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59d1c80 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 | import OAI.Geometry.IsometricImmersion.Pulses.PulseProfiles
import OAI.Geometry.IsometricImmersion.Flows.FlowImageFirstBounds
import OAI.Geometry.IsometricImmersion.Caps.FixedMetricLowerCap
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology BigOperators
namespace SmoothLocal.Pulse
open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.ODE SmoothLocal.Weighted
theorem scalar_fderiv_coordinate_bound {q : Coord → ℝ} {p v : Coord} {M : ℝ}
(_hM : 0 ≤ M) (hpartial : ∀ i : Fin 2, |coordPartial i q p| ≤ M) :
|fderiv ℝ q p v| ≤ M*(|v 0|+|v 1|) := by
have he : v = (v 0) • Pi.single 0 (1 : ℝ) + (v 1) • Pi.single 1 (1 : ℝ) := by
ext i
fin_cases i <;> simp
conv_lhs => rw [he, map_add, map_smul, map_smul]
change |v 0 * coordPartial 0 q p + v 1 * coordPartial 1 q p| ≤ _
calc
_ ≤ |v 0 * coordPartial 0 q p| + |v 1 * coordPartial 1 q p| := abs_add_le _ _
_ = |v 0| * |coordPartial 0 q p|+|v 1| * |coordPartial 1 q p| := by rw [abs_mul,abs_mul]
_ ≤ |v 0| * M+|v 1| * M := add_le_add
(mul_le_mul_of_nonneg_left (hpartial 0) (abs_nonneg _))
(mul_le_mul_of_nonneg_left (hpartial 1) (abs_nonneg _))
_ = _ := by ring
theorem quotient_change_from_origin
{q : Coord → ℝ} {U : Set Coord} {M : ℝ}
(hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) (hM : 0 ≤ M)
(hpartial : ∀ p ∈ modelSquare, ∀ i : Fin 2, |coordPartial i q p| ≤ M)
{p : Coord} (hp : p ∈ modelSquare) : |q p-q 0| ≤ M*(|p 0|+|p 1|) := by
have hz : (0 : Coord) ∈ modelSquare := by constructor <;> intro i <;> norm_num
have hconv : Convex ℝ modelSquare := convex_Icc _ _
have hseg (t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) : t • p ∈ modelSquare := by
have h := hconv hp hz ht.1 (sub_nonneg.mpr ht.2) (by ring)
simpa only [smul_zero, add_zero] using h
have hder (t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) :
HasDerivAt (fun s : ℝ => q (s • p)) (fderiv ℝ q (t • p) p) t := by
have hdq := ((hq.contDiffAt (hU.mem_nhds (hSU (hseg t ht)))).differentiableAt (by simp)).hasFDerivAt
have hdp : HasDerivAt (fun s : ℝ => s • p) p t := by
have ht := (hasDerivAt_id t).smul_const p
change HasDerivAt (fun s : ℝ => s • p) ((1 : ℝ) • p) t at ht
rw [one_smul] at ht
exact ht
exact hdq.comp_hasDerivAt t hdp
have hb (t : ℝ) (ht : t ∈ Icc (0 : ℝ) 1) :
‖deriv (fun s : ℝ => q (s • p)) t‖ ≤ M*(|p 0|+|p 1|) := by
rw [(hder t ht).deriv, Real.norm_eq_abs]
exact scalar_fderiv_coordinate_bound hM (hpartial (t • p) (hseg t ht))
have h := (convex_Icc (0 : ℝ) 1).norm_image_sub_le_of_norm_deriv_le
(fun t ht => (hder t ht).differentiableAt) hb
(by simp : (0 : ℝ) ∈ Icc (0 : ℝ) 1) (by simp : (1 : ℝ) ∈ Icc (0 : ℝ) 1)
simpa only [zero_smul, one_smul, sub_zero, norm_one, mul_one, Real.norm_eq_abs] using h
variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ}
variable (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U)
variable (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2))
variable (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, Y s t ∈ Icc (-3 : ℝ) 3)
variable (hstart : ∀ s ∈ Icc (-2 : ℝ) 2, Y s 0 = s)
variable (hode : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2,
HasDerivWithinAt (Y s) (-q (coordinatePoint t (Y s t))) (Icc (-2 : ℝ) 2) t)
include hq hU hSU hY hrange hstart hode
theorem flow_segment_coordinate_sum
(hsmall : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ)/100)
{q0 T r s t u : ℝ} (hq0 : |q0| ≤ 1/20) (_hT : 0 < T) (hT2 : T < 2)
(hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Icc (-T) T) (hu : u ∈ Icc (-T) T)
(htheta : |Y s t+q0*t| ≤ r) :
