File size: 12,372 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 | import OAI.Geometry.CylinderCovering.Radial.PairedSweepDefinitions
namespace OAI
open Set
noncomputable section
namespace CylinderCovering.RadialCapCoverage
theorem quadratic_crossing {a b y : ℝ} (hay : |y| ≤ a) :
∃ q : ℝ, |q| ≤ 1 ∧ y = a * q - b * (1 - q ^ 2) / 2 ∧ 0 ≤ a + b * q := by
have ha : 0 ≤ a := (abs_nonneg y).trans hay
by_cases hb : b = 0
· subst b
by_cases hza : a = 0
· subst a
have hy : y = 0 := abs_eq_zero.mp (le_antisymm hay (abs_nonneg y))
subst y
exact ⟨0, by norm_num, by norm_num, by norm_num⟩
· refine ⟨y / a, ?_, ?_, ?_⟩
· rw [abs_div, abs_of_nonneg ha]
exact (div_le_one (lt_of_le_of_ne ha (Ne.symm hza))).mpr hay
· simp only [zero_mul]
field_simp; ring
· simpa using ha
· have hab : |b * y| ≤ |b| * a := by
rw [abs_mul]
exact mul_le_mul_of_nonneg_left hay (abs_nonneg b)
have hlo := (abs_le.mp hab).1
have hhi := (abs_le.mp hab).2
have hb2 : |b| ^ 2 = b ^ 2 := sq_abs b
have hdisc : 0 ≤ a ^ 2 + b ^ 2 + 2 * b * y := by
nlinarith [sq_nonneg (a - |b|)]
let r := Real.sqrt (a ^ 2 + b ^ 2 + 2 * b * y)
have hr : 0 ≤ r := Real.sqrt_nonneg _
have hr2 : r ^ 2 = a ^ 2 + b ^ 2 + 2 * b * y := Real.sq_sqrt hdisc
have hupper : r ≤ a + |b| := by nlinarith [abs_nonneg b]
have hlower : a - |b| ≤ r := by nlinarith [sq_nonneg (a - |b|)]
have habs : |r - a| ≤ |b| := abs_le.mpr ⟨by linarith, by linarith⟩
refine ⟨(r - a) / b, ?_, ?_, ?_⟩
· rw [abs_div]
exact (div_le_one (abs_pos.mpr hb)).mpr habs
· field_simp
nlinarith [hr2]
· have heq : a + b * ((r - a) / b) = r := by field_simp; ring
rw [heq]
exact hr
private theorem abs_mul_bound {x y A B : ℝ} (hx : |x| ≤ A) (hy : |y| ≤ B)
(hA : 0 ≤ A) : |x * y| ≤ A * B := by
rw [abs_mul]
exact mul_le_mul hx hy (abs_nonneg y) hA
theorem no_double_failure {eps x y t p q bp bq : ℝ}
(heps : 0 < eps) (hx : |x| ≤ 2) (hy : |y| ≤ 2)
(hp : |p| ≤ 1) (hq : |q| ≤ 1)
(hrootq : y = (1 - t) * q - eps * x * (1 - q ^ 2) / 2)
(hrootp : x = (1 + t) * p + eps * y * (1 - p ^ 2) / 2)
(hbp : 0 ≤ bp) (hbq : 0 ≤ bq) (hpadding : 9 * eps ≤ bp + bq) :
1 - t + eps * x * q ≤ 1 + eps ^ 2 * (RadialCaps.H q + bq) ∨
1 + t - eps * y * p ≤ 1 + eps ^ 2 * (RadialCaps.H p + bp) := by
by_contra hfail
push Not at hfail
obtain ⟨hfq, hfp⟩ := hfail
have hep : 0 ≤ eps := heps.le
have hp2 : p ^ 2 ≤ 1 := by nlinarith only [hp, sq_abs p, abs_nonneg p]
have hq2 : q ^ 2 ≤ 1 := by nlinarith only [hq, sq_abs q, abs_nonneg q]
let dp := (1 - p ^ 2) / 2
let dq := (1 - q ^ 2) / 2
have hdp : 0 ≤ dp ∧ dp ≤ 1 / 2 := by dsimp [dp]; constructor <;> nlinarith only [hp2, sq_nonneg p]
have hdq : 0 ≤ dq ∧ dq ≤ 1 / 2 := by dsimp [dq]; constructor <;> nlinarith only [hq2, sq_nonneg q]
have hHp : 0 ≤ RadialCaps.H p := by unfold RadialCaps.H; positivity
have hHq : 0 ≤ RadialCaps.H q := by unfold RadialCaps.H; positivity
