| import OAI.Geometry.CylinderCovering.Physical.MinimumDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.TetraGeometry |
| import OAI.Geometry.CylinderCovering.Convexity.CylinderSimilarity |
|
|
| namespace OAI |
|
|
| open Set |
| open scoped InnerProductSpace |
| noncomputable section |
| namespace RadialPhysical.PhysicalMinimum |
| open CylinderCovering |
|
|
| private abbrev canonicalBody := RadialPairedSweep.tetrahedron (Real.sqrt 2) |
|
|
| lemma centered_similarity (hb : 0 < b) : |
| modelBody true b = Similarity.map (0 : Space) b |
| (LinearIsometryEquiv.refl ℝ Space) '' canonicalBody := by |
| ext z |
| constructor |
| · rintro ⟨x, y, t, hx, hy, rfl⟩ |
| refine ⟨RadialPairedSweep.point (x/b) (y/b) (Real.sqrt 2*t), ?_, ?_⟩ |
| · refine ⟨x/b, y/b, t, ?_, ?_, rfl⟩ |
| · rw [abs_div, abs_of_pos hb] |
| exact (div_le_iff₀ hb).2 (by nlinarith only [hx]) |
| · rw [abs_div, abs_of_pos hb] |
| exact (div_le_iff₀ hb).2 (by nlinarith only [hy]) |
| · ext i |
| fin_cases i <;> simp [Similarity.map, RadialPairedSweep.point, point, height] <;> |
| field_simp |
| · rintro ⟨_, ⟨x, y, t, hx, hy, rfl⟩, rfl⟩ |
| refine ⟨b*x, b*y, t, ?_, ?_, ?_⟩ |
| · rw [abs_mul, abs_of_pos hb] |
| exact mul_le_mul_of_nonneg_left hx hb.le |
| · rw [abs_mul, abs_of_pos hb] |
| exact mul_le_mul_of_nonneg_left hy hb.le |
| · ext i |
| fin_cases i <;> simp [Similarity.map, RadialPairedSweep.point, point, height] |
| ring |
|
|
| lemma halfheight_similarity (hb : 0 < b) : |
| modelBody false b = Similarity.map (point b 0 0 (1/2)) (b/2) |
| (LinearIsometryEquiv.refl ℝ Space) '' canonicalBody := by |
| ext z |
| constructor |
| · rintro ⟨x, y, t, ht0, ht1, hx, hy, rfl⟩ |
| refine ⟨RadialPairedSweep.point (2*x/b) (2*y/b) (Real.sqrt 2*(2*t-1)), ?_, ?_⟩ |
| · refine ⟨2*x/b, 2*y/b, 2*t-1, ?_, ?_, rfl⟩ |
| · rw [abs_div, abs_of_pos hb, abs_mul, abs_of_pos (by norm_num : (0:ℝ)<2)] |
| exact (div_le_iff₀ hb).2 (by nlinarith only [hx]) |
| · rw [abs_div, abs_of_pos hb, abs_mul, abs_of_pos (by norm_num : (0:ℝ)<2)] |
| exact (div_le_iff₀ hb).2 (by nlinarith only [hy]) |
| · ext i |
| fin_cases i <;> simp [Similarity.map, RadialPairedSweep.point, point, height, |
| ne_of_gt hb] |
| field_simp |
| ring |
| · rintro ⟨_, ⟨x, y, t, hx, hy, rfl⟩, rfl⟩ |
| have ht0 : -1 ≤ t := by linarith [abs_nonneg x] |
| have ht1 : t ≤ 1 := by linarith [abs_nonneg y] |
| refine ⟨b/2*x, b/2*y, (t+1)/2, by linarith, by linarith, ?_, ?_, ?_⟩ |
| · rw [abs_mul, abs_of_pos (by positivity : 0 < b/2)] |
| nlinarith only [mul_le_mul_of_nonneg_left hx (show 0 ≤ b/2 by positivity)] |
| · rw [abs_mul, abs_of_pos (by positivity : 0 < b/2)] |
| nlinarith only [mul_le_mul_of_nonneg_left hy (show 0 ≤ b/2 by positivity)] |
| · ext i |
| fin_cases i <;> simp [Similarity.map, RadialPairedSweep.point, point, height] |
| ring |
|
|
| theorem minimum_modelBody (centered : Bool) (hb : 0 < b) : |
| Similarity.minimumArea (modelBody centered b) = |
