| import OAI.Geometry.CylinderCovering.Singular.EdgeOneGeometryDefinitions |
| import OAI.Geometry.CylinderCovering.Singular.EdgeOne |
| import OAI.Geometry.RuledCovering.Main |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped Topology |
| namespace SingularEdgeOne |
| open RuledApproximation |
|
|
| lemma fieldP_eq_ruled (p b r : ℝ) : |
| fieldP p b r = singularField p (vec2 b r) 0 := by |
| unfold fieldP singularField |
| simp only [vec2_apply_zero, vec2_apply_one] |
| rw [show r / (1 / 2 : ℝ) = 2 * r by ring] |
| ring |
|
|
| lemma fieldQ_eq_ruled (p b r : ℝ) : |
| fieldQ p b r = singularField p (vec2 b r) 1 := by |
| unfold fieldQ singularField |
| simp only [vec2_apply_zero, vec2_apply_one] |
| rw [show r / (1 / 2 : ℝ) = 2 * r by ring] |
|
|
| lemma ceiling_eq_ruled (p b : ℝ) : ceiling p b = ceilingPolynomial p b := by |
| unfold ceiling ceilingPolynomial |
| ring |
|
|
| lemma coefficient_eq_ruled (p : ℝ) : coefficient p = singularCoefficient p := rfl |
|
|
| lemma edgeDomain0_isCompact : IsCompact edgeDomain0 := by |
| apply (coneTriangle_isCompact (1 / 2)).of_isClosed_subset |
| · unfold edgeDomain0 |
| exact (isClosed_le (f := fun _ : Plane => (0 : ℝ)) (g := fun z => z 1) |
| (by fun_prop) (by fun_prop)).inter |
| ((isClosed_le (f := fun z : Plane => z 1) (g := fun _ => (1 / 2 : ℝ)) |
| (by fun_prop) (by fun_prop)).inter |
| (isClosed_le (f := fun z : Plane => |z 0|) (g := fun z => z 1 / 2) |
| (by fun_prop) (by fun_prop))) |
| · intro z hz |
| exact ⟨hz.1, hz.2.1, hz.2.2.trans (by linarith [hz.1])⟩ |
|
|
| lemma edgeEnergy_eq_ruled (p : ℝ) {w l : ℝ} (hl : 0 < l) : |
| edgeEnergy p w l = singularEnergy p (vec2 (2 * w) l) / 32 := by |
| unfold edgeEnergy singularEnergy |
| rw [edgeU_eq, edgeV_eq p w hl, fieldP_eq_ruled, fieldQ_eq_ruled] |
| ring |
|
|
| lemma edgeEnergy_nonneg (p w l : ℝ) : 0 ≤ edgeEnergy p w l := by |
| unfold edgeEnergy |
| positivity |
|
|
| lemma edgeEnergy_measurable (p : ℝ) : |
| Measurable (fun z : Plane => edgeEnergy p (z 0) (z 1)) := by |
| unfold edgeEnergy edgeU edgeV |
| fun_prop |
|
|
| lemma edgeEnergy_on_domain_eq {p : ℝ} (hp : 1 < p) {z : Plane} (hz : z ∈ edgeDomain0) : |
| edgeEnergy p (z 0) (z 1) = singularEnergy p (vec2 (2 * z 0) (z 1)) / 32 := by |
| by_cases hl : z 1 = 0 |
| · have hw : z 0 = 0 := abs_nonpos_iff.mp (by simpa only [hl, zero_div] using hz.2.2) |
| simp [hl, hw, edgeEnergy, edgeU, edgeV, singularEnergy, singularField, vec2, |
| Real.zero_rpow (show p - 1 ≠ 0 by linarith)] |
| · exact edgeEnergy_eq_ruled p (lt_of_le_of_ne hz.1 (Ne.symm hl)) |
|
|
| lemma edgeEnergy_integrable {p : ℝ} (hp : 1 < p) : |
| IntegrableOn (fun z : Plane => edgeEnergy p (z 0) (z 1)) edgeDomain0 := by |
| obtain ⟨C, hC, hbound⟩ := singularEnergy_bounded hp |
| apply IntegrableOn.of_bound edgeDomain0_isCompact.measure_lt_top |
| (edgeEnergy_measurable p).aestronglyMeasurable (C / 32) |
| filter_upwards [ae_restrict_mem edgeDomain0_isCompact.measurableSet] with z hz |
| rw [Real.norm_eq_abs, abs_of_nonneg (edgeEnergy_nonneg p _ _), edgeEnergy_on_domain_eq hp hz] |
| apply div_le_div_of_nonneg_right _ (by norm_num) |
| apply hbound |
| change 0 ≤ z 1 ∧ z 1 ≤ 2 ∧ |2 * z 0| ≤ z 1 |
| refine ⟨hz.1, hz.2.1.trans (by norm_num), ?_⟩ |
| rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| linarith [hz.2.2] |