have htl : eps * y * p < t := by nlinarith only [hfp, mul_nonneg (sq_nonneg eps) (add_nonneg hHp hbp)]
have htu : t < eps * x * q := by nlinarith only [hfq, mul_nonneg (sq_nonneg eps) (add_nonneg hHq hbq)]
have hxp0 : |x * q| ≤ 2 := by simpa using abs_mul_bound hx hq (by norm_num)
have hyp0 : |y * p| ≤ 2 := by simpa using abs_mul_bound hy hp (by norm_num)
have ht : |t| ≤ 2 * eps := by
have h1 := (abs_le.mp hyp0).1
have h2 := (abs_le.mp hxp0).2
rw [abs_le]
constructor <;> nlinarith only [htl, htu, mul_le_mul_of_nonneg_left h1 hep,
mul_le_mul_of_nonneg_left h2 hep]
have hxp : |x - p| ≤ 3 * eps := by
have ht1 : |t * p| ≤ 2 * eps := by simpa using abs_mul_bound ht hp (by positivity)
have hy1 : |y * dp| ≤ 1 := by
have hh := abs_mul_bound hy (show |dp| ≤ 1 / 2 by rw [abs_of_nonneg hdp.1]; exact hdp.2) (by norm_num)
norm_num at hh ⊢
exact hh
have hy2 : |eps * (y * dp)| ≤ eps := by
simpa using abs_mul_bound (show |eps| ≤ eps by rw [abs_of_nonneg hep]) hy1 hep
have heq : x - p = t * p + eps * (y * dp) := by dsimp [dp]; nlinarith only [hrootp]
rw [heq]
exact (abs_add_le _ _).trans (by linarith only [ht1, hy2])
have hyq : |y - q| ≤ 3 * eps := by
have ht1 : |t * q| ≤ 2 * eps := by simpa using abs_mul_bound ht hq (by positivity)
have hx1 : |x * dq| ≤ 1 := by
have hh := abs_mul_bound hx (show |dq| ≤ 1 / 2 by rw [abs_of_nonneg hdq.1]; exact hdq.2) (by norm_num)
norm_num at hh ⊢
exact hh
have hx2 : |eps * (x * dq)| ≤ eps := by
simpa using abs_mul_bound (show |eps| ≤ eps by rw [abs_of_nonneg hep]) hx1 hep
have heq : y - q = -(t * q + eps * (x * dq)) := by dsimp [dq]; nlinarith only [hrootq]
rw [heq, abs_neg]
exact (abs_add_le _ _).trans (by linarith only [ht1, hx2])
have hpq : |p * q| ≤ 1 := by simpa using abs_mul_bound hp hq (by norm_num)
have hxe : |eps * q * (x - p)| ≤ 3 * eps ^ 2 := by
have h1 : |eps * q| ≤ eps := by
simpa using abs_mul_bound (show |eps| ≤ eps by rw [abs_of_nonneg hep]) hq hep
convert abs_mul_bound h1 hxp hep using 1
ring
have hye : |eps * p * (y - q)| ≤ 3 * eps ^ 2 := by
have h1 : |eps * p| ≤ eps := by
simpa using abs_mul_bound (show |eps| ≤ eps by rw [abs_of_nonneg hep]) hp hep
convert abs_mul_bound h1 hyq hep using 1
ring
have htinterface : |t - eps * p * q| ≤ 3 * eps ^ 2 := by
rw [abs_le]
constructor
· have hh := (abs_le.mp hye).1
nlinarith only [hh, htl]
· have hh := (abs_le.mp hxe).2
nlinarith only [hh, htu]
have htcoeff : |2 * eps * (p * q)| ≤ 2 * eps := by
simpa using abs_mul_bound (show |2 * eps| ≤ 2 * eps by rw [abs_of_nonneg (by positivity)]) hpq (by positivity)
have hterr : |2 * eps * (p * q) * (t - eps * p * q)| ≤ 6 * eps ^ 3 := by
convert abs_mul_bound htcoeff htinterface (by positivity) using 1
ring
have hqD : |q * dp| ≤ 1 / 2 := by
