| import OAI.Geometry.RuledCovering.SingularGeometry |
| import OAI.Geometry.RuledCovering.SingularAsymptotics |
| import OAI.Geometry.RuledCovering.TipCylinders |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open Set MeasureTheory Filter Metric |
| open scoped Topology ContDiff Pointwise |
|
|
| namespace RuledApproximation |
|
|
| lemma singular_hyperbolic_eventually {p gamma : ℝ} (hp : 1 < p) (hp2 : p < 2) |
| (hpg : 0 < 1+gamma*(p-2)) : |
| ∀ᶠ c : ℝ in 𝓝[>] 0, ∀ z ∈ singularDomain p gamma c, |
| HyperbolicAt 1 (fun w => c • singularField p w) z ∧ |
| HyperbolicAt 1 (fun w => -c • singularField p w) z := by |
| obtain ⟨C,hC,hbound⟩ := singularField_homogeneous_bounds p |
| have hlim : Tendsto (fun c : ℝ => C*(c*(c^gamma)^(p-2))) (𝓝[>] 0) (𝓝 0) := by |
| simpa only [mul_zero] using (singular_cutoff_small hpg).const_mul C |
| filter_upwards [self_mem_nhdsWithin,hlim.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1))] |
| with c hc hsmall |
| have hc : 0 < c := hc |
| intro z hz |
| have hd : 0 < c^gamma := Real.rpow_pos_of_pos hc gamma |
| have hz1 : 0 < z 1 := hd.trans_le hz.1 |
| have hb : |z 0| ≤ 2*z 1 := by linarith [hz.2.2.1] |
| have hn := (hbound z hz1 hb).2 |
| have hr := Real.rpow_le_rpow_of_nonpos hd hz.1 (by linarith : p-2 ≤ 0) |
| have hclt : c*‖fderiv ℝ (singularField p) z‖ < 1 := calc |
| _ ≤ c*(C*(z 1)^(p-2)) := mul_le_mul_of_nonneg_left hn hc.le |
| _ ≤ c*(C*(c^gamma)^(p-2)) := mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_left hr (by linarith)) hc.le |
| _ = C*(c*(c^gamma)^(p-2)) := by ring |
| _ < 1 := hsmall |
| exact ⟨singularField_scaled_hyperbolic hp hp2 hc.ne' hz1 hz.2.2.1 |
| (by simpa only [abs_of_pos hc] using hclt), |
| singularField_scaled_hyperbolic hp hp2 (neg_ne_zero.mpr hc.ne') hz1 hz.2.2.1 |
| (by simpa only [abs_neg,abs_of_pos hc] using hclt)⟩ |
|
|
| lemma singular_scaled_smoothNear (p gamma k : ℝ) {c : ℝ} (hc : 0 < c) : |
| SmoothNear 1 (singularDomain p gamma c) (fun z => k • singularField p z) := by |
| obtain ⟨U,hU,hDU,hF⟩ := singularDomain_smoothNear p gamma hc |
| exact ⟨U,hU,hDU,(hF.of_le (by norm_num : (1:WithTop ℕ∞) ≤ ∞)).const_smul k⟩ |
|
|
| lemma physicalWeight_neg (h : ℝ) (g : Plane) : physicalWeight h (-g) = physicalWeight h g := by |
| simp [physicalWeight] |
|
|
| lemma singularUpper_image (p gamma c : ℝ) : |
| (fun z => vec3 0 0 (Real.sqrt 2)+z) '' |
| (swapReflect '' physicalRuled (Real.sqrt 2) (singularDomain p gamma c) |
| (fun z => -c • singularField p z) 1) = singularUpper p gamma c := by |
| ext z |
| constructor |
| · rintro ⟨_,⟨_,⟨b,hb,s,hs,rfl⟩,rfl⟩,rfl⟩ |
| refine ⟨b,hb,s,hs,?_⟩ |
| ext i |
| fin_cases i <;> simp [swapReflect,vec3] <;> ring |
