hsmall : (η + 3) * τ ≤ 1 / 4)
(hmargin : (24 * (η + 3) ^ 3 + 2 * (η + 2) * (η + 4)) * τ ≤ η / 2) :
1 < Real.sqrt (1 / 4 + τ * (x * y) + τ ^ 2 * (D y + η / 2 - x ^ 2 / 4)) +
Real.sqrt (1 / 4 - τ * (x * y) + τ ^ 2 * (D x + η / 2 - y ^ 2 / 4)) := by
have h := (abs_le.mp (cartesian_pair_expansion hτ.le hτ1 hη.le hx hy hsmall)).1
have hm := mul_le_mul_of_nonneg_right hmargin (sq_nonneg τ)
have hp := mul_pos (sq_pos_of_pos hτ) hη
have hq := mul_nonneg (sq_nonneg τ) (sq_nonneg (x ^ 2 - y ^ 2))
nlinarith only [h, hm, hp, hq]
theorem cartesian_edge_one_obstruction_identity (ε x y : ℝ) :
(ε / 2) ^ 2 * (((2 * x) ^ 2 - (2 * y) ^ 2) ^ 2 + 1 / 1000) =
ε ^ 2 * (4 * (x ^ 2 - y ^ 2) ^ 2 + 1 / 4000) := by ring
end CylinderCovering.CartesianCaps
end OAI
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