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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 | import OAI.Geometry.IsometricImmersion.Flows.FlowImageFirstBounds
import OAI.Geometry.IsometricImmersion.Flows.FlowCurvature
import OAI.Geometry.IsometricImmersion.Energy.RectangleSobolev
import Mathlib.Topology.Order.IntermediateValue
namespace OAI
noncomputable section
open Set Function Filter
open scoped ContDiff Topology
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
def openCoordinateRectangle (tl tr sb st : ℝ) : Set Coord :=
{p | p 0 ∈ Ioo tl tr ∧ p 1 ∈ Ioo sb st}
def flowInteriorRadius (M dt ds : ℝ) : ℝ :=
min (dt / 2) (((50 : ℝ) / 101) * Real.exp (-2 * M) * ds)
theorem flowInteriorRadius_pos (M : ℝ) {dt ds : ℝ} (ht : 0 < dt) (hs : 0 < ds) :
0 < flowInteriorRadius M dt ds := by
apply lt_min
· exact half_pos ht
· exact mul_pos (mul_pos (by norm_num) (Real.exp_pos _)) hs
theorem flowInteriorRadius_lt_time (M : ℝ) {dt ds : ℝ} (ht : 0 < dt) :
flowInteriorRadius M dt ds < dt := by
have h := min_le_left (dt / 2) (((50 : ℝ) / 101) * Real.exp (-2 * M) * ds)
change flowInteriorRadius M dt ds ≤ dt / 2 at h
linarith
theorem flowInteriorRadius_vertical_margin (M : ℝ) {dt ds : ℝ} (hs : 0 < ds) :
((101 : ℝ) / 100) * flowInteriorRadius M dt ds < Real.exp (-2 * M) * ds := by
have h := min_le_right (dt / 2) (((50 : ℝ) / 101) * Real.exp (-2 * M) * ds)
change flowInteriorRadius M dt ds ≤ ((50 : ℝ) / 101) * Real.exp (-2 * M) * ds at h
have hp := mul_pos (Real.exp_pos (-2 * M)) hs
nlinarith
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 actual_flow_ordinary_rectangle_inclusion
{M tl tr sb st dt ds t0 s0 : ℝ} (hM : 0 ≤ M)
(hq0 : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100)
(hq1 : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
(htl : -2 < tl) (htr : tr < 2) (hsb : -2 < sb) (hst : st < 2)
(hdt : 0 < dt) (hds : 0 < ds)
(ht0 : t0 ∈ Icc (tl + dt) (tr - dt))
(hs0 : s0 ∈ Icc (sb + ds) (st - ds)) :
closedRectangle (t0 - flowInteriorRadius M dt ds) (t0 + flowInteriorRadius M dt ds)
(Y s0 t0 - flowInteriorRadius M dt ds) (Y s0 t0 + flowInteriorRadius M dt ds) ⊆
capChart Y '' openCoordinateRectangle tl tr sb st := by
intro p hp
let r := flowInteriorRadius M dt ds
have hr : 0 < r := flowInteriorRadius_pos M hdt hds
have hrt : r < dt := flowInteriorRadius_lt_time M hdt
have hrs : ((101 : ℝ) / 100) * r < Real.exp (-2 * M) * ds :=
flowInteriorRadius_vertical_margin M hds
have hpx : p 0 ∈ Icc (t0 - r) (t0 + r) := hp.1
have hpy : p 1 ∈ Icc (Y s0 t0 - r) (Y s0 t0 + r) := hp.2
have htime : p 0 ∈ Ioo tl tr := by constructor <;> linarith [ht0.1, ht0.2, hpx.1, hpx.2]
have htimeGlobal : p 0 ∈ Ioo (-2 : ℝ) 2 := ⟨htl.trans htime.1, htime.2.trans htr⟩
have ht0Global : t0 ∈ Ioo (-2 : ℝ) 2 := by constructor <;> linarith [ht0.1, ht0.2]
have hs0Global : s0 ∈ Ioo (-2 : ℝ) 2 := by constructor <;> linarith [hs0.1, hs0.2]
have hsbGlobal : sb ∈ Ioo (-2 : ℝ) 2 := by constructor <;> linarith [hs0.1, hs0.2]
have hstGlobal : st ∈ Ioo (-2 : ℝ) 2 := by constructor <;> linarith [hs0.1, hs0.2]
