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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 | import OAI.Geometry.CylinderCovering.Radial.OverlapDefinitions
namespace OAI
noncomputable section
namespace CylinderCovering.RadialOverlapQuantitative
open RadialOverlap
def CanonicalStatement : Prop :=
∀ eps x y t p q : ℝ, 0 < eps → |x| ≤ 2 → |y| ≤ 2 →
|p| ≤ 1 → |q| ≤ 1 → RootEquations eps x y t p q →
1 ≤ rho eps x t q → 1 ≤ sigma eps y t p →
|t| ≤ 2 * eps ∧ |x - p| ≤ 3 * eps ∧ |y - q| ≤ 3 * eps ∧
rho eps x t q + sigma eps y t p - 2 =
eps * ((x - p) * q - (y - q) * p) ∧
0 ≤ rho eps x t q - 1 ∧
rho eps x t q - 1 ≤ rho eps x t q + sigma eps y t p - 2 ∧
0 ≤ sigma eps y t p - 1 ∧
sigma eps y t p - 1 ≤ rho eps x t q + sigma eps y t p - 2 ∧
rho eps x t q + sigma eps y t p - 2 ≤ 6 * eps ^ 2 ∧
|rho eps x t q - sigma eps y t p| ≤ 6 * eps ^ 2 ∧
|t - eps * p * q| ≤ 3 * eps ^ 2 ∧
eps * (x * q - y * p) ≤
eps ^ 2 * (RadialCaps.H p + RadialCaps.H q) + 9 * eps ^ 3 ∧
eps * (x * q - y * p) <
(RadialCaps.cubicCap 12 eps p - 1) + (RadialCaps.cubicCap 12 eps q - 1)
def PhysicalStatement : Prop :=
∀ tau x y z p q : ℝ, 0 < tau →
|x| ≤ 1 - (1 / Real.sqrt 2) * z →
|y| ≤ 1 + (1 / Real.sqrt 2) * z → |p| ≤ 1 → |q| ≤ 1 →
x = (1 - (1 / Real.sqrt 2) * z) * q -
((1 / Real.sqrt 2) * tau) * y * (1 - q ^ 2) / 2 →
y = (1 + (1 / Real.sqrt 2) * z) * p +
((1 / Real.sqrt 2) * tau) * x * (1 - p ^ 2) / 2 →
tau * x * p ≤ z → z ≤ tau * y * q →
(tau * (y * q - x * p) ≤
(1 / Real.sqrt 2) * tau ^ 2 * (RadialCaps.H p + RadialCaps.H q) +
9 * (1 / Real.sqrt 2) ^ 2 * tau ^ 3) ∧
(tau * (y * q - x * p) <
(tau ^ 2 * (1 / Real.sqrt 2) * RadialCaps.H p + 10 * tau ^ 3) +
(tau ^ 2 * (1 / Real.sqrt 2) * RadialCaps.H q + 10 * tau ^ 3))
def Statement : Prop := CanonicalStatement ∧ PhysicalStatement
end CylinderCovering.RadialOverlapQuantitative
end
end OAI
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