simpa using abs_mul_bound hq (show |dp| ≤ 1 / 2 by rw [abs_of_nonneg hdp.1]; exact hdp.2) (by norm_num)
have hpD : |p * dq| ≤ 1 / 2 := by
simpa using abs_mul_bound hp (show |dq| ≤ 1 / 2 by rw [abs_of_nonneg hdq.1]; exact hdq.2) (by norm_num)
have herrq : |(y - q) * (q * dp)| ≤ 3 * eps / 2 := by
convert abs_mul_bound hyq hqD (by positivity) using 1
ring
have herrp : |(x - p) * (p * dq)| ≤ 3 * eps / 2 := by
convert abs_mul_bound hxp hpD (by positivity) using 1
ring
have heq : eps * (x * q - y * p) = 2 * eps * p * q * t +
eps ^ 2 * (y * q * dp + x * p * dq) := by
dsimp [dp, dq]
linear_combination eps * q * hrootp - eps * p * hrootq
have hmain : eps * (x * q - y * p) ≤
eps ^ 2 * ((p ^ 2 + q ^ 2) / 2 + p ^ 2 * q ^ 2) + 9 * eps ^ 3 := by
have hte := (abs_le.mp hterr).2
have hqe := mul_le_mul_of_nonneg_left (abs_le.mp herrq).2 (sq_nonneg eps)
have hpe := mul_le_mul_of_nonneg_left (abs_le.mp herrp).2 (sq_nonneg eps)
dsimp [dp, dq] at heq hqe hpe
nlinarith only [heq, hte, hqe, hpe]
have hpoly : (p ^ 2 + q ^ 2) / 2 + p ^ 2 * q ^ 2 ≤
RadialCaps.H p + RadialCaps.H q := by
unfold RadialCaps.H
nlinarith only [sq_nonneg (p ^ 2 - q ^ 2)]
have hfinal : eps * (x * q - y * p) ≤
eps ^ 2 * (RadialCaps.H p + RadialCaps.H q) + 9 * eps ^ 3 := hmain.trans (by
have hh := mul_le_mul_of_nonneg_left hpoly (sq_nonneg eps)
linarith only [hh])
have hpad := mul_le_mul_of_nonneg_left hpadding (sq_nonneg eps)
nlinarith only [hfq, hfp, hfinal, hpad]
theorem padding_coverage {eps : ℝ} {R b : ℝ → ℝ}
(heps : 0 < eps)
(hcap : ∀ r ∈ Icc (-1 : ℝ) 1, R r = 1 + eps ^ 2 * (RadialCaps.H r + b r))
(hb : ∀ r ∈ Icc (-1 : ℝ) 1, 0 ≤ b r)
(hpadding : ∀ p ∈ Icc (-1 : ℝ) 1, ∀ q ∈ Icc (-1 : ℝ) 1,
9 * eps ≤ b p + b q)
{x y t : ℝ} (hx : |x| ≤ 1 + t) (hy : |y| ≤ 1 - t) :
(∃ q rho : ℝ, |q| ≤ 1 ∧ 0 ≤ rho ∧ rho ≤ R q ∧
y = rho * q - eps * x * RadialCaps.G q ∧ t = 1 - rho + eps * x * q) ∨
(∃ p sigma : ℝ, |p| ≤ 1 ∧ 0 ≤ sigma ∧ sigma ≤ R p ∧
x = sigma * p + eps * y * RadialCaps.G p ∧ t = -1 + sigma + eps * y * p) := by
obtain ⟨q, hq, hrootq, hrho⟩ := quadratic_crossing (b := eps * x) hy
obtain ⟨p, hp, hrootp, hsigma⟩ := quadratic_crossing (b := -(eps * y)) hx
have hrootp' : x = (1 + t) * p + eps * y * (1 - p ^ 2) / 2 := by
nlinarith only [hrootp]
have hx2 : |x| ≤ 2 := by linarith only [hx, hy, abs_nonneg y]
have hy2 : |y| ≤ 2 := by linarith only [hx, hy, abs_nonneg x]
have hpI : p ∈ Icc (-1 : ℝ) 1 := abs_le.mp hp
have hqI : q ∈ Icc (-1 : ℝ) 1 := abs_le.mp hq
rcases no_double_failure heps hx2 hy2 hp hq hrootq hrootp'
(hb p hpI) (hb q hqI) (hpadding p hpI q hqI) with hu | hl
· left
refine ⟨q, 1 - t + eps * x * q, hq, hrho, ?_, ?_, ?_⟩
· simpa only [hcap q hqI] using hu
· unfold RadialCaps.G
nlinarith only [hrootq]
· ring
· right
refine ⟨p, 1 + t - eps * y * p, hp, ?_, ?_, ?_, ?_⟩
· nlinarith only [hsigma]
