| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace SlopeRational | |
| abbrev Plane := EuclideanSpace ℝ (Fin 2) | |
| def vec2 (x y : ℝ) : Plane := WithLp.toLp 2 ![x, y] | |
| def radius (T : ℝ) : ℝ := 1 / 2 - T | |
| def potential (d Y T : ℝ) : ℝ := | |
| (Y ^ 2 - radius T ^ 2 / 4) / (4 * (radius T + d)) | |
| def a (d Y T : ℝ) : ℝ := | |
| -Y ^ 2 / (4 * (radius T + d) ^ 2) - | |
| radius T * (radius T + 2 * d) / (16 * (radius T + d) ^ 2) | |
| def b (d Y T : ℝ) : ℝ := Y / (2 * (radius T + d)) | |
| def field (d : ℝ) (z : Plane) : Plane := vec2 (a d (z 0) (z 1)) (b d (z 0) (z 1)) | |
| def wedge : Set Plane := {z | 2 * |z 0| ≤ radius (z 1)} | |
| def alpha (d : ℝ) : ℝ := 1 / (1 + 2 * d) | |
| def ceilingConstant (d : ℝ) : ℝ := | |
| ((1 / 2 : ℝ) * (1 / 2 + 2 * d)) / (16 * (1 / 2 + d) ^ 2) | |
| def ceilingQuadratic (d : ℝ) : ℝ := 1 / (4 * (1 / 2 + d) ^ 2) | |
| def ceilingCoefficient (d Y : ℝ) : ℝ := ceilingConstant d + ceilingQuadratic d * Y ^ 2 | |
| def ceilingPolynomial (d Y : ℝ) : ℝ := alpha d * ceilingCoefficient d Y * Y ^ 2 | |
| def labelDomain (d ε q : ℝ) : Set Plane := | |
| {z | z ∈ wedge ∧ -(1 / 4 : ℝ) ≤ z 1 ∧ | |
| -ε ^ 2 * ceilingPolynomial d (z 0) - q ≤ z 1} | |
| end SlopeRational | |
| end | |
| end OAI | |