openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Radial /OverlapQuantitativeDefinitions.lean
| 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 | |