import OAI.Geometry.RuledCovering.Model

namespace OAI

noncomputable section
open Set MeasureTheory Filter
open scoped Topology ContDiff
namespace RuledApproximation

def NilpotentC1Statement : Prop :=
  ∀ (D : Set Plane) (G : Plane → Plane), IsCompact D → SmoothNear 1 D G →
    (∀ b ∈ D, (fderiv ℝ G b).comp (fderiv ℝ G b) = 0) →
    ∀ h eps : ℝ, 0 < h → 0 < eps →
    ∃ n : ℕ, ∃ T : Fin n → Set Plane, ∃ g : Fin n → Plane,
      (∀ i, IsSquareTile (T i)) ∧
      physicalRuled 1 D G 1 ⊆ ⋃ i, auxiliaryCylinder (T i) (g i) ∧
      (∑ i, volume.real (T i) * physicalWeight h (g i)) ≤
        (∫ b in D, physicalWeight h (G b)) + eps ∧
      ∀ a : ℝ, ∃ C₁ C₂ : Fin n → Cylinder,
        (∀ i, (C₁ i).parallelogram ∧ (C₂ i).parallelogram ∧
          (C₁ i).carrier = firstPhysicalMap h '' auxiliaryCylinder (T i) (g i) ∧
          (C₂ i).carrier = secondPhysicalMap h a '' auxiliaryCylinder (T i) (g i) ∧
          (C₁ i).area = h * volume.real (T i) * physicalWeight h (g i) ∧
          (C₂ i).area = h * volume.real (T i) * physicalWeight h (g i))

end RuledApproximation

end

end OAI