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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 | 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
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