| (if centered then 4 else 1) * b * height b := by |
| have hmin : Similarity.minimumArea canonicalBody = 4 * Real.sqrt 2 := |
| RadialPairedSweep.reference_minimum |
| cases centered |
| · rw [halfheight_similarity hb, Similarity.minimumArea_map _ _ |
| (by positivity : b/2 ≠ 0) (LinearIsometryEquiv.refl ℝ Space), hmin] |
| simp only [Bool.false_eq_true, ite_false, one_mul, height] |
| ring |
| · rw [centered_similarity hb, Similarity.minimumArea_map _ _ |
| (ne_of_gt hb) (LinearIsometryEquiv.refl ℝ Space), hmin] |
| simp only [ite_true, height] |
| ring |
|
|
| theorem shadow_x_modelBody (centered : Bool) (hb : 0 < b) : |
| Similarity.shadowArea (modelBody centered b) (TriangularCovering.tiltV 0 0 0) = |
| (if centered then 4 else 1)*b*height b := by |
| have hx : Similarity.shadowArea canonicalBody (TriangularCovering.tiltV 0 0 0) = |
| 4 * Real.sqrt 2 := by |
| change CylinderCovering.shadowArea canonicalBody (TriangularCovering.tiltV 0 0 0) = _ |
| rw [RadialPairedSweep.shadowArea_eq_cubic, Cubic.K4_shadow, |
| TriangularCovering.xAxis_shadow] |
| rfl |
| cases centered |
| · rw [halfheight_similarity hb] |
| rw [show Similarity.shadowArea |
| (Similarity.map (point b 0 0 (1/2)) (b/2) |
| (LinearIsometryEquiv.refl ℝ Space) '' canonicalBody) |
| (TriangularCovering.tiltV 0 0 0) = |
| (b/2)^2 * Similarity.shadowArea canonicalBody (TriangularCovering.tiltV 0 0 0) from |
| Similarity.shadowArea_map _ _ (by positivity : b/2 ≠ 0) _ _] |
| rw [hx] |
| simp only [Bool.false_eq_true, ite_false, height] |
| ring |
| · rw [centered_similarity hb] |
| rw [show Similarity.shadowArea |
| (Similarity.map 0 b (LinearIsometryEquiv.refl ℝ Space) '' canonicalBody) |
| (TriangularCovering.tiltV 0 0 0) = |
| b^2 * Similarity.shadowArea canonicalBody (TriangularCovering.tiltV 0 0 0) from |
| Similarity.shadowArea_map _ _ (ne_of_gt hb) _ _] |
| rw [hx] |
| simp only [ite_true, height] |
| ring |
|
|
| theorem minimum_attained (centered : Bool) (hb : 0 < b) : |
| ∃ u : Space, ‖u‖ = 1 ∧ |
| Similarity.shadowArea (modelBody centered b) u = |
| Similarity.minimumArea (modelBody centered b) := by |
| refine ⟨TriangularCovering.tiltV 0 0 0, TriangularCovering.xAxis_unit, ?_⟩ |
| rw [shadow_x_modelBody centered hb, minimum_modelBody centered hb] |
|
|
| theorem nonorthonormal_minimum : |
| Similarity.minimumArea |
| = 2*Real.sqrt 2 := by |
| have hs : Real.sqrt 2 ≠ 0 := by positivity |
| have hpoint (p q t : ℝ) : |
| Similarity.map 0 (1/Real.sqrt 2) (LinearIsometryEquiv.refl ℝ Space) |
| (RadialPairedSweep.point p q (Real.sqrt 2*t)) = nonorthPoint p q t := by |
| ext i |
| fin_cases i <;> simp [Similarity.map, RadialPairedSweep.point, nonorthPoint, hs] <;> |
| field_simp |
| have hset : |
| = |
| Similarity.map 0 (1/Real.sqrt 2) (LinearIsometryEquiv.refl ℝ Space) '' canonicalBody := by |
| ext z |
| constructor |
| · rintro ⟨p,q,t,hp,hq,rfl⟩ |
| exact ⟨RadialPairedSweep.point p q (Real.sqrt 2*t), ⟨p,q,t,hp,hq,rfl⟩, |
| hpoint p q t⟩ |
