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695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | 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
/-- Singular-field approximation. The positive parameter is fixed before
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
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