| import OAI.Geometry.CylinderCovering.Rational.DictionaryStatementDefinitions | |
| import OAI.Geometry.CylinderCovering.Rational.CostDictionary | |
| import OAI.Geometry.CylinderCovering.Rational.Realization | |
| namespace OAI | |
| noncomputable section | |
| open Set MeasureTheory | |
| namespace RationalDictionary | |
| theorem centeredTetrahedron_eq_reference : | |
| RationalSimilarity.centeredTetrahedron = SlopeRational.referenceTetrahedron := by | |
| ext z | |
| constructor | |
| · rintro ⟨t, hz, ht₁, ht₂, hx, hy⟩ | |
| exact ⟨t, ht₁, ht₂, hx, hy, hz⟩ | |
| · rintro ⟨t, ht₁, ht₂, hx, hy, hz⟩ | |
| exact ⟨t, hz, ht₁, ht₂, hx, hy⟩ | |
| theorem centeredFirstSweep_eq_lowerSweep (d eps q : ℝ) : | |
| centeredFirstSweep d eps q = SlopeRational.lowerSweep d eps q := by | |
| ext z | |
| rw [SlopeRational.lowerSweep_mem_iff] | |
| rfl | |
| theorem centeredSecondSweep_eq_upperSweep (d eps q : ℝ) : | |
| centeredSecondSweep d eps q = SlopeRational.upperSweep d eps q := by | |
| ext z | |
| rw [SlopeRational.upperSweep_mem_iff] | |
| rfl | |
| theorem centeredWeight_eq_physicalWeight (d eps : ℝ) (p : Plane) : | |
| centeredWeight d eps p = SlopeRational.physicalScale * | |
| RuledApproximation.physicalWeight SlopeRational.physicalScale | |
| (eps • SlopeRational.field d p) := by | |
| unfold centeredWeight RuledApproximation.physicalWeight SlopeRational.field | |
| simp only [PiLp.smul_apply, smul_eq_mul] | |
| change c0 / Real.sqrt (1 + eps ^ 2 * | |
| (SlopeRational.a d (p 0) (p 1) ^ 2 + c0 ^ 2 * SlopeRational.b d (p 0) (p 1) ^ 2)) = | |
| c0 * (1 / Real.sqrt (1 + (eps * SlopeRational.a d (p 0) (p 1)) ^ 2 + | |
| c0 ^ 2 * (eps * SlopeRational.b d (p 0) (p 1)) ^ 2)) | |
| rw [mul_one_div] | |
| congr 2 | |
| ring | |
| theorem centeredCost_eq_weightedCost (d eps q : ℝ) : | |
| centeredCost d eps q = SlopeRational.weightedCost d eps q := by | |
| unfold centeredCost SlopeRational.weightedCost | |
| rw [← integral_const_mul] | |
| apply integral_congr_ae | |
| exact Filter.Eventually.of_forall (centeredWeight_eq_physicalWeight d eps) | |
| theorem actual_coordinate_identity {d0 : ℝ} (hd : 0 < d0) (eta0 k : ℝ) : | |
| RationalSimilarity.phi '' RationalSimilarity.uncenteredTetrahedron = | |
| SlopeRational.referenceTetrahedron ∧ | |
| RationalSimilarity.phi '' uncenteredFirstSweep d0 eta0 k = | |
| SlopeRational.lowerSweep (centeredRegularization d0) (centeredTilt d0 k) | |
| (centeredPadding eta0 k) ∧ | |
| RationalSimilarity.phi '' uncenteredSecondSweep d0 eta0 k = | |
| SlopeRational.upperSweep (centeredRegularization d0) (centeredTilt d0 k) | |
| (centeredPadding eta0 k) ∧ | |
| SlopeRational.weightedCost (centeredRegularization d0) (centeredTilt d0 k) | |
| (centeredPadding eta0 k) = uncenteredCost d0 eta0 k / 8 ∧ | |
| centeredPadding eta0 k = centeredMargin d0 eta0 * centeredTilt d0 k ^ 2 ∧ | |
| ∀ S : Set Space, | |
| (Measure.euclideanHausdorffMeasure 2 (RationalSimilarity.phi '' S)).toReal = | |
| (1 / 8 : ℝ) * (Measure.euclideanHausdorffMeasure 2 S).toReal := by | |
| refine ⟨?_, ?_, ?_, ?_, | |
| centeredPadding_eq_margin_mul_tilt_sq (by linarith) eta0 k, | |
| RationalSimilarity.phi_area⟩ | |
| · rw [RationalSimilarity.phi_image_tetrahedron, centeredTetrahedron_eq_reference] | |
| · rw [phi_image_firstSweep hd eta0 k, centeredFirstSweep_eq_lowerSweep] | |
| · rw [phi_image_secondSweep hd eta0 k, centeredSecondSweep_eq_upperSweep] | |
| · rw [← centeredCost_eq_weightedCost] | |
| exact cost_dictionary hd eta0 k | |
| theorem coordinate_cost_dictionary : CoordinateCostStatement := by | |
| intro d0 hd eta0 k | |
| exact actual_coordinate_identity hd eta0 k | |
| end RationalDictionary | |
| end | |
| end OAI | |