| import OAI.Geometry.IsometricImmersion.Caps.ActualPulseCapSeparation |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology BigOperators |
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| namespace SmoothLocal.Pulse |
| open SmoothLocal.Geometry SmoothLocal.Flow SmoothLocal.ODE SmoothLocal.Weighted |
| open SmoothLocal.HighEquation |
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| theorem actual_flow_preimage_near_center |
| {g : MetricField} {U W : Set Coord} {G Z d c e0 kappa : ℝ} |
| {z : Coord → ℝ} {Y : ℝ → ℝ → ℝ} |
| (hf : CapInductionFlow g U G Z d c e0 kappa z Y W) |
| {t y : ℝ} (ht : |t| ≤ 1) (hy : |y| ≤ 3/4) : |
| ∃ s ∈ Ioo (-1 : ℝ) 1, Y s t = y := by |
| have ht2 : t ∈ Ioo (-2 : ℝ) 2 := by |
| constructor <;> linarith [(abs_le.mp ht).1,(abs_le.mp ht).2] |
| have hnegp : boxPoint t (-1) ∈ capChartDomain := by |
| simpa [capChartDomain,coordinateRectangle,boxPoint] using And.intro ht2 (show (-1 : ℝ) ∈ Ioo (-2 : ℝ) 2 by norm_num) |
| have hposp : boxPoint t 1 ∈ capChartDomain := by |
| simpa [capChartDomain,coordinateRectangle,boxPoint] using And.intro ht2 (show (1 : ℝ) ∈ Ioo (-2 : ℝ) 2 by norm_num) |
| have hneg : |Y (-1) t+1| ≤ (1 : ℝ)/50 := by |
| simpa [capFlowHeight,boxPoint] using hf.displacement (boxPoint t (-1)) hnegp |
| have hpos : |Y 1 t-1| ≤ (1 : ℝ)/50 := by |
| simpa [capFlowHeight,boxPoint] using hf.displacement (boxPoint t 1) hposp |
| have hlo : Y (-1) t < y := by linarith [(abs_le.mp hneg).2,(abs_le.mp hy).1] |
| have hhi : y < Y 1 t := by linarith [(abs_le.mp hpos).1,(abs_le.mp hy).2] |
| have hmap : MapsTo (fun s : ℝ => (s,t)) (Icc (-1 : ℝ) 1) |
| (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2) := by |
| intro s hs |
| exact ⟨⟨by linarith [hs.1],by linarith [hs.2]⟩,⟨ht2.1.le,ht2.2.le⟩⟩ |
| have hcont : ContinuousOn (fun s : ℝ => Y s t) (Icc (-1 : ℝ) 1) := |
| hf.continuous.comp (by fun_prop : ContinuousOn (fun s : ℝ => (s,t)) (Icc (-1 : ℝ) 1)) hmap |
| obtain ⟨s,hs,hse⟩ := intermediate_value_Icc (by norm_num : (-1 : ℝ) ≤ 1) hcont ⟨hlo.le,hhi.le⟩ |
| refine ⟨s,⟨?_,?_⟩,hse⟩ |
| · by_contra hn |
| have he : s = -1 := le_antisymm (le_of_not_gt hn) hs.1 |
| subst s |
| linarith |
| · by_contra hn |
| have he : s = 1 := le_antisymm hs.2 (le_of_not_gt hn) |
| subst s |
| linarith |
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| def lowerCutRectangle (L r : ℝ) (hL : 0 < L) (hr : 0 < r) (hrhalf : r < 1/2) |
| (hLr : L*r ≤ 1/20) : LowerCapRectangle (-r/2) where |
| left := -(2*L*r) |
| right := 2*L*r |
| bottom := -(3*r/2) |
| top := -(3*r/4) |
| left_gt := by nlinarith |
| right_lt := by nlinarith |
| bottom_gt := by linarith |
| horizontal := by nlinarith [mul_pos hL hr] |
| vertical := by linarith |
| top_lt := by linarith |
| ceiling_le := by linarith |
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| theorem actual_lower_cut_in_cap_image |
| {g : MetricField} {U W : Set Coord} {G Z d c e0 kappa q0 L r : ℝ} |
| {z : Coord → ℝ} {Y : ℝ → ℝ → ℝ} |
