| import OAI.Geometry.CylinderCovering.Nilpotent.C1 |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory |
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| namespace RuledApproximation |
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| theorem empty_physical_pair (G : Plane → Plane) (h a L : ℝ) : |
| ∃ C₁ C₂ : Fin 0 → Cylinder, |
| (∀ i, (C₁ i).parallelogram ∧ (C₂ i).parallelogram) ∧ |
| physicalRuled h ∅ G L ⊆ familySet C₁ ∧ |
| secondPhysicalMap h a '' physicalRuled 1 ∅ G L ⊆ familySet C₂ ∧ |
| familyArea C₁ = 0 ∧ familyArea C₂ = 0 := by |
| refine ⟨Fin.elim0, Fin.elim0, fun i => i.elim0, ?_, ?_, ?_, ?_⟩ |
| · simp [physicalRuled_empty] |
| · simp [physicalRuled_empty] |
| · simp [familyArea] |
| · simp [familyArea] |
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| theorem null_domain_physical_pair {D : Set Plane} {G : Plane → Plane} |
| (hD : IsCompact D) (hnull : volume D = 0) (hG : SmoothNear 1 D G) |
| (hnil : ∀ b ∈ D, (fderiv ℝ G b).comp (fderiv ℝ G b) = 0) |
| {h ε : ℝ} (hh : 0 < h) (hε : 0 < ε) (a : ℝ) : |
| ∃ n : ℕ, ∃ C₁ C₂ : Fin n → Cylinder, |
| (∀ i, (C₁ i).parallelogram ∧ (C₂ i).parallelogram ∧ |
| IsCompact (C₁ i).base ∧ IsCompact (C₂ i).base ∧ |
| Convex ℝ (C₁ i).base ∧ Convex ℝ (C₂ i).base) ∧ |
| physicalRuled h D G 1 ⊆ familySet C₁ ∧ |
| secondPhysicalMap h a '' physicalRuled 1 D G 1 ⊆ familySet C₂ ∧ |
| familyArea C₁ = familyArea C₂ ∧ familyArea C₁ + familyArea C₂ < ε := by |
| obtain ⟨n, T, g, _hT, hcover, hcost, hphysical⟩ := |
| nilpotentC1 D G hD hG hnil h (ε / (4 * h)) hh (by positivity) |
| rw [setIntegral_measure_zero _ hnull, zero_add] at hcost |
| obtain ⟨C₁, C₂, hC⟩ := hphysical a |
| have harea₁ : familyArea C₁ = h * ∑ i, volume.real (T i) * physicalWeight h (g i) := by |
| unfold familyArea |
| simp_rw [(hC _).2.2.2.2.1, mul_assoc] |
| rw [Finset.mul_sum] |
| have harea₂ : familyArea C₂ = h * ∑ i, volume.real (T i) * physicalWeight h (g i) := by |
| unfold familyArea |
| simp_rw [(hC _).2.2.2.2.2, mul_assoc] |
| rw [Finset.mul_sum] |
| refine ⟨n, C₁, C₂, ?_, ?_, ?_, harea₁.trans harea₂.symm, ?_⟩ |
| · intro i |
| obtain ⟨hpar₁, hpar₂, _⟩ := hC i |
| exact ⟨hpar₁, hpar₂, hpar₁.isCompact, hpar₂.isCompact, |
| hpar₁.convex, hpar₂.convex⟩ |
| · rw [← firstPhysicalMap_ruled] |
| rintro _ ⟨z, hz, rfl⟩ |
| obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hz) |
| exact mem_iUnion.mpr ⟨i, (hC i).2.2.1 ▸ mem_image_of_mem _ hi⟩ |
| · rintro _ ⟨z, hz, rfl⟩ |
| obtain ⟨i, hi⟩ := mem_iUnion.mp (hcover hz) |
| exact mem_iUnion.mpr ⟨i, (hC i).2.2.2.1 ▸ mem_image_of_mem _ hi⟩ |
| · rw [harea₁, harea₂] |
| have hc := mul_le_mul_of_nonneg_left hcost hh.le |
| have he : h * (ε / (4 * h)) = ε / 4 := by field_simp |
| rw [he] at hc |
| linarith |
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| theorem null_domain_physical_pair_of_fderiv_zero |
| {D : Set Plane} {G : Plane → Plane} |
| (hD : IsCompact D) (hnull : volume D = 0) (hG : SmoothNear 1 D G) |
| (hzero : ∀ b ∈ D, fderiv ℝ G b = 0) |
| {h ε : ℝ} (hh : 0 < h) (hε : 0 < ε) (a : ℝ) : |
| ∃ n : ℕ, ∃ C₁ C₂ : Fin n → Cylinder, |
| (∀ i, (C₁ i).parallelogram ∧ (C₂ i).parallelogram ∧ |
| IsCompact (C₁ i).base ∧ IsCompact (C₂ i).base ∧ |
| Convex ℝ (C₁ i).base ∧ Convex ℝ (C₂ i).base) ∧ |
| physicalRuled h D G 1 ⊆ familySet C₁ ∧ |
| secondPhysicalMap h a '' physicalRuled 1 D G 1 ⊆ familySet C₂ ∧ |
| familyArea C₁ = familyArea C₂ ∧ familyArea C₁ + familyArea C₂ < ε := by |
| apply null_domain_physical_pair hD hnull hG _ hh hε a |
| intro b hb |
| simp [hzero b hb] |
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| end RuledApproximation |
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| end |
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| end OAI |
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