|u|+|Y s u| ≤ r+107*T/100 := by
have ht' : t ∈ Ioo (-2 : ℝ) 2 := ⟨by linarith [ht.1],by linarith [ht.2]⟩
have hu' : u ∈ Ioo (-2 : ℝ) 2 := ⟨by linarith [hu.1],by linarith [hu.2]⟩
have habst : |t| ≤ T := abs_le.mpr ht
have habsu : |u| ≤ T := abs_le.mpr hu
have hdt : |u-t| ≤ 2*T := (abs_sub u t).trans (by linarith)
have hmove := (cap_flow_time_lipschitz hq hU hSU hY hrange hstart hode hsmall hs ht' hu').trans
(mul_le_mul_of_nonneg_left hdt (by norm_num : (0 : ℝ) ≤ 1/100))
have hprod : |q0*t| ≤ (1/20)*T := by
rw [abs_mul]
exact mul_le_mul hq0 habst (abs_nonneg _) (by norm_num)
have hyt : |Y s t| ≤ r+(1/20)*T := by
calc
|Y s t| = |(Y s t+q0*t)-q0*t| := by congr 1; ring
_ ≤ |Y s t+q0*t|+|q0*t| := abs_sub _ _
_ ≤ _ := add_le_add htheta hprod
have hyu : |Y s u| ≤ |Y s u-Y s t|+|Y s t| := by
calc
|Y s u| = |(Y s u-Y s t)+Y s t| := by congr 1; ring
_ ≤ _ := abs_add_le _ _
linarith
theorem flow_segment_quotient_near_center
(hsmall : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ)/100)
{M q0 L T r s t : ℝ} (hM : 0 ≤ M) (hL : 0 < L)
(hpartial : ∀ p ∈ modelSquare, ∀ i : Fin 2, |coordPartial i q p| ≤ M)
(hq0 : |q0| ≤ 1/20) (hcenter : |q 0-q0| ≤ 1/(100*L))
(hT : 0 < T) (hT2 : T < 2) (hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Icc (-T) T)
(htheta : |Y s t+q0*t| ≤ r)
(hrsmall : M*(r+107*T/100) ≤ 9/(100*L)) :
∀ u ∈ Icc (-T) T, |q (coordinatePoint u (Y s u))-q0| ≤ 1/(10*L) := by
intro u hu
have hu2 : u ∈ Icc (-2 : ℝ) 2 := ⟨by linarith [hu.1],by linarith [hu.2]⟩
have hp : coordinatePoint u (Y s u) ∈ modelSquare := coordinatePoint_mem_modelSquare
⟨by linarith [hu2.1],by linarith [hu2.2]⟩
(hrange s ⟨hs.1.le,hs.2.le⟩ u hu2)
have hnorm := flow_segment_coordinate_sum hq hU hSU hY hrange hstart hode
hsmall hq0 hT hT2 hs ht hu htheta
have hlocal : |q (coordinatePoint u (Y s u))-q 0| ≤ M*(|u|+|Y s u|) := by
simpa [coordinatePoint] using quotient_change_from_origin hq hU hSU hM hpartial hp
have hsum : |q (coordinatePoint u (Y s u))-q0| ≤
|q (coordinatePoint u (Y s u))-q 0|+|q 0-q0| := by
calc
_ = |(q (coordinatePoint u (Y s u))-q 0)+(q 0-q0)| := by congr 1; ring
_ ≤ _ := abs_add_le _ _
have ha := hsum.trans (add_le_add (hlocal.trans
(mul_le_mul_of_nonneg_left hnorm hM)) hcenter)
have heq : 9/(100*L)+1/(100*L) = 1/(10*L) := by field_simp [hL.ne']; ring
exact ha.trans ((add_le_add hrsmall le_rfl).trans_eq heq)
theorem actual_shear_seed_distance
{q0 epsilon T s t : ℝ} (hT : 0 < T) (hT2 : T < 2)
(hs : s ∈ Ioo (-2 : ℝ) 2) (ht : t ∈ Icc (-T) T)
(hclose : ∀ u ∈ Icc (-T) T, |q (coordinatePoint u (Y s u))-q0| ≤ epsilon) :
|Y s t+q0*t-s| ≤ epsilon*|t| := by
have hd (u : ℝ) (hu : u ∈ Icc (-T) T) :
HasDerivAt (fun a => Y s a+q0*a) (-q (coordinatePoint u (Y s u))+q0) u := by
have hu2 : u ∈ Ioo (-2 : ℝ) 2 := ⟨by linarith [hu.1],by linarith [hu.2]⟩
have hh := (cap_flow_time_hasDerivAt hq hU hSU hY hrange hstart hode hs hu2).add
((hasDerivAt_id u).const_mul q0)
change HasDerivAt (fun a : ℝ => Y s a + q0 * a)
(-q (coordinatePoint u (Y s u)) + q0 * 1) u at hh
rw [mul_one] at hh
exact hh
have hb (u : ℝ) (hu : u ∈ Icc (-T) T) :
‖deriv (fun a => Y s a+q0*a) u‖ ≤ epsilon := by
rw [(hd u hu).deriv, Real.norm_eq_abs]
have heq : -q (coordinatePoint u (Y s u))+q0 = -(q (coordinatePoint u (Y s u))-q0) := by ring
rw [heq,abs_neg]
exact hclose u hu
have h := (convex_Icc (-T) T).norm_image_sub_le_of_norm_deriv_le
(fun u hu => (hd u hu).differentiableAt) hb
(show (0 : ℝ) ∈ Icc (-T) T by constructor <;> linarith) ht
simpa only [hstart s ⟨hs.1.le,hs.2.le⟩, mul_zero, add_zero, sub_zero, Real.norm_eq_abs] using h
end SmoothLocal.Pulse
end
end OAI
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