| import OAI.Geometry.RuledCovering.SingularField |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Filter Metric |
| open scoped Topology ContDiff Pointwise |
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| namespace RuledApproximation |
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| lemma plane_norm_sq (z : Plane) : ‖z‖^2 = z 0^2+z 1^2 := by |
| rw [EuclideanSpace.norm_eq, Real.sq_sqrt (Finset.sum_nonneg (fun i _ => sq_nonneg ‖z i‖))] |
| simp [Fin.sum_univ_succ,Real.norm_eq_abs] |
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| lemma plane_coordinate_le (z : Plane) (i : Fin 2) : |z i| ≤ ‖z‖ := by |
| simpa only [Real.norm_eq_abs] using PiLp.norm_apply_le z i |
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| lemma plane_norm_le_sum (z : Plane) : ‖z‖ ≤ |z 0|+|z 1| := by |
| have h := plane_norm_sq z |
| have h0 := abs_nonneg (z 0) |
| have h1 := abs_nonneg (z 1) |
| nlinarith [sq_abs (z 0),sq_abs (z 1)] |
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| lemma singularField_homogeneous_bounds (p : ℝ) : |
| ∃ C : ℝ, 1 ≤ C ∧ ∀ z : Plane, 0 < z 1 → |z 0| ≤ 2*z 1 → |
| ‖singularField p z‖ ≤ C*(z 1)^(p-1) ∧ |
| ‖fderiv ℝ (singularField p) z‖ ≤ C*(z 1)^(p-2) := by |
| let S : Set Plane := (fun b : ℝ => vec2 b 1) '' Icc (-2 : ℝ) 2 |
| have hS : IsCompact S := isCompact_Icc.image (by unfold vec2; fun_prop) |
| have hpos : ∀ z ∈ S, 0 < z 1 := by |
| rintro z ⟨b,hb,rfl⟩ |
| norm_num [vec2] |
| obtain ⟨C₁,hC₁⟩ := hS.exists_bound_of_continuousOn |
| (fun z hz => (singularField_contDiffAt p (hpos z hz)).continuousAt.continuousWithinAt) |
| obtain ⟨C₂,hC₂⟩ := hS.exists_bound_of_continuousOn |
| (fun z hz => ((singularField_contDiffAt p (hpos z hz)).continuousAt_fderiv |
| (by norm_num)).continuousWithinAt) |
| refine ⟨max 1 (max C₁ C₂),le_max_left _ _,?_⟩ |
| intro z hz hcone |
| let u := vec2 (z 0/z 1) 1 |
| have hu : u ∈ S := by |
| refine ⟨z 0/z 1,?_,rfl⟩ |
| have ht : |z 0/z 1| ≤ 2 := by |
| rw [abs_div,abs_of_pos hz,div_le_iff₀ hz] |
| exact hcone |
| exact abs_le.mp ht |
| have hue : z 1 • u = z := by |
| ext i |
| fin_cases i <;> simp [u,vec2] |
| field_simp |
| have hf := singularField_smul p hz (z := u) (by norm_num [u,vec2]) |
| have hd := singularField_fderiv_smul p hz (z := u) (by norm_num [u,vec2]) |
| rw [hue] at hf hd |
| constructor |
| · rw [hf,norm_smul,Real.norm_eq_abs,abs_of_pos (Real.rpow_pos_of_pos hz _)] |
| calc |
| _ ≤ (z 1)^(p-1)*C₁ := mul_le_mul_of_nonneg_left (hC₁ u hu) (by positivity) |
| _ ≤ (z 1)^(p-1)*max 1 (max C₁ C₂) := mul_le_mul_of_nonneg_left |
| ((le_max_left _ _).trans (le_max_right _ _)) (by positivity) |
| _ = _ := mul_comm _ _ |
