| import OAI.Geometry.CylinderCovering.Singular.EdgeOneNormalizationDefinitions |
| import OAI.Geometry.CylinderCovering.Rational.Normalization |
| import OAI.Geometry.CylinderCovering.Ruled.Similarity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory |
|
|
| namespace CylinderCovering.SingularEdgeOneCover |
| open RuledApproximation |
|
|
| lemma height_pos : 0 < height := by |
| unfold height |
| positivity |
|
|
| theorem tetrahedron_eq_centered_translation : |
| tetrahedron = |
| (fun z : Space => z + vec3 0 0 (height / 2)) '' SlopeRational.referenceTetrahedron := by |
| ext z |
| constructor |
| · rintro ⟨s, hs0, hs1, hx, hy, hz⟩ |
| refine ⟨vec3 (z 0) (z 1) (z 2 - height / 2), ?_, ?_⟩ |
| · refine ⟨s - 1 / 2, by linarith, by linarith, ?_, ?_, ?_⟩ |
| · change |z 0| ≤ (1 / 2 + (s - 1 / 2)) / 2 |
| simpa only [show (1 / 2 : ℝ) + (s - 1 / 2) = s by ring] using hx |
| · change |z 1| ≤ (1 / 2 - (s - 1 / 2)) / 2 |
| simpa only [show (1 / 2 : ℝ) - (s - 1 / 2) = 1 - s by ring] using hy |
| · change z 2 - height / 2 = height * (s - 1 / 2) |
| rw [hz] |
| ring |
| · ext i |
| fin_cases i <;> simp [vec3] |
| · rintro ⟨w, ⟨t, ht0, ht1, hx, hy, hz⟩, rfl⟩ |
| refine ⟨t + 1 / 2, by linarith, by linarith, ?_, ?_, ?_⟩ |
| · change |w 0 + 0| ≤ (t + 1 / 2) / 2 |
| simpa [add_comm] using hx |
| · change |w 1 + 0| ≤ (1 - (t + 1 / 2)) / 2 |
| simpa only [add_zero, show (1 : ℝ) - (t + 1 / 2) = 1 / 2 - t by ring] using hy |
| · change w 2 + height / 2 = height * (t + 1 / 2) |
| change w 2 = height * t at hz |
| rw [hz] |
| ring |
|
|
| theorem shadowArea_eq_centered (u : Space) : |
| shadowArea tetrahedron u = shadowArea SlopeRational.referenceTetrahedron u := by |
| rw [tetrahedron_eq_centered_translation] |
| have hmap : (fun z : Space => z + vec3 0 0 (height / 2)) = |
| Similarity.map (vec3 0 0 (height / 2)) 1 (LinearIsometryEquiv.refl ℝ Space) := by |
| funext z |
| simp [Similarity.map, add_comm] |
| rw [hmap] |
| change Similarity.shadowArea |
| (Similarity.map (vec3 0 0 (height / 2)) 1 (LinearIsometryEquiv.refl ℝ Space) '' |
| SlopeRational.referenceTetrahedron) ((LinearIsometryEquiv.refl ℝ Space) u) = _ |
| rw [Similarity.shadowArea_map _ _ (by norm_num)] |
| simp only [one_pow, one_mul] |
| rfl |
|
|
| theorem tetrahedron_minimumShadowArea : minimumShadowArea tetrahedron = height / 2 := by |
| have heq : minimumShadowArea tetrahedron = |
| minimumShadowArea SlopeRational.referenceTetrahedron := by |
| unfold minimumShadowArea |
| simp only [shadowArea_eq_centered] |
| rw [heq, SlopeRational.referenceTetrahedron_minimumShadowArea] |
| rfl |
|
|
| lemma minimumShadowArea_pos : 0 < minimumShadowArea tetrahedron := by |
| rw [tetrahedron_minimumShadowArea] |
| exact div_pos height_pos (by norm_num) |
|
|
| theorem same_family_half_bounds {n m : ℕ} (C : Fin n → Cylinder) |
| (D : Fin m → Cylinder) (B₁ B₂ : Cylinder) |
| (hcost : familyArea C + familyArea D + B₁.area + B₂.area < height / 4) : |
| familyArea C + familyArea D + B₁.area + B₂.area < |
| minimumShadowArea tetrahedron / 2 ∧ |
| relativeCost tetrahedron C + relativeCost tetrahedron D + |
| B₁.area / shadowArea tetrahedron B₁.axis + |
| B₂.area / shadowArea tetrahedron B₂.axis < 1 / 2 := by |
| have hmin := minimumShadowArea_pos |
| have htotal : familyArea C + familyArea D + B₁.area + B₂.area < |
| minimumShadowArea tetrahedron / 2 := by |
| rw [tetrahedron_minimumShadowArea] |
| linarith only [hcost] |
| refine ⟨htotal, ?_⟩ |
| have hC : relativeCost tetrahedron C ≤ familyArea C / minimumShadowArea tetrahedron := |
| relativeCost_le_familyArea_div_minimum tetrahedron hmin C |
| have hD : relativeCost tetrahedron D ≤ familyArea D / minimumShadowArea tetrahedron := |
| relativeCost_le_familyArea_div_minimum tetrahedron hmin D |
| have hB (B : Cylinder) : B.area / shadowArea tetrahedron B.axis ≤ |
| B.area / minimumShadowArea tetrahedron := |
| div_le_div_of_nonneg_left ENNReal.toReal_nonneg hmin |
| (Similarity.minimumArea_le_shadowArea tetrahedron B.unit_axis) |
| have hsum : relativeCost tetrahedron C + relativeCost tetrahedron D + |
| B₁.area / shadowArea tetrahedron B₁.axis + |
| B₂.area / shadowArea tetrahedron B₂.axis ≤ |
| (familyArea C + familyArea D + B₁.area + B₂.area) / |
| minimumShadowArea tetrahedron := by |
| rw [add_div, add_div, add_div] |
| linarith only [hC, hD, hB B₁, hB B₂] |
| apply hsum.trans_lt |
| apply (div_lt_iff₀ hmin).mpr |
| linarith only [htotal] |
|
|
| theorem normalization : NormalizationStatement := |
| ⟨tetrahedron_eq_centered_translation, tetrahedron_minimumShadowArea, |
| fun _ _ C D B₁ B₂ hcost => same_family_half_bounds C D B₁ B₂ hcost⟩ |
|
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| end CylinderCovering.SingularEdgeOneCover |
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|
| end |
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| end OAI |
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