| · rintro ⟨b,hb,s,hs,rfl⟩ |
| refine ⟨swapReflect (vec3 s (b 0+s*(-c • singularField p b) 0) |
| (Real.sqrt 2*(b 1+s*(-c • singularField p b) 1))), |
| ⟨_,⟨b,hb,s,hs,rfl⟩,rfl⟩,?_⟩ |
| ext i |
| fin_cases i <;> simp [swapReflect,vec3] <;> ring |
|
|
| / |
| choosing either finite approximation, and both errors use its strict saving. -/ |
| theorem singular : SingularStatement := by |
| intro p gamma hp hp' hg hg' |
| obtain ⟨hp2,hpg,_,hk⟩ := singular_parameter_bounds hp hp' hg hg' |
| refine ⟨hk,singular_asymptotic hp hp2.le hg,?_⟩ |
| filter_upwards [self_mem_nhdsWithin,singular_geometric_cover hp hp2 hpg, |
| singular_hyperbolic_eventually hp hp2 hpg,singular_totalCost_small hp hp2.le hg hk] |
| with c hc hgeom hhyp hcost |
| have hc : 0 < c := hc |
| let eps : ℝ := (Real.sqrt 2/2-(2*singularCost p gamma c+8*Real.sqrt 2*(c^gamma)^2))/4 |
| have heps : 0 < eps := div_pos (sub_pos.mpr hcost) (by norm_num) |
| have hroot : 0 < Real.sqrt 2 := Real.sqrt_pos.mpr (by norm_num) |
| have hD := singularDomain_isCompact p gamma c |
| obtain ⟨n,C,hC,hcovC,hcostC⟩ := physical_hyperbolic_cover hroot (by norm_num : (0:ℝ)<1) |
| hD (singular_scaled_smoothNear p gamma c hc) (fun z hz => (hhyp z hz).1) heps |
| have hneg := physical_hyperbolic_cover hroot (by norm_num : (0:ℝ)<1) |
| hD (singular_scaled_smoothNear p gamma (-c) hc) (fun z hz => (hhyp z hz).2) heps |
| simp only [neg_smul,physicalWeight_neg] at hneg |
| have hupper := (hneg.isometry swapReflect.toIsometryEquiv).isometry |
| (IsometryEquiv.addLeft (vec3 0 0 (Real.sqrt 2))) |
| simp only [← neg_smul] at hupper |
| change HasFiniteCover Cylinder.parallelogram |
| ((fun z => vec3 0 0 (Real.sqrt 2)+z) '' |
| (swapReflect '' physicalRuled (Real.sqrt 2) (singularDomain p gamma c) |
| (fun z => -c • singularField p z) 1)) (singularCost p gamma c+eps) at hupper |
| rw [singularUpper_image] at hupper |
| obtain ⟨m,D,hD,hcovD,hcostD⟩ := hupper |
| refine ⟨n,m,C,D,lowerTipCylinder (c^gamma),upperTipCylinder (c^gamma),hC,hD, |
| lowerTipCylinder_triangle (Real.rpow_pos_of_pos hc gamma), |
| upperTipCylinder_triangle (Real.rpow_pos_of_pos hc gamma),hcovC,hcovD, |
| lowerTipCylinder_carrier _,upperTipCylinder_carrier _,?_,?_⟩ |
| · rw [lowerTipCylinder_carrier,upperTipCylinder_carrier] |
| exact hgeom.trans (union_subset_union (union_subset_union (union_subset_union hcovC hcovD) Subset.rfl) Subset.rfl) |
| · rw [lowerTipCylinder_area (Real.rpow_nonneg hc.le gamma), |
| upperTipCylinder_area (Real.rpow_nonneg hc.le gamma)] |
| change familyArea C ≤ singularCost p gamma c+eps at hcostC |
| dsimp only [eps] at hcostC hcostD |
| linarith |
|
|
| end RuledApproximation |
|
|
| end |
|
|
| end OAI |
|
|