· simpa only [hcap p hpI] using hl
· unfold RadialCaps.G
nlinarith only [hrootp']
· ring
theorem padding_covers {h eps : ℝ} {R b : ℝ → ℝ}
(heps : 0 < eps)
(hcap : ∀ r ∈ Icc (-1 : ℝ) 1, R r = 1 + eps ^ 2 * (RadialCaps.H r + b r))
(hb : ∀ r ∈ Icc (-1 : ℝ) 1, 0 ≤ b r)
(hpadding : ∀ p ∈ Icc (-1 : ℝ) 1, ∀ q ∈ Icc (-1 : ℝ) 1,
9 * eps ≤ b p + b q) :
RadialPairedSweep.tetrahedron h ⊆
RadialPairedSweep.upperSweep h eps R ∪ RadialPairedSweep.lowerSweep h eps R := by
rintro z ⟨x, y, t, hx, hy, rfl⟩
rcases padding_coverage heps hcap hb hpadding hx hy with hu | hl
· obtain ⟨q, rho, hq, hrho, hR, hyrep, htrep⟩ := hu
left
refine ⟨q, abs_le.mp hq, rho, ⟨hrho, hR⟩, x, ?_, ?_⟩
· linarith only [hx, hy, abs_nonneg y]
· rw [← hyrep, ← htrep]
· obtain ⟨p, sigma, hp, hsigma, hR, hxrep, htrep⟩ := hl
right
refine ⟨p, abs_le.mp hp, sigma, ⟨hsigma, hR⟩, y, ?_, ?_⟩
· linarith only [hx, hy, abs_nonneg x]
· rw [← hxrep, ← htrep]
theorem linear_covers {h beta eps : ℝ} (hbeta : 0 ≤ beta)
(heps : 0 < eps) (hsmall : eps ≤ 2 * beta / 9) :
RadialPairedSweep.tetrahedron h ⊆
RadialPairedSweep.upperSweep h eps (RadialCaps.linearCap beta eps) ∪
RadialPairedSweep.lowerSweep h eps (RadialCaps.linearCap beta eps) := by
apply padding_covers (b := fun _ => beta) heps
· intro r hr
rfl
· intro r hr
exact hbeta
· intro p hp q hq
linarith only [hsmall]
theorem cubic_covers {h c eps : ℝ} (hc : (9 : ℝ) / 2 ≤ c) (heps : 0 < eps) :
RadialPairedSweep.tetrahedron h ⊆
RadialPairedSweep.upperSweep h eps (RadialCaps.cubicCap c eps) ∪
RadialPairedSweep.lowerSweep h eps (RadialCaps.cubicCap c eps) := by
apply padding_covers (b := fun _ => c * eps) heps
· intro r hr
unfold RadialCaps.cubicCap
ring
· intro r hr
exact mul_nonneg (by linarith only [hc]) heps.le
· intro p hp q hq
nlinarith only [mul_le_mul_of_nonneg_right hc heps.le]
theorem fractional_power (eps : ℝ) (heps : 0 < eps) :
eps ^ (5 / 2 : ℝ) = eps ^ 2 * Real.sqrt eps := by
rw [show (5 / 2 : ℝ) = 2 + 1 / 2 by norm_num, Real.rpow_add heps]
rw [← Real.sqrt_eq_rpow]
norm_num
theorem fractional_covers {h c eps : ℝ} (hc : 0 < c) (heps : 0 < eps)
(hsmall : eps ≤ (2 * c / 9) ^ 2) :
RadialPairedSweep.tetrahedron h ⊆
RadialPairedSweep.upperSweep h eps (RadialCaps.fractionalCap c eps) ∪
RadialPairedSweep.lowerSweep h eps (RadialCaps.fractionalCap c eps) := by
have hsqrt : Real.sqrt eps ≤ 2 * c / 9 := by
nlinarith only [Real.sq_sqrt heps.le, Real.sqrt_nonneg eps, hc, hsmall]
apply padding_covers (b := fun _ => c * Real.sqrt eps) heps
· intro r hr
unfold RadialCaps.fractionalCap
rw [fractional_power eps heps]
ring
· intro r hr
exact mul_nonneg hc.le (Real.sqrt_nonneg eps)
· intro p hp q hq
have hh := mul_le_mul_of_nonneg_right hsqrt (Real.sqrt_nonneg eps)
nlinarith only [hh, Real.sq_sqrt heps.le]
end CylinderCovering.RadialCapCoverage
end
end OAI
|