| · rw [hd,norm_smul,Real.norm_eq_abs,abs_of_pos (Real.rpow_pos_of_pos hz _)] |
| calc |
| _ ≤ (z 1)^(p-2)*C₂ := mul_le_mul_of_nonneg_left (hC₂ u hu) (by positivity) |
| _ ≤ (z 1)^(p-2)*max 1 (max C₁ C₂) := mul_le_mul_of_nonneg_left |
| ((le_max_right _ _).trans (le_max_right _ _)) (by positivity) |
| _ = _ := mul_comm _ _ |
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| lemma singularDomain_isCompact (p gamma c : ℝ) : IsCompact (singularDomain p gamma c) := by |
| have hb : Continuous (fun z : Plane => z 0) := by fun_prop |
| have hr : Continuous (fun z : Plane => z 1) := by fun_prop |
| have hcap : Continuous (fun z : Plane => (1/2:ℝ)+c^2*ceilingPolynomial p (z 0)+c^(5/2:ℝ)) := by |
| unfold ceilingPolynomial |
| fun_prop |
| have hclosed : IsClosed (singularDomain p gamma c) := |
| (isClosed_le continuous_const hr).inter ((isClosed_le hr continuous_const).inter |
| ((isClosed_le hb.abs hr).inter (isClosed_le hr hcap))) |
| apply (isCompact_closedBall (0 : Plane) 4).of_isClosed_subset hclosed |
| intro z hz |
| have hz0 : |z 0| ≤ 2 := hz.2.2.1.trans hz.2.1 |
| have hz1 : |z 1| ≤ 2 := (abs_of_nonneg ((abs_nonneg _).trans hz.2.2.1)).le.trans hz.2.1 |
| rw [Metric.mem_closedBall,dist_zero_right] |
| exact (plane_norm_le_sum z).trans (by linarith) |
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| lemma singularDomain_smoothNear (p gamma : ℝ) {c : ℝ} (hc : 0 < c) : |
| SmoothNear ∞ (singularDomain p gamma c) (singularField p) := by |
| refine ⟨{z | 0 < z 1},isOpen_lt continuous_const (by fun_prop),?_,singularField_contDiffOn p⟩ |
| intro z hz |
| exact (Real.rpow_pos_of_pos hc gamma).trans_le hz.1 |
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| lemma singular_cutoff_small {p gamma : ℝ} (hpg : 0 < 1+gamma*(p-2)) : |
| Tendsto (fun c : ℝ => c*(c^gamma)^(p-2)) (𝓝[>] 0) (𝓝 0) := by |
| have h : Tendsto (fun c : ℝ => c^(1+gamma*(p-2))) (𝓝[>] 0) |
| (𝓝 ((0:ℝ)^(1+gamma*(p-2)))) := |
| ((Real.continuous_rpow_const hpg.le).tendsto 0).mono_left nhdsWithin_le_nhds |
| rw [Real.zero_rpow hpg.ne'] at h |
| apply h.congr' |
| filter_upwards [self_mem_nhdsWithin] with c hc |
| have hc : 0 < c := hc |
| rw [← Real.rpow_mul hc.le,Real.rpow_add hc,Real.rpow_one] |
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| lemma singular_tip_small {gamma : ℝ} (hg : 1 < gamma) : |
| Asymptotics.IsLittleO (𝓝[>] (0:ℝ)) (fun c : ℝ => (c^gamma)^2) (fun c => c^2) := by |
| rw [Asymptotics.isLittleO_iff_tendsto'] |
| · have h : Tendsto (fun c : ℝ => c^(2*gamma-2)) (𝓝[>] 0) |
| (𝓝 ((0:ℝ)^(2*gamma-2))) := |
| ((Real.continuous_rpow_const (show 0 ≤ 2*gamma-2 by linarith)).tendsto 0).mono_left nhdsWithin_le_nhds |
