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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.Geometry.RuledCovering.SingularBounds
namespace OAI
noncomputable section
open Set MeasureTheory Filter Metric
open scoped Topology ContDiff Pointwise
namespace RuledApproximation
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)
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⟩
end RuledApproximation
end
end OAI
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