| import OAI.Geometry.CylinderCovering.Rational.Domain |
| import OAI.Geometry.RuledCovering.Main |
|
|
| namespace OAI |
|
|
| open Set MeasureTheory |
| noncomputable section |
|
|
| namespace SlopeRational |
|
|
| lemma labelDomain_volume_integral (d ε q : ℝ) : |
| volume.real (labelDomain d ε q) = |
| ∫ y : ℝ, max ((1 / 2 - 2 * |y|) - |
| max (-(1 / 4 : ℝ)) (-ε ^ 2 * ceilingPolynomial d y - q)) 0 := by |
| simpa only [Measure.restrict_univ, RuledApproximation.vec2, vec2] using |
| RuledApproximation.plane_volume_vertical_interval |
| (labelDomain_measurableSet d ε q) (labelDomain_volume_ne_top d ε q) |
| (s := Set.univ) MeasurableSet.univ |
| (l := fun y => max (-(1 / 4 : ℝ)) (-ε ^ 2 * ceilingPolynomial d y - q)) |
| (u := fun y => 1 / 2 - 2 * |y|) |
| (by intro z; simpa only [mem_univ, true_and] using labelDomain_mem_iff d ε q z) |
|
|
| lemma labelDomain_setIntegral_vertical {d ε q : ℝ} {f : Plane → ℝ} |
| (hf : IntegrableOn f (labelDomain d ε q)) : |
| (∫ z in labelDomain d ε q, f z) = |
| ∫ y : ℝ, ∫ t in Icc |
| (max (-(1 / 4 : ℝ)) (-ε ^ 2 * ceilingPolynomial d y - q)) |
| (1 / 2 - 2 * |y|), f (vec2 y t) := by |
| simpa only [Measure.restrict_univ, RuledApproximation.vec2, vec2] using |
| RuledApproximation.plane_setIntegral_vertical_interval |
| (labelDomain_measurableSet d ε q) hf |
| (s := Set.univ) MeasurableSet.univ |
| (l := fun y => max (-(1 / 4 : ℝ)) (-ε ^ 2 * ceilingPolynomial d y - q)) |
| (u := fun y => 1 / 2 - 2 * |y|) |
| (by intro z; simpa only [mem_univ, true_and] using labelDomain_mem_iff d ε q z) |
|
|
| end SlopeRational |
|
|
| end |
|
|
| end OAI |
|
|
