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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 | import OAI.Geometry.IsometricImmersion.Calculus.FinitePrimitive
namespace OAI
noncomputable section
open Set Filter MeasureTheory Function
open scoped ContDiff Topology Interval
namespace SmoothLocal.Weighted
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Flow
variable {B : Coord → ℝ} {R a b : ℝ}
theorem coefficient_pair_contDiffOn
(hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b)) :
ContDiffOn ℝ ∞ (fun p : ℝ × ℝ => B (coordinatePoint p.1 p.2)) (pairRectangle R a b) := by
have hpoint : ContDiff ℝ ∞ (fun p : ℝ × ℝ => coordinatePoint p.1 p.2) := by
unfold coordinatePoint
fun_prop
exact hB.comp hpoint.contDiffOn (fun p hp => point_mem_rectangle hp.1 hp.2)
theorem coefficient_pairPartialY
(hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b))
{x y : ℝ} (hx : x ∈ Ioo (-R) R) (hy : y ∈ Ioo a b) :
pairPartialY (fun p : ℝ × ℝ => B (coordinatePoint p.1 p.2)) (x, y) =
coordPartial 1 B (coordinatePoint x y) := by
have hdB : DifferentiableAt ℝ B (coordinatePoint x y) :=
((hB.contDiffAt ((coordinateRectangle_isOpen R a b).mem_nhds
(point_mem_rectangle hx hy))).differentiableAt (by simp))
have hdPair : DifferentiableAt ℝ (fun p : ℝ × ℝ => B (coordinatePoint p.1 p.2)) (x, y) :=
(((coefficient_pair_contDiffOn hB).contDiffAt
((pairRectangle_isOpen R a b).mem_nhds ⟨hx, hy⟩)).differentiableAt (by simp))
have hcoord : HasDerivAt (fun t => B (coordinatePoint x t))
(coordPartial 1 B (coordinatePoint x y)) y :=
hdB.hasFDerivAt.comp_hasDerivAt y (coordinatePoint_hasDerivAt_y x y)
have hpair : HasDerivAt (fun t => B (coordinatePoint x t))
(pairPartialY (fun p : ℝ × ℝ => B (coordinatePoint p.1 p.2)) (x, y)) y :=
hdPair.hasFDerivAt.comp_hasDerivAt y
((hasDerivAt_const y x).prodMk (hasDerivAt_id y))
exact hpair.unique hcoord
theorem coordinatePrimitive_partial_s
(hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b))
(hR : 0 < R) {p : Coord} (hp : p ∈ coordinateRectangle R a b) :
coordPartial 1 (coordinatePrimitive B) p =
∫ t in 0..p 0, coordPartial 1 B (coordinatePoint t (p 1)) := by
have hpairC1 : ContDiffOn ℝ 1 (fun q : ℝ × ℝ => B (coordinatePoint q.1 q.2))
(pairRectangle R a b) := (coefficient_pair_contDiffOn hB).of_le (by simp)
have hftc := pairPrimitive_hasDerivAt_y hpairC1 hp.1 hp.2
have hseg : uIcc 0 (p 0) ⊆ Ioo (-R) R := by
have hz : (0 : ℝ) ∈ Ioo (-R) R := ⟨by linarith, hR⟩
intro t ht
exact ⟨lt_of_lt_of_le (lt_min hz.1 hp.1.1) ht.1,
lt_of_le_of_lt ht.2 (max_lt hz.2 hp.1.2)⟩
have he : pairPrimitive
(pairPartialY (fun q : ℝ × ℝ => B (coordinatePoint q.1 q.2))) (p 0, p 1) =
∫ t in 0..p 0, coordPartial 1 B (coordinatePoint t (p 1)) := by
unfold pairPrimitive
apply intervalIntegral.integral_congr
intro t ht
exact coefficient_pairPartialY hB (hseg ht) hp.2
rw [he] at hftc
have hftc' : HasDerivAt (fun s => coordinatePrimitive B (coordinatePoint (p 0) s))