|
|
| lemma edgeG_integral {p : ℝ} (hp : 1 < p) : |
| (∫ z in edgeDomain0, edgeEnergy p (z 0) (z 1)) = edgeG p := by |
| have heq : ∀ z : Plane, z ∈ edgeDomain0 ↔ |
| z 1 ∈ Icc (0 : ℝ) (1 / 2) ∧ z 0 ∈ Icc (-z 1 / 2) (z 1 / 2) := by |
| intro z |
| simp only [edgeDomain0, mem_ofPred_eq, mem_Icc, abs_le] |
| constructor |
| · rintro ⟨h0, h1, h2, h3⟩ |
| exact ⟨⟨h0, h1⟩, ⟨by linarith, h3⟩⟩ |
| · rintro ⟨⟨h0, h1⟩, ⟨h2, h3⟩⟩ |
| exact ⟨h0, h1, ⟨by linarith, h3⟩⟩ |
| rw [plane_setIntegral_horizontal_interval edgeDomain0_isCompact.measurableSet |
| (edgeEnergy_integrable hp) (s := Icc (0 : ℝ) (1 / 2)) |
| (l := fun l => -l / 2) (u := fun l => l / 2) measurableSet_Icc heq] |
| have he : (∫ l in Icc (0 : ℝ) (1 / 2), |
| ∫ w in Icc (-l / 2) (l / 2), edgeEnergy p w l) = |
| (∫ l in (0 : ℝ)..(1 / 2 : ℝ), |
| ∫ w in (-l / 2)..(l / 2), edgeEnergy p w l) := by |
| rw [intervalIntegral.integral_of_le (by norm_num : (0 : ℝ) ≤ 1 / 2), |
| ← integral_Icc_eq_integral_Ioc] |
| apply setIntegral_congr_fun measurableSet_Icc |
| intro l hl |
| dsimp only |
| rw [intervalIntegral.integral_of_le (by linarith [hl.1] : -l / 2 ≤ l / 2), |
| ← integral_Icc_eq_integral_Ioc] |
| simpa only [vec2_apply_zero, vec2_apply_one] using he.trans (edgeG_iterated_integral (by linarith)) |
|
|
| lemma height_factor : Real.sqrt 2 / 2 = (1 / Real.sqrt 2 : ℝ) := by |
| have hs : Real.sqrt 2 ≠ 0 := (Real.sqrt_pos.mpr (by norm_num : (0 : ℝ) < 2)).ne' |
| field_simp |
| nlinarith [Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 2)] |
| |
| lemma physicalSimilarity_vec3 (x y z : ℝ) : |
| physicalSimilarity (vec3 x y z) = vec3 (y / 2) (x / 2) (z / 2) := rfl |
| |
| lemma physicalSimilarity_tetrahedron : physicalSimilarity '' tetrahedron = edgeTetrahedron := by |
| ext z |
| constructor |
| · rintro ⟨x, ⟨t, ht0, ht1, hx, hy, hz⟩, rfl⟩ |
| refine ⟨t, ht0, ht1, ?_, ?_, ?_⟩ |
| · change |x 1 / 2| ≤ t / 2 |
| exact (cone_dictionary (x 1) t).2 hy |
| · change |x 0 / 2| ≤ (1 - t) / 2 |
| exact (cone_dictionary (x 0) (1 - t)).2 hx |
| · change x 2 / 2 = (1 / Real.sqrt 2) * t |
| rw [hz, show Real.sqrt 2 * t / 2 = (Real.sqrt 2 / 2) * t by ring, height_factor] |
| · rintro ⟨t, ht0, ht1, hx, hy, hz⟩ |
| refine ⟨vec3 (2 * z 1) (2 * z 0) (2 * z 2), ?_, ?_⟩ |
| · refine ⟨t, ht0, ht1, ?_, ?_, ?_⟩ |
| · change |2 * z 1| ≤ 1 - t |
| rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| linarith |
| · change |2 * z 0| ≤ t |
| rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| linarith |
| · change 2 * z 2 = Real.sqrt 2 * t |
| rw [hz, ← height_factor] |
| ring |
| · ext i |
| fin_cases i <;> simp [physicalSimilarity, vec3] |
| |
| lemma edgeU_labelMap (p : ℝ) (z : Plane) : |
| edgeU p (labelMap z 0) (labelMap z 1) = singularField p z 0 / 4 := by |
| change edgeU p (z 0 / 2) (z 1) = _ |
| rw [edgeU_eq, show 2 * (z 0 / 2) = z 0 by ring, fieldP_eq_ruled, vec2_coordinates] |
| |
| lemma edgeV_labelMap (p : ℝ) {z : Plane} (hz : 0 < z 1) : |
| edgeV p (labelMap z 0) (labelMap z 1) = singularField p z 1 / 2 := by |
| change edgeV p (z 0 / 2) (z 1) = _ |
| rw [edgeV_eq p (z 0 / 2) hz, show 2 * (z 0 / 2) = z 0 by ring, |
| fieldQ_eq_ruled, vec2_coordinates] |
| |
| lemma physicalSimilarity_lower (p c : ℝ) {D : Set Plane} |