| rw [Real.zero_rpow (by linarith : 2*gamma-2 ≠ 0)] at h |
| apply h.congr' |
| filter_upwards [self_mem_nhdsWithin] with c hc |
| have hc : 0 < c := hc |
| rw [← Real.rpow_natCast,← Real.rpow_mul hc.le,← Real.rpow_natCast,← Real.rpow_sub hc] |
| congr 1 |
| ring |
| · filter_upwards [self_mem_nhdsWithin] with c hc |
| exact fun h => False.elim ((ne_of_gt (pow_pos hc 2)) h) |
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| lemma vec2_coordinates (z : Plane) : vec2 (z 0) (z 1) = z := by |
| ext i |
| fin_cases i <;> rfl |
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| lemma singular_wall_coefficient_bound (p : ℝ) {z : Plane} |
| (hz : 0 < z 1) (hb : |z 0| ≤ 2*z 1) {e : ℝ} (he : |e| ≤ 1) : |
| |(z 1/(1/2:ℝ))^(p-2)/2*((2-p)*(z 0/z 1)+e*p)| ≤ |
| ((2:ℝ)^(p-2)/2*(2*|2-p|+|p|))*(z 1)^(p-2) := by |
| have hq : |z 0/z 1| ≤ 2 := by |
| rw [abs_div,abs_of_pos hz,div_le_iff₀ hz] |
| exact hb |
| have ha : |(2-p)*(z 0/z 1)+e*p| ≤ 2*|2-p|+|p| := by |
| calc |
| _ ≤ |(2-p)*(z 0/z 1)|+|e*p| := abs_add_le _ _ |
| _ = |2-p| *|z 0/z 1|+|e| *|p| := by rw [abs_mul,abs_mul] |
| _ ≤ |2-p| *2+1*|p| := add_le_add |
| (mul_le_mul_of_nonneg_left hq (abs_nonneg _)) |
| (mul_le_mul_of_nonneg_right he (abs_nonneg _)) |
| _ = _ := by ring |
| rw [abs_mul,abs_of_nonneg (by positivity : 0 ≤ (z 1/(1/2:ℝ))^(p-2)/2)] |
| calc |
| _ ≤ (z 1/(1/2:ℝ))^(p-2)/2*(2*|2-p|+|p|) := |
| mul_le_mul_of_nonneg_left ha (by positivity) |
| _ = _ := by |
| rw [show z 1/(1/2:ℝ)=2*z 1 by ring,Real.mul_rpow (by norm_num) hz.le] |
| ring |
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| lemma singular_uniform_bounds (p : ℝ) : |
| ∃ C : ℝ, 1 ≤ C ∧ ∀ z : Plane, 0 < z 1 → |z 0| ≤ 2*z 1 → |
| ‖singularField p z‖ ≤ C*(z 1)^(p-1) ∧ |
| ‖fderiv ℝ (singularField p) z‖ ≤ C*(z 1)^(p-2) ∧ |
| ∀ e : ℝ, |e| ≤ 1 → |(z 1/(1/2:ℝ))^(p-2)/2*((2-p)*(z 0/z 1)+e*p)| ≤ |
| C*(z 1)^(p-2) := by |
| obtain ⟨C,hC,hbound⟩ := singularField_homogeneous_bounds p |
| let W := (2:ℝ)^(p-2)/2*(2*|2-p|+|p|) |
| refine ⟨max C W,hC.trans (le_max_left _ _),?_⟩ |
| intro z hz hb |
| obtain ⟨h0,h1⟩ := hbound z hz hb |
| refine ⟨h0.trans (mul_le_mul_of_nonneg_right (le_max_left _ _) (by positivity)), |
| h1.trans (mul_le_mul_of_nonneg_right (le_max_left _ _) (by positivity)),?_⟩ |
| intro e he |
| exact (singular_wall_coefficient_bound p hz hb he).trans |
| (mul_le_mul_of_nonneg_right (le_max_right _ _) (by positivity)) |
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| end RuledApproximation |
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| end |
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| end OAI |
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