| import OAI.Geometry.RuledCovering.SingularBounds |
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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 inverse_in_closedBall {V : Plane → Plane} {q : Plane} {R k : ℝ} (hR : 0 ≤ R) |
| (hcont : ∀ z ∈ Metric.closedBall q R, DifferentiableAt ℝ V z) |
| (hsize : ∀ z ∈ Metric.closedBall q R, |k| *‖V z‖ ≤ R) |
| (hder : ∀ z ∈ Metric.closedBall q R, |k| *‖fderiv ℝ V z‖ ≤ 1/2) : |
| ∃ z ∈ Metric.closedBall q R, q = z+k • V z := by |
| let S := Metric.closedBall q R |
| let F : S → S := fun z => ⟨q-k • V z,by |
| rw [Metric.mem_closedBall,dist_eq_norm,sub_sub_cancel_left,norm_neg,norm_smul,Real.norm_eq_abs] |
| exact hsize z z.2⟩ |
| have hL : ContractingWith (1/2 : NNReal) F := by |
| refine ⟨by norm_num,LipschitzWith.of_dist_le_mul ?_⟩ |
| intro x y |
| have he := Convex.norm_image_sub_le_of_norm_fderiv_le |
| (f := fun z => k • V z) (C := 1/2) |
| (fun z hz => (hcont z hz).const_smul k) |
| (fun z hz => by |
| rw [fderiv_fun_const_smul (hcont z hz),norm_smul,Real.norm_eq_abs] |
| exact hder z hz) |
| (convex_closedBall q R) y.2 x.2 |
| change ‖(q-k • V x)-(q-k • V y)‖ ≤ (1/2)*‖(x:Plane)-(y:Plane)‖ |
| rw [sub_sub_sub_cancel_left,norm_sub_rev] |
| exact he |
| have : Nonempty S := ⟨⟨q,Metric.mem_closedBall_self hR⟩⟩ |
| have : IsClosed S := Metric.isClosed_closedBall |
| let z := ContractingWith.fixedPoint F hL |
| have hz := congrArg Subtype.val hL.fixedPoint_isFixedPt |
| refine ⟨z,z.2,?_⟩ |
| change q-k • V z = z at hz |
| exact (sub_eq_iff_eq_add.mp hz) |
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| lemma singular_inverse_small {p : ℝ} (hp2 : p < 2) : |
| ∃ C : ℝ, 1 ≤ C ∧ ∀ (d c B T k : ℝ), 0 < d → 0 ≤ c → |
| c*C*d^(p-2) ≤ 1/8 → 2*d ≤ T → T ≤ 1 → |B| ≤ T → |k| ≤ c → |
| ∃ z : Plane, d ≤ z 1 ∧ z 1 ≤ 2 ∧ |z 0| ≤ z 1 ∧ |
| vec2 B T = z+k • singularField p z := by |
| obtain ⟨C,hC,hbound⟩ := singular_uniform_bounds p |
| refine ⟨C,hC,?_⟩ |
| intro d c B T k hd hc hsmall hT hT1 hB hk |
| have hTp : 0 < T := by linarith |
| have hCp : 0 ≤ C := by linarith |
| have hball : ∀ z ∈ Metric.closedBall (vec2 B T) (T/4), |
| d ≤ z 1 ∧ z 1 ≤ 5*T/4 ∧ |z 0| ≤ 2*z 1 := by |
| intro z hz |
| rw [Metric.mem_closedBall,dist_eq_norm] at hz |
| have hr := (plane_coordinate_le (z-vec2 B T) 1).trans hz |
| have hb := (plane_coordinate_le (z-vec2 B T) 0).trans hz |
| change |z 1-T| ≤ T/4 at hr |
| change |z 0-B| ≤ T/4 at hb |
| obtain ⟨hrl,hru⟩ := abs_le.mp hr |
| have hb' : |z 0| ≤ T/4+T := by |
| calc |
| _ = |(z 0-B)+B| := by congr 1; ring |
| _ ≤ |z 0-B|+|B| := abs_add_le _ _ |
| _ ≤ T/4+T := add_le_add hb hB |
| exact ⟨by linarith,by linarith,by linarith⟩ |
| have hprod : ∀ z ∈ Metric.closedBall (vec2 B T) (T/4), |k| * C*(z 1)^(p-2) ≤ 1/8 := by |
| intro z hz |
| have hz' := hball z hz |
| have hzp : 0 < z 1 := hd.trans_le hz'.1 |
| calc |
| _ ≤ c*C*(z 1)^(p-2) := mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_right hk hCp) (by positivity) |