have hsb0 : sb ≤ s0 := by linarith [hs0.1]
have hs0t : s0 ≤ st := by linarith [hs0.2]
have hlow := cap_flow_seed_gap hq hU hSU hY hrange hstart hode hM hq1
hsbGlobal hs0Global htimeGlobal hsb0
have hupp := cap_flow_seed_gap hq hU hSU hY hrange hstart hode hM hq1
hs0Global hstGlobal htimeGlobal hs0t
have hgap0 : Real.exp (-2 * M) * ds ≤ Real.exp (-2 * M) * (s0 - sb) :=
mul_le_mul_of_nonneg_left (by linarith [hs0.1]) (Real.exp_pos _).le
have hgap1 : Real.exp (-2 * M) * ds ≤ Real.exp (-2 * M) * (st - s0) :=
mul_le_mul_of_nonneg_left (by linarith [hs0.2]) (Real.exp_pos _).le
have hxdist : |p 0 - t0| ≤ r := abs_le.mpr ⟨by linarith [hpx.1], by linarith [hpx.2]⟩
have hmove : |Y s0 (p 0) - Y s0 t0| ≤ ((1 : ℝ) / 100) * r :=
(cap_flow_time_lipschitz hq hU hSU hY hrange hstart hode hq0
hs0Global ht0Global htimeGlobal).trans
(mul_le_mul_of_nonneg_left hxdist (by norm_num))
have hlo : Y sb (p 0) < p 1 := by
have hm := (abs_le.mp hmove).2
nlinarith [hlow, hgap0, hpy.1, hrs]
have hhi : p 1 < Y st (p 0) := by
have hm := (abs_le.mp hmove).1
nlinarith [hupp, hgap1, hpy.2, hrs]
have hsb_st : sb ≤ st := hsb0.trans hs0t
have hseedCont : ContinuousOn (fun s : ℝ => (s, p 0)) (Icc sb st) := by fun_prop
have hseedMap : MapsTo (fun s : ℝ => (s, p 0)) (Icc sb st)
(Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2) := by
intro s hs
exact ⟨⟨by linarith [hs.1], by linarith [hs.2]⟩,
⟨htimeGlobal.1.le, htimeGlobal.2.le⟩⟩
have hcont : ContinuousOn (fun s => Y s (p 0)) (Icc sb st) := by
convert hY.comp hseedCont hseedMap using 1; rfl
obtain ⟨s, hs, hse⟩ := intermediate_value_Icc hsb_st hcont ⟨hlo.le, hhi.le⟩
have hsStrict : s ∈ Ioo sb st := by
constructor
· by_contra hnot
have he : s = sb := le_antisymm (le_of_not_gt hnot) hs.1
subst s
linarith
· by_contra hnot
have he : s = st := le_antisymm hs.2 (le_of_not_gt hnot)
subst s
linarith
refine ⟨boxPoint (p 0) s, ?_, ?_⟩
· simpa [openCoordinateRectangle, boxPoint] using And.intro htime hsStrict
· ext i
fin_cases i <;> simp [capChart, capFlowHeight, boxPoint, coordinatePoint, hse]
theorem actual_flow_uniform_image_margin
{M tl tr sb st dt ds : ℝ} (hM : 0 ≤ M)
(hq0 : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100)
(hq1 : ∀ p ∈ modelSquare, |coordPartial 1 q p| ≤ M)
(htl : -2 < tl) (htr : tr < 2) (hsb : -2 < sb) (hst : st < 2)
(hdt : 0 < dt) (hds : 0 < ds) :
0 < flowInteriorRadius M dt ds ∧
∀ p ∈ capChart Y '' closedRectangle (tl + dt) (tr - dt) (sb + ds) (st - ds),
closedRectangle (p 0 - flowInteriorRadius M dt ds) (p 0 + flowInteriorRadius M dt ds)
(p 1 - flowInteriorRadius M dt ds) (p 1 + flowInteriorRadius M dt ds) ⊆
capChart Y '' openCoordinateRectangle tl tr sb st := by
refine ⟨flowInteriorRadius_pos M hdt hds, ?_⟩
rintro p ⟨a, ha, rfl⟩
simpa [capChart, capFlowHeight, coordinatePoint] using
actual_flow_ordinary_rectangle_inclusion hq hU hSU hY hrange hstart hode
hM hq0 hq1 htl htr hsb hst hdt hds ha.1 ha.2
end SmoothLocal.Flow
end
end OAI
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