| · rintro ⟨_,⟨p,q,t,hp,hq,rfl⟩,rfl⟩ |
| exact ⟨p,q,t,hp,hq,hpoint p q t⟩ |
| rw [hset, Similarity.minimumArea_map _ _ (one_div_ne_zero hs) |
| (LinearIsometryEquiv.refl ℝ Space)] |
| have hmin : Similarity.minimumArea canonicalBody = 4*Real.sqrt 2 := |
| RadialPairedSweep.reference_minimum |
| rw [hmin, div_pow, one_pow, Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 2)] |
| ring |
|
|
| theorem halfheight_reference (hb : 0 < b) : |
| Similarity.minimumArea (tetrahedron b) / 2 = b * height b / 2 := by |
| have h := minimum_modelBody false hb |
| simpa [modelBody] using congrArg (fun x : ℝ => x/2) h |
|
|
| theorem centered_reference (hb : 0 < b) : |
| Similarity.minimumArea (modelBody true b) / 2 = 2 * b * height b := by |
| rw [minimum_modelBody true hb] |
| simp only [ite_true] |
| ring |
|
|
| lemma reference_pos (centered : Bool) (hb : 0 < b) : |
| 0 < (if centered then (4:ℝ) else 1) * b * height b := by |
| cases centered <;> simp only [Bool.false_eq_true, ite_false, ite_true, height] <;> positivity |
|
|
| theorem shadow_lower (centered : Bool) (hb : 0 < b) |
| (u : Space) (hu : ‖u‖ = 1) : |
| (if centered then 4 else 1) * b * height b ≤ |
| CylinderCovering.shadowArea (modelBody centered b) u := by |
| rw [← minimum_modelBody centered hb] |
| exact Similarity.minimumArea_le_shadowArea _ hu |
|
|
| theorem relativeCost_le (centered : Bool) (hb : 0 < b) |
| (C : Fin n → TriangularCovering.TriCylinder) : |
| triangularRelativeCost (modelBody centered b) C ≤ |
| TriangularCovering.totalArea C / ((if centered then 4 else 1)*b*height b) := by |
| unfold triangularRelativeCost TriangularCovering.totalArea |
| rw [Finset.sum_div] |
| apply Finset.sum_le_sum |
| intro i _ |
| exact div_le_div_of_nonneg_left ENNReal.toReal_nonneg (reference_pos centered hb) |
| (shadow_lower centered hb (C i).axis (C i).unit_axis) |
|
|
| theorem both_half_bounds (centered : Bool) (hb : 0 < b) |
| (C : Fin n → TriangularCovering.TriCylinder) |
| (hcost : TriangularCovering.totalArea C < |
| ((if centered then 4 else 1)*b*height b)/2) : |
| TriangularCovering.totalArea C < minimumShadowArea (modelBody centered b)/2 ∧ |
| triangularRelativeCost (modelBody centered b) C < 1/2 := by |
| refine ⟨?_, (relativeCost_le centered hb C).trans_lt ?_⟩ |
| · change TriangularCovering.totalArea C < Similarity.minimumArea (modelBody centered b)/2 |
| rwa [minimum_modelBody centered hb] |
| · exact (div_lt_iff₀ (reference_pos centered hb)).2 (by linarith) |
|
|
| theorem baseRelativeCost_le [Fintype ι] |
| (centered : Bool) (hb : 0 < b) |
| (B : ι → Set Space) (u : ι → Space) (hu : ∀ i, ‖u i‖ = 1) : |
| (∑ i, TriangularCovering.area (B i) / |
| CylinderCovering.shadowArea (modelBody centered b) (u i)) ≤ |
| (∑ i, TriangularCovering.area (B i)) / |
| ((if centered then 4 else 1)*b*height b) := by |
| rw [Finset.sum_div] |
| exact Finset.sum_le_sum fun i _ => |
| div_le_div_of_nonneg_left ENNReal.toReal_nonneg (reference_pos centered hb) |
