| 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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|
