| import OAI.Geometry.CylinderCovering.Radial.CapsDefinitions |
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| namespace OAI |
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| noncomputable section |
| namespace CylinderCovering.RadialOverlap |
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| def RootEquations (ε x y t p q : ℝ) : Prop := |
| y = (1 - t) * q - ε * x * (1 - q ^ 2) / 2 ∧ |
| x = (1 + t) * p + ε * y * (1 - p ^ 2) / 2 |
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| def rho (ε x t q : ℝ) : ℝ := 1 - t + ε * x * q |
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| def sigma (ε y t p : ℝ) : ℝ := 1 + t - ε * y * p |
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| def T (ε x t q : ℝ) : ℝ := t - ε * x * q |
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| def S (ε y t p : ℝ) : ℝ := -t + ε * y * p |
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| def IdentitiesStatement : Prop := |
| ∀ ε x y t p q : ℝ, RootEquations ε x y t p q → |
| ((1 + ε * p * q) * rho ε x t q + (1 - ε * p * q) * sigma ε y t p = |
| 2 + ε ^ 2 * (q * RadialCaps.G p * y + p * RadialCaps.G q * x)) ∧ |
| ((1 + ε * x * q) * T ε x t q = t - ε * x * y - ε ^ 2 * x ^ 2 * RadialCaps.G q) ∧ |
| ((1 - ε * y * p) * S ε y t p = -t + ε * x * y - ε ^ 2 * y ^ 2 * RadialCaps.G p) ∧ |
| ((rho ε x t q) ^ 2 = (1 - t) ^ 2 + 2 * ε * x * y + ε ^ 2 * x ^ 2) ∧ |
| ((sigma ε y t p) ^ 2 = (1 + t) ^ 2 - 2 * ε * x * y + ε ^ 2 * y ^ 2) |
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| def ComplementaryBoundStatement : Prop := |
| ∀ ε x y t p q : ℝ, 0 < ε → ε ≤ 1 / 4 → |
| |x| ≤ 2 → |y| ≤ 2 → |p| ≤ 1 → |q| ≤ 1 → |
| RootEquations ε x y t p q → T ε x t q ≤ 0 → S ε y t p ≤ 0 → |
| |T ε x t q| ≤ 16 * ε ^ 2 ∧ |S ε y t p| ≤ 16 * ε ^ 2 |
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| end CylinderCovering.RadialOverlap |
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| end |
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| end OAI |
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