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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 | import OAI.Geometry.IsometricImmersion.Flows.FlowParameter
namespace OAI
noncomputable section
open Set Metric Filter Function MeasureTheory
open scoped ContDiff Topology Interval
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
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']
abbrev CapInterval := Icc (-2 : ℝ) 2
abbrev VerticalInterval := Icc (-3 : ℝ) 3
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))
def flowAverageCoefficient (q : Coord → ℝ) (Y : ℝ → ℝ → ℝ) (s0 s1 t : ℝ) : ℝ :=
verticalAverageDerivative q t (Y s0 t) (Y s1 t)
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
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
end SmoothLocal.Flow
end
end OAI
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