| (hh : CapInductionHeight g U Z c e0 z) (hf : CapInductionFlow g U G Z d c e0 kappa z Y W) |
| (hG : 0 ≤ G) (hZ : 0 ≤ Z) (hd : 0 < d) (hc : 0 < c) |
| (hL : 0 < L) (hr : 0 < r) (hrhalf : r < 1/2) (hLr : L*r ≤ 1/20) |
| (hq0 : |q0| ≤ 1/20) (hcenter : |hessianQuotient g z 0-q0| ≤ 1/(100*L)) |
| (hrsmall : heightQuotientJetBound G Z d c*(r+107*(L*r)/100) ≤ 9/(100*L)) |
| {t : ℝ} (ht : |t| ≤ L*r) : |
| inverseShearCoordinates q0 (boxPoint t (-r)) ∈ (lowerCutRectangle L r hL hr hrhalf hLr).image Y := by |
| let M := heightQuotientJetBound G Z d c |
| have hM : 0 ≤ M := heightQuotientJetBound_nonneg hG hZ hd hc |
| have hpartial (p : Coord) (hp : p ∈ modelSquare) (i : Fin 2) : |
| |coordPartial i (hessianQuotient g z) p| ≤ M := hf.quotientBound [i] (by norm_num) p hp |
| have hTpos : 0 < L*r := mul_pos hL hr |
| have ht1 : |t| ≤ 1 := ht.trans (by linarith) |
| have hqt : |q0*t| ≤ (1/20)*(L*r) := by |
| rw [abs_mul] |
| exact mul_le_mul hq0 ht (abs_nonneg _) (by norm_num) |
| have hy : |-r-q0*t| ≤ 3/4 := by |
| have h := abs_sub (-r) (q0*t) |
| rw [abs_neg,abs_of_pos hr] at h |
| linarith |
| obtain ⟨s,hs,hse⟩ := actual_flow_preimage_near_center hf ht1 hy |
| have hs2 : s ∈ Ioo (-2 : ℝ) 2 := ⟨by linarith [hs.1],by linarith [hs.2]⟩ |
| have htt : t ∈ Icc (-(L*r)) (L*r) := abs_le.mp ht |
| have htheta : |Y s t+q0*t| ≤ r := by |
| rw [hse] |
| have he : -r-q0*t+q0*t = -r := by ring |
| rw [he,abs_neg,abs_of_pos hr] |
| have hclose := flow_segment_quotient_near_center hf.quotientSmooth hf.domainOpen hf.squareSubset |
| hf.continuous hf.range hf.start hf.ode hh.quotient hM hL hpartial hq0 hcenter |
| hTpos (by linarith : L*r < 2) hs2 htt htheta hrsmall |
| have hdist := actual_shear_seed_distance hf.quotientSmooth hf.domainOpen hf.squareSubset |
| hf.continuous hf.range hf.start hf.ode hTpos (by linarith : L*r < 2) hs2 htt hclose |
| have hdist' : |-r-s| ≤ r/10 := by |
| have he : (1/(10*L))*(L*r) = r/10 := by |
| rw [div_mul_eq_mul_div₀, one_mul, mul_comm (10 : ℝ) L, mul_div_mul_left _ _ hL.ne'] |
| have hbound := hdist.trans ((mul_le_mul_of_nonneg_left ht (by positivity)).trans_eq he) |
| simpa only [hse,sub_add_cancel] using hbound |
| refine ⟨boxPoint t s,?_,?_⟩ |
| · change (boxPoint t s) ∈ closedRectangle (-(2*L*r)) (2*L*r) (-(3*r/2)) (-(3*r/4)) |
| constructor |
| · simpa [boxPoint] using (show t ∈ Icc (-(2*L*r)) (2*L*r) by |
| constructor <;> nlinarith [(abs_le.mp ht).1,(abs_le.mp ht).2]) |
| · simpa [boxPoint] using (show s ∈ Icc (-(3*r/2)) (-(3*r/4)) by |
| constructor <;> linarith [(abs_le.mp hdist').1,(abs_le.mp hdist').2]) |
| · ext i |
| fin_cases i <;> simp [capChart,capFlowHeight,coordinatePoint,boxPoint,inverseShearCoordinates,hse] |
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| end SmoothLocal.Pulse |
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| end |
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| end OAI |
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