| import OAI.Geometry.CylinderCovering.Singular.Cost |
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| namespace OAI |
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| noncomputable section |
| open Set MeasureTheory Filter Topology |
| open scoped ContDiff |
| namespace CylinderCovering.FlexibleSingular |
| open RuledApproximation |
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|
| lemma hyperbolic_eventually {p : ℝ} {ρ q : ℝ → ℝ} (hp : 1 < p) (hp2 : p < 2) |
| (hadm : SingularCutoff.Admissible p ρ q) : |
| ∀ᶠ c : ℝ in 𝓝[>] 0, ∀ z ∈ labelDomain p c (ρ c) (q 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*(ρ c)^(p-2))) (𝓝[>] 0) (𝓝 0) := by |
| simpa only [mul_zero] using (hadm.2.2.1).const_mul C |
| filter_upwards [self_mem_nhdsWithin,hlim.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1)),hadm.1] |
| with c hc hsmall hpos |
| have hc : 0 < c := hc |
| intro z hz |
| have hd : 0 < ρ c := hpos.1 |
| 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*(ρ c)^(p-2)) := mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_left hr (by linarith)) hc.le |
| _ = C*(c*(ρ c)^(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)⟩ |
|
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| lemma scaled_smoothNear (p c q k : ℝ) {d : ℝ} (hd : 0 < d) : |
| SmoothNear 1 (labelDomain p c d q) (fun z => k • singularField p z) := by |
| obtain ⟨U,hU,hDU,hF⟩ := labelDomain_smoothNear p c q hd |
| exact ⟨U,hU,hDU,(hF.of_le (by norm_num : (1:WithTop ℕ∞) ≤ ∞)).const_smul k⟩ |
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| lemma upperFamily_image (p c d q : ℝ) : |
| (fun z => vec3 0 0 (Real.sqrt 2)+z) '' |
| (swapReflect '' physicalRuled (Real.sqrt 2) (labelDomain p c d q) |
| (fun z => -c • singularField p z) 1) = upperFamily p c d q := 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 |
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| theorem flexible_singular : Statement := by |
| intro p ρ q hp hp' hadm |
| have hs : (Real.sqrt 51)^2 = (51:ℝ) := Real.sq_sqrt (by norm_num) |
| have hsn := Real.sqrt_nonneg (51:ℝ) |
| have hs9 : Real.sqrt 51 < 9 := by nlinarith |
| have hp2 : p < 2 := by linarith |
| have hroot : 2*p+5 < Real.sqrt 51 := by linarith |
| have hnum : 2*p^2+10*p-13 < 0 := by nlinarith |
| have hk : singularCoefficient p < 0 := |
| div_neg_of_neg_of_pos (mul_neg_of_pos_of_neg (by norm_num) hnum) (by linarith) |
| refine ⟨hk,asymptotic hp hp2.le hadm,?_⟩ |
| filter_upwards [self_mem_nhdsWithin,geometric_cover hp hp2 hadm, |
| hyperbolic_eventually hp hp2 hadm,totalCost_small hp hp2.le hadm hk,hadm.1] |
| with c hc hgeom hhyp hcost hpos |
| have hc : 0 < c := hc |
| let eps : ℝ := (Real.sqrt 2/2-(2*cost p c (ρ c) (q c)+8*Real.sqrt 2*(ρ c)^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 := labelDomain_isCompact p c (ρ c) (q c) |
| obtain ⟨n,C,hC,hcovC,hcostC⟩ := physical_hyperbolic_cover hroot (by norm_num : (0:ℝ)<1) |
| hD (scaled_smoothNear p c (q c) c hpos.1) (fun z hz => (hhyp z hz).1) heps |
| have hneg := physical_hyperbolic_cover hroot (by norm_num : (0:ℝ)<1) |
| hD (scaled_smoothNear p c (q c) (-c) hpos.1) (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) (labelDomain p c (ρ c) (q c)) |
| (fun z => -c • singularField p z) 1)) (cost p c (ρ c) (q c)+eps) at hupper |
| rw [upperFamily_image] at hupper |
| obtain ⟨m,D,hD,hcovD,hcostD⟩ := hupper |
| refine ⟨n,m,C,D,lowerTipCylinder (ρ c),upperTipCylinder (ρ c),hC,hD, |
| lowerTipCylinder_triangle (hpos.1), |
| upperTipCylinder_triangle (hpos.1),hcovC,hcovD, |
| lowerTipCylinder_carrier _,upperTipCylinder_carrier _, |
| lowerTipCylinder_area hpos.1.le,upperTipCylinder_area hpos.1.le,?_,?_⟩ |
| · 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 (hpos.1.le), |
| upperTipCylinder_area (hpos.1.le)] |
| change familyArea C ≤ cost p c (ρ c) (q c)+eps at hcostC |
| dsimp only [eps] at hcostC hcostD |
| linarith |
| |
| end CylinderCovering.FlexibleSingular |
| |
| end |
| |
| end OAI |
| |