| (shadow_lower centered hb (u i) (hu i)) |
|
|
| lemma upperAxis_unit (b eta s : ℝ) : ‖upperPhysicalAxis b eta s‖ = 1 := by |
| have hv : point b 1 (-eta*(s^2+b^2)/2) (eta*s) ≠ 0 := by |
| intro h |
| have h0 := congrArg (fun z : Space => z 0) h |
| norm_num [point] at h0 |
| unfold upperPhysicalAxis |
| rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel₀ (norm_ne_zero_iff.mpr hv)] |
|
|
| lemma lowerAxis_unit (b eta s : ℝ) : ‖lowerPhysicalAxis b eta s‖ = 1 := by |
| have hv : point b (eta*(s^2+b^2)/2) 1 (eta*s) ≠ 0 := by |
| intro h |
| have h1 := congrArg (fun z : Space => z 1) h |
| norm_num [point] at h1 |
| unfold lowerPhysicalAxis |
| rw [norm_smul, norm_inv, norm_norm, inv_mul_cancel₀ (norm_ne_zero_iff.mpr hv)] |
|
|
| theorem actual_relativeCost_le (centered : Bool) (hb : 0 < b) |
| (eta : ℝ) (T : ℝ → ℝ) (N : ℕ) (tagU tagL : Fin N → ℝ) : |
| (∑ i, TriangularCovering.area (upperPhysicalBase centered b eta T N tagU i) / |
| CylinderCovering.shadowArea (modelBody centered b) (upperPhysicalAxis b eta (tagU i))) + |
| (∑ i, TriangularCovering.area (lowerPhysicalBase centered b eta T N tagL i) / |
| CylinderCovering.shadowArea (modelBody centered b) (lowerPhysicalAxis b eta (tagL i))) ≤ |
| physicalActualCost centered b eta T N tagU tagL / |
| ((if centered then 4 else 1)*b*height b) := by |
| unfold physicalActualCost |
| rw [add_div] |
| exact add_le_add |
| (baseRelativeCost_le centered hb _ _ (fun i => upperAxis_unit b eta (tagU i))) |
| (baseRelativeCost_le centered hb _ _ (fun i => lowerAxis_unit b eta (tagL i))) |
|
|
| theorem actual_half_bound (centered : Bool) (hb : 0 < b) |
| (eta : ℝ) (T : ℝ → ℝ) (N : ℕ) (tagU tagL : Fin N → ℝ) |
| (hcost : physicalActualCost centered b eta T N tagU tagL < |
| ((if centered then 4 else 1)*b*height b)/2) : |
| (∑ i, TriangularCovering.area (upperPhysicalBase centered b eta T N tagU i) / |
| CylinderCovering.shadowArea (modelBody centered b) (upperPhysicalAxis b eta (tagU i))) + |
| (∑ i, TriangularCovering.area (lowerPhysicalBase centered b eta T N tagL i) / |
| CylinderCovering.shadowArea (modelBody centered b) (lowerPhysicalAxis b eta (tagL i))) < 1/2 := by |
| apply (actual_relativeCost_le centered hb eta T N tagU tagL).trans_lt |
| exact (div_lt_iff₀ (reference_pos centered hb)).2 (by linarith) |
|
|
| theorem physical_minimum : Statement := by |
| refine ⟨?_, nonorthonormal_minimum, ?_⟩ |
| · intro centered b hb |
| exact ⟨minimum_modelBody centered hb, minimum_attained centered hb, |
| shadow_lower centered hb, fun _ C hC => both_half_bounds centered hb C hC⟩ |
| · intro centered b hb eta T N tagU tagL hcost |
| refine ⟨?_, actual_half_bound centered hb eta T N tagU tagL hcost⟩ |
| change physicalActualCost centered b eta T N tagU tagL < |
| Similarity.minimumArea (modelBody centered b)/2 |
| rwa [minimum_modelBody centered hb] |
|
|
| end RadialPhysical.PhysicalMinimum |
|
|
| end |
|
|
| end OAI |
|
|