(∫ t in 0..p 0, coordPartial 1 B (coordinatePoint t (p 1))) (p 1) := by
simpa [pairPrimitive, coordinatePrimitive, coordinatePoint] using hftc
have hdI : DifferentiableAt ℝ (coordinatePrimitive B) (coordinatePoint (p 0) (p 1)) := by
rw [point_eta]
exact ((coordinatePrimitive_contDiffOn hB).contDiffAt
((coordinateRectangle_isOpen R a b).mem_nhds hp)).differentiableAt (by simp)
have hchain := hdI.hasFDerivAt.comp_hasDerivAt (p 1) (coordinatePoint_hasDerivAt_y (p 0) (p 1))
have hcoord : HasDerivAt (fun s => coordinatePrimitive B (coordinatePoint (p 0) s))
(coordPartial 1 (coordinatePrimitive B) p) (p 1) := by
simpa only [point_eta, Function.comp_def, coordPartial] using hchain
exact hcoord.unique hftc'
theorem coordinatePrimitive_segment_intervalIntegrable
(hB : ContinuousOn B (coordinateRectangle R a b))
(hR : 0 < R) {p : Coord} (hp : p ∈ coordinateRectangle R a b) :
IntervalIntegrable (fun t => B (coordinatePoint t (p 1))) volume 0 (p 0) := by
have hc : ContinuousOn (fun t => B (coordinatePoint t (p 1))) (Ioo (-R) R) :=
hB.comp (by unfold coordinatePoint; fun_prop)
(fun t ht => point_mem_rectangle ht hp.2)
have hz : (0 : ℝ) ∈ Ioo (-R) R := ⟨by linarith, hR⟩
have hseg : uIcc 0 (p 0) ⊆ Ioo (-R) R := by
intro t ht
exact ⟨lt_of_lt_of_le (lt_min hz.1 hp.1.1) ht.1,
lt_of_le_of_lt ht.2 (max_lt hz.2 hp.1.2)⟩
exact (hc.mono hseg).intervalIntegrable
theorem coordinatePrimitive_abs_bound
(hB : ContinuousOn B (coordinateRectangle R a b))
(hR : 0 < R) {p : Coord} (hp : p ∈ coordinateRectangle R a b)
{M : ℝ} (hbound : ∀ q ∈ coordinateRectangle R a b, |B q| ≤ M) :
|coordinatePrimitive B p| ≤ R * M := by
have hM : 0 ≤ M := (abs_nonneg _).trans (hbound p hp)
have hz : (0 : ℝ) ∈ Ioo (-R) R := ⟨by linarith, hR⟩
have hseg : uIcc 0 (p 0) ⊆ Ioo (-R) R := by
intro t ht
exact ⟨lt_of_lt_of_le (lt_min hz.1 hp.1.1) ht.1,
lt_of_le_of_lt ht.2 (max_lt hz.2 hp.1.2)⟩
have hi := coordinatePrimitive_segment_intervalIntegrable hB hR hp
have hb := intervalIntegral.norm_integral_le_of_norm_le_const (a := 0) (b := p 0) (C := M)
(f := fun t => B (coordinatePoint t (p 1)))
(fun t ht => by
rw [Real.norm_eq_abs]
exact hbound _ (point_mem_rectangle (hseg (uIoc_subset_uIcc ht)) hp.2))
calc
_ ≤ M * |p 0| := by simpa only [coordinatePrimitive, Real.norm_eq_abs, sub_zero] using hb
_ ≤ M * R := mul_le_mul_of_nonneg_left (abs_lt.mpr hp.1).le hM
_ = R * M := mul_comm _ _
theorem coordinatePrimitive_partial_s_bound
(hB : ContDiffOn ℝ ∞ B (coordinateRectangle R a b))
(hR : 0 < R) {p : Coord} (hp : p ∈ coordinateRectangle R a b)
{M : ℝ} (hBs : ∀ q ∈ coordinateRectangle R a b, |coordPartial 1 B q| ≤ M) :
|coordPartial 1 (coordinatePrimitive B) p| ≤ R * M := by
rw [coordinatePrimitive_partial_s hB hR hp]
exact coordinatePrimitive_abs_bound
(partial_contDiffOn hB (coordinateRectangle_isOpen R a b) 1).continuousOn hR hp hBs
end SmoothLocal.Weighted
end
end OAI
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