| import OAI.Geometry.IsometricImmersion.Flows.FlowParameter |
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| namespace OAI |
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| noncomputable section |
| open Set Metric Filter Function MeasureTheory |
| open scoped ContDiff Topology Interval |
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|
| namespace SmoothLocal.Flow |
| open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted |
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|
| theorem continuous_intervalIntegral_of_joint |
| {X : Type*} [TopologicalSpace X] [FirstCountableTopology X] [LocallyCompactSpace X] |
| {a b : ℝ} (hab : a ≤ b) {f : X → ℝ → ℝ} |
| (hf : Continuous (fun p : X × Icc a b => f p.1 p.2)) : |
| Continuous (fun x => ∫ u in a..b, f x u) := by |
| let g : X → ℝ → ℝ := fun x u => f x (projIcc a b hab u) |
| have hg : Continuous (uncurry g) := |
| hf.comp (continuous_fst.prodMk ((continuous_projIcc (h := hab)).comp continuous_snd)) |
| have hc : Continuous (fun x => ∫ u in Icc a b, g x u) := |
| continuous_parametric_integral_of_continuous hg isCompact_Icc |
| apply hc.congr |
| intro x |
| calc |
| (∫ u in Icc a b, g x u) = ∫ u in a..b, g x u := by |
| rw [intervalIntegral.integral_of_le hab, integral_Icc_eq_integral_Ioc] |
| _ = ∫ u in a..b, f x u := by |
| apply intervalIntegral.integral_congr |
| intro u hu |
| have hu' : u ∈ Icc a b := by simpa only [uIcc_of_le hab] using hu |
| simp only [g, projIcc_of_mem hab hu'] |
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|
| abbrev CapInterval := Icc (-2 : ℝ) 2 |
| abbrev VerticalInterval := Icc (-3 : ℝ) 3 |
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| theorem verticalAverageDerivative_continuous {q : Coord → ℝ} {U : Set Coord} |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) : |
| Continuous (fun p : CapInterval × (VerticalInterval × VerticalInterval) => |
| verticalAverageDerivative q p.1 p.2.1 p.2.2) := by |
| apply continuous_intervalIntegral_of_joint (show (0 : ℝ) ≤ 1 by norm_num) |
| have hmap : Continuous |
| (fun p : (CapInterval × (VerticalInterval × VerticalInterval)) × Icc (0 : ℝ) 1 => |
| coordinatePoint p.1.1 ((p.1.2.1 : ℝ) + p.2 * ((p.1.2.2 : ℝ) - p.1.2.1))) := by |
| unfold coordinatePoint |
| fun_prop |
| exact (partial_contDiffOn hq hU 1).continuousOn.comp_continuous hmap |
| (fun p => hSU (verticalSegment_mem_modelSquare p.1.1.2 p.1.2.1.2 p.1.2.2.2 p.2.2)) |
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| def flowAverageCoefficient (q : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (s0 s1 t : ℝ) : ℝ := |
| verticalAverageDerivative q t (Y s0 t) (Y s1 t) |
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|
| theorem flowAverageCoefficient_continuous {q : Coord → ℝ} {U : Set Coord} |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| {Y : ℝ → ℝ → ℝ} |
| (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) : |
| Continuous (fun p : CapInterval × (CapInterval × CapInterval) => |
| flowAverageCoefficient q Y p.1 p.2.1 p.2.2) := by |
| have hy0 : Continuous (fun p : CapInterval × (CapInterval × CapInterval) => |
| Y p.1 p.2.2) := |
| hY.comp_continuous |
| (f := fun p : CapInterval × (CapInterval × CapInterval) => ((p.1 : ℝ), (p.2.2 : ℝ))) |
| (by fun_prop) (fun p => ⟨p.1.2, p.2.2.2⟩) |
| have hy1 : Continuous (fun p : CapInterval × (CapInterval × CapInterval) => |
| Y p.2.1 p.2.2) := |
| hY.comp_continuous |
| (f := fun p : CapInterval × (CapInterval × CapInterval) => ((p.2.1 : ℝ), (p.2.2 : ℝ))) |
| (by fun_prop) (fun p => ⟨p.2.1.2, p.2.2.2⟩) |
| have hmap : Continuous |
| (fun p : CapInterval × (CapInterval × CapInterval) => |
| (p.2.2, ((⟨Y p.1 p.2.2, hrange _ p.1.2 _ p.2.2.2⟩, |
| ⟨Y p.2.1 p.2.2, hrange _ p.2.1.2 _ p.2.2.2⟩) : |
| VerticalInterval × VerticalInterval))) := |
| (continuous_snd.snd).prodMk |
| ((hy0.subtype_mk _).prodMk (hy1.subtype_mk _)) |
| exact (verticalAverageDerivative_continuous hq hU hSU).comp hmap |
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|
| theorem flowAverageIntegral_continuous {q : Coord → ℝ} {U : Set Coord} |
| (hq : ContDiffOn ℝ ∞ q U) (hU : IsOpen U) (hSU : modelSquare ⊆ U) |
| {Y : ℝ → ℝ → ℝ} |
| (hY : ContinuousOn (uncurry Y) (Icc (-2 : ℝ) 2 ×ˢ Icc (-2 : ℝ) 2)) |
| (hrange : ∀ s ∈ Icc (-2 : ℝ) 2, ∀ t ∈ Icc (-2 : ℝ) 2, |
| Y s t ∈ Icc (-3 : ℝ) 3) {t : ℝ} (ht : t ∈ Icc (-2 : ℝ) 2) : |
| Continuous (fun p : CapInterval × CapInterval => |
| ∫ u in 0..t, flowAverageCoefficient q Y p.1 p.2 u) := by |
| have hcoeff := flowAverageCoefficient_continuous hq hU hSU hY hrange |
| have haux (a b : ℝ) (hab : a ≤ b) (ha : -2 ≤ a) (hb : b ≤ 2) : |
| Continuous (fun p : CapInterval × CapInterval => |
| ∫ u in a..b, flowAverageCoefficient q Y p.1 p.2 u) := by |
| apply continuous_intervalIntegral_of_joint hab |
| have hmap : Continuous |
| (fun p : (CapInterval × CapInterval) × Icc a b => |
| (p.1.1, (p.1.2, (⟨p.2, ⟨ha.trans p.2.2.1, p.2.2.2.trans hb⟩⟩ : CapInterval)))) := by |
| fun_prop |
| exact hcoeff.comp hmap |
| by_cases ht0 : 0 ≤ t |
| · exact haux 0 t ht0 (by norm_num) ht.2 |
| · have hc := (haux t 0 (le_of_not_ge ht0) ht.1 (by norm_num)).neg |
| apply hc.congr |
| intro p |
| exact (intervalIntegral.integral_symm _ _).symm |
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|
| end SmoothLocal.Flow |
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| end |
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| end OAI |
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