| _ ≤ c*C*d^(p-2) := mul_le_mul_of_nonneg_left |
| (Real.rpow_le_rpow_of_nonpos hd hz'.1 (by linarith)) (mul_nonneg hc hCp) |
| _ ≤ _ := hsmall |
| obtain ⟨z,hz,heq⟩ := inverse_in_closedBall (V := singularField p) |
| (k := k) (show 0 ≤ T/4 by positivity) |
| (fun z hz => (singularField_contDiffAt p (hd.trans_le (hball z hz).1)).differentiableAt (by norm_num)) |
| (fun z hz => by |
| have hz' := hball z hz |
| have hzp := hd.trans_le hz'.1 |
| have hV := (hbound z hzp hz'.2.2).1 |
| calc |
| _ ≤ |k| * (C*(z 1)^(p-1)) := mul_le_mul_of_nonneg_left hV (abs_nonneg _) |
| _ = (|k| * C*(z 1)^(p-2))*z 1 := by |
| rw [show p-1=p-2+1 by ring,Real.rpow_add_one hzp.ne'] |
| ring |
| _ ≤ (1/8)*z 1 := mul_le_mul_of_nonneg_right (hprod z hz) hzp.le |
| _ ≤ T/4 := by linarith) |
| (fun z hz => by |
| have hz' := hball z hz |
| have hV := (hbound z (hd.trans_le hz'.1) hz'.2.2).2.1 |
| exact ((mul_le_mul_of_nonneg_left hV (abs_nonneg _)).trans |
| (by simpa only [mul_assoc] using hprod z hz)).trans (by norm_num)) |
| have hz' := hball z hz |
| have hzp := hd.trans_le hz'.1 |
| have htheta : ∀ e : ℝ, |e| ≤ 1 → |
| 0 < 1+k*((z 1/(1/2:ℝ))^(p-2)/2*((2-p)*(z 0/z 1)+e*p)) := by |
| intro e he |
| have ht := mul_le_mul_of_nonneg_left ((hbound z hzp hz'.2.2).2.2 e he) (abs_nonneg k) |
| rw [← abs_mul] at ht |
| have ht' : |k*((z 1/(1/2:ℝ))^(p-2)/2*((2-p)*(z 0/z 1)+e*p))| ≤ 1/8 := by |
| exact ht.trans (by simpa only [mul_assoc] using hprod z hz) |
| have := (abs_le.mp ht').1 |
| linarith |
| have he0 := congrArg (fun w : Plane => w 0) heq |
| have he1 := congrArg (fun w : Plane => w 1) heq |
| change B = z 0+k*singularField p z 0 at he0 |
| change T = z 1+k*singularField p z 1 at he1 |
| have hw := singularField_wall p (b := z 0) hzp |
| rw [vec2_coordinates] at hw |
| have hwplus : T+B = (z 1+z 0)*(1+k*((z 1/(1/2:ℝ))^(p-2)/2*((2-p)*(z 0/z 1)+1*p))) := by |
| calc |
| _ = z 1+z 0+k*(singularField p z 1+singularField p z 0) := by rw [he0,he1]; ring |
| _ = _ := by rw [hw.1]; ring |
| have hwminus : T-B = (z 1-z 0)*(1+k*((z 1/(1/2:ℝ))^(p-2)/2*((2-p)*(z 0/z 1)+(-1)*p))) := by |
| calc |
| _ = z 1-z 0+k*(singularField p z 1-singularField p z 0) := by rw [he0,he1]; ring |
| _ = _ := by rw [hw.2]; ring |
| have hp0 : 0 ≤ z 1+z 0 := (mul_nonneg_iff_of_pos_right (htheta 1 (by norm_num))).mp |
| (by rw [← hwplus]; linarith [(abs_le.mp hB).1]) |
| have hm0 : 0 ≤ z 1-z 0 := (mul_nonneg_iff_of_pos_right (htheta (-1) (by norm_num))).mp |
| (by rw [← hwminus]; linarith [(abs_le.mp hB).2]) |
| exact ⟨z,hz'.1,by linarith [hz'.2.1],abs_le.mpr ⟨by linarith,by linarith⟩,heq⟩ |
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| end RuledApproximation |
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| end |
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| end OAI |
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