| (hD : ∀ b ∈ D, 0 < b 1) : |
| physicalSimilarity '' physicalRuled (Real.sqrt 2) D |
| (fun b => c • singularField p b) 1 = edgeLower p (4 * c) (labelMap '' D) := by |
| ext z |
| constructor |
| · rintro ⟨q, ⟨b, hb, s, hs, rfl⟩, rfl⟩ |
| refine ⟨labelMap b, ⟨b, hb, rfl⟩, s / 2, ?_, ?_⟩ |
| · exact (cone_dictionary s 1).2 hs |
| · ext i |
| fin_cases i |
| · change (b 0 + s * (c * singularField p b 0)) / 2 = |
| b 0 / 2 + (4 * c) * (s / 2) * edgeU p (labelMap b 0) (labelMap b 1) |
| rw [edgeU_labelMap] |
| ring |
| · change s / 2 = s / 2 |
| rfl |
| · change Real.sqrt 2 * (b 1 + s * (c * singularField p b 1)) / 2 = |
| (1 / Real.sqrt 2) * |
| (b 1 + (4 * c) * (s / 2) * edgeV p (labelMap b 0) (labelMap b 1)) |
| rw [edgeV_labelMap p (hD b hb), ← height_factor] |
| ring |
| · rintro ⟨b', ⟨b, hb, rfl⟩, y, hy, rfl⟩ |
| let q := vec3 (2 * y) (b 0 + (2 * y) * (c * singularField p b 0)) |
| (Real.sqrt 2 * (b 1 + (2 * y) * (c * singularField p b 1))) |
| refine ⟨q, ⟨b, hb, 2 * y, ?_, rfl⟩, ?_⟩ |
| · rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| linarith |
| · ext i |
| fin_cases i |
| · change (b 0 + (2 * y) * (c * singularField p b 0)) / 2 = |
| b 0 / 2 + (4 * c) * y * edgeU p (labelMap b 0) (labelMap b 1) |
| rw [edgeU_labelMap] |
| ring |
| · change (2 * y) / 2 = y |
| ring |
| · change Real.sqrt 2 * (b 1 + (2 * y) * (c * singularField p b 1)) / 2 = |
| (1 / Real.sqrt 2) * |
| (b 1 + (4 * c) * y * edgeV p (labelMap b 0) (labelMap b 1)) |
| rw [edgeV_labelMap p (hD b hb), ← height_factor] |
| ring |
|
|
| lemma physicalSimilarity_upper (p c : ℝ) {D : Set Plane} |
| (hD : ∀ b ∈ D, 0 < b 1) : |
| physicalSimilarity '' upperSegments p c D = edgeUpper p (4 * c) (labelMap '' D) := by |
| ext z |
| constructor |
| · rintro ⟨q, ⟨b, hb, s, hs, rfl⟩, rfl⟩ |
| refine ⟨labelMap b, ⟨b, hb, rfl⟩, s / 2, (cone_dictionary s 1).2 hs, ?_⟩ |
| ext i |
| fin_cases i |
| · change s / 2 = s / 2 |
| rfl |
| · change (b 0 - c * s * singularField p b 0) / 2 = |
| b 0 / 2 - (4 * c) * (s / 2) * edgeU p (labelMap b 0) (labelMap b 1) |
| rw [edgeU_labelMap] |
| ring |
| · change Real.sqrt 2 * (1 - (b 1 - c * s * singularField p b 1)) / 2 = |
| (1 / Real.sqrt 2) * |
| (1 - (b 1 - (4 * c) * (s / 2) * edgeV p (labelMap b 0) (labelMap b 1))) |
| rw [edgeV_labelMap p (hD b hb), ← height_factor] |
| ring |
| · rintro ⟨b', ⟨b, hb, rfl⟩, x, hx, rfl⟩ |
| let q := vec3 (b 0 - c * (2 * x) * singularField p b 0) (2 * x) |
| (Real.sqrt 2 * (1 - (b 1 - c * (2 * x) * singularField p b 1))) |
| refine ⟨q, ⟨b, hb, 2 * x, ?_, rfl⟩, ?_⟩ |
| · rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2)] |
| linarith |
| · ext i |
| fin_cases i |
| · change (2 * x) / 2 = x |
| ring |
| · change (b 0 - c * (2 * x) * singularField p b 0) / 2 = |
| b 0 / 2 - (4 * c) * x * edgeU p (labelMap b 0) (labelMap b 1) |
| rw [edgeU_labelMap] |
| ring |
| · change Real.sqrt 2 * (1 - (b 1 - c * (2 * x) * singularField p b 1)) / 2 = |
| (1 / Real.sqrt 2) * |
| (1 - (b 1 - (4 * c) * x * edgeV p (labelMap b 0) (labelMap b 1))) |
| rw [edgeV_labelMap p (hD b hb), ← height_factor] |
| ring |
| |
| lemma upperSegments_canonical (p gamma c : ℝ) : |
| upperSegments p c (singularDomain p gamma c) = singularUpper p gamma c := rfl |
| |
| end SingularEdgeOne |
| |
| end |
| |
| end OAI |
| |
