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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 | import OAI.Geometry.IsometricImmersion.Caps.CapFlowCoordinateJets
namespace OAI
noncomputable section
open Set Filter Function
open scoped ContDiff Topology
namespace SmoothLocal.Flow
open SmoothLocal.Geometry SmoothLocal.ODE SmoothLocal.Weighted
def flowCoordinateBound (M : ℝ) : ℝ :=
4 + Real.exp (2 * M) + 2 * M + M * Real.exp (2 * M) + 2 * M * Real.exp (4 * M) +
(4 * M + 2 * M^2) + (2 * M + M^2) * Real.exp (2 * M) +
(M + 2 * M^2) * Real.exp (4 * M) + (2 * M + 8 * M^2) * Real.exp (6 * M)
theorem flowCoordinateBound_dominates {M : ℝ} (hM : 0 ≤ M) :
3 ≤ flowCoordinateBound M ∧ 1 ≤ flowCoordinateBound M ∧
Real.exp (2 * M) ≤ flowCoordinateBound M ∧ 2 * M ≤ flowCoordinateBound M ∧
M * Real.exp (2 * M) ≤ flowCoordinateBound M ∧ 2 * M * Real.exp (4 * M) ≤ flowCoordinateBound M ∧
4 * M + 2 * M^2 ≤ flowCoordinateBound M ∧
(2 * M + M^2) * Real.exp (2 * M) ≤ flowCoordinateBound M ∧
(M + 2 * M^2) * Real.exp (4 * M) ≤ flowCoordinateBound M ∧
(2 * M + 8 * M^2) * Real.exp (6 * M) ≤ flowCoordinateBound M := by
have h1 : 0 ≤ Real.exp (2 * M) := (Real.exp_pos _).le
have h2 : 0 ≤ 2 * M := by positivity
have h3 : 0 ≤ M * Real.exp (2 * M) := by positivity
have h4 : 0 ≤ 2 * M * Real.exp (4 * M) := by positivity
have h5 : 0 ≤ 4 * M + 2 * M^2 := by positivity
have h6 : 0 ≤ (2 * M + M^2) * Real.exp (2 * M) := by positivity
have h7 : 0 ≤ (M + 2 * M^2) * Real.exp (4 * M) := by positivity
have h8 : 0 ≤ (2 * M + 8 * M^2) * Real.exp (6 * M) := by positivity
dsimp [flowCoordinateBound]
refine ⟨?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_, ?_⟩ <;> linarith
theorem flowCoordinateBound_nonneg {M : ℝ} (hM : 0 ≤ M) : 0 ≤ flowCoordinateBound M :=
(by norm_num : (0 : ℝ) ≤ 3).trans (flowCoordinateBound_dominates hM).1
theorem coordinateProjection_partial (i j : Fin 2) :
coordPartial i (fun p : Coord => p j) = (fun _ : Coord => (Pi.single i (1 : ℝ) : Coord) j) := by
funext p
unfold coordPartial
rw [(hasFDerivAt_apply (𝕜 := ℝ) j p).fderiv]
rfl
theorem coordinateProjection_bound (j : Fin 2) :
CoordinateBound (fun p : Coord => p j) capChartDomain 3 3 := by
intro ds hds p hp
cases ds with
| nil =>
have hh : |p j| < 2 := by
fin_cases j <;> simp only [Fin.mk_zero, Fin.mk_one]
· exact abs_lt.mpr hp.1
· exact abs_lt.mpr hp.2
exact hh.le.trans (by norm_num)
| cons i ds =>
cases ds with
| nil =>
change |coordPartial i (fun p : Coord => p j) p| ≤ 3
rw [coordinateProjection_partial]
fin_cases i <;> fin_cases j <;> norm_num [Pi.single_apply]
| cons k ds =>
cases ds with
| nil =>
change |coordPartial i (coordPartial k (fun p : Coord => p j)) p| ≤ 3
rw [coordinateProjection_partial]
simp [coordPartial]
| cons l ds =>
cases ds with
| nil =>
change |coordPartial i (coordPartial k (coordPartial l (fun p : Coord => p j))) p| ≤ 3
rw [coordinateProjection_partial]
have hc : coordPartial k (fun _ : Coord => (Pi.single l (1 : ℝ) : Coord) j) =
(fun _ : Coord => 0) := by
funext x
simp [coordPartial]
rw [hc]
simp [coordPartial]
| cons m ds => simp only [List.length_cons] at hds; omega
section ActualFlow
variable {q : Coord → ℝ} {U : Set Coord} {Y : ℝ → ℝ → ℝ} {M : 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)
variable (hM : 0 ≤ M)
variable (hq0 : ∀ p ∈ modelSquare, |q p| ≤ (1 : ℝ) / 100)
variable (hq1 : ∀ i : Fin 2, ∀ p ∈ modelSquare, |coordPartial i q p| ≤ M)
variable (hq2 : ∀ i j : Fin 2, ∀ p ∈ modelSquare, |coordPartial i (coordPartial j q) p| ≤ M)
variable (hq3 : ∀ i j k : Fin 2, ∀ p ∈ modelSquare, |coordPartial i (coordPartial j (coordPartial k q)) p| ≤ M)
include hq hU hSU hY hrange hstart hode hM hq0 hq1 hq2 hq3
omit hq2 hq3 in
theorem capFlowHeight_word_one_bound
(_ : ∀ first second : Fin 2, ∀ point ∈ modelSquare,
|coordPartial first (coordPartial second q) point| ≤ M)
(_ : ∀ first second third : Fin 2, ∀ point ∈ modelSquare,
|coordPartial first (coordPartial second (coordPartial third q)) point| ≤ M)
{p : Coord} (hp : p ∈ capChartDomain) (i : Fin 2) :
|iteratedCoordPartial [i] (capFlowHeight Y) p| ≤ flowCoordinateBound M := by
have hs := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
obtain ⟨_, h1, hV, _⟩ := flowCoordinateBound_dominates hM
fin_cases i <;> simp only [Fin.mk_zero, Fin.mk_one]
· rw [show iteratedCoordPartial [0] (capFlowHeight Y) p = timeFlowDerivative Y (p 0, p 1) from
capFlowHeight_partial_zero_eq hs hp]
exact (cap_flow_time_bound hq hU hSU hY hrange hstart hode hq0 hp.2 hp.1).trans
((by norm_num : (1 : ℝ) / 100 ≤ 1).trans h1)
· rw [show iteratedCoordPartial [1] (capFlowHeight Y) p = initialFlowDerivative Y (p 0, p 1) from
capFlowHeight_partial_one_eq hs hp, initialFlowDerivative,
abs_of_pos (cap_flow_initial_deriv_pos hq hU hSU hY hrange hstart hode hp.2 hp.1)]
exact (cap_flow_initial_deriv_bounds hq hU hSU hY hrange hstart hode hM (hq1 1) hp.2 hp.1).2.trans hV
omit hq3 in
theorem capFlowHeight_word_two_bound
(_ : ∀ first second third : Fin 2, ∀ point ∈ modelSquare,
|coordPartial first (coordPartial second (coordPartial third q)) point| ≤ M)
{p : Coord} (hp : p ∈ capChartDomain) (i j : Fin 2) :
|iteratedCoordPartial [i, j] (capFlowHeight Y) p| ≤ flowCoordinateBound M := by
have hs := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
have hf := capFlowHeight_contDiffOn hs
obtain ⟨_, _, _, htt, hts, hss, _⟩ := flowCoordinateBound_dominates hM
have hqweak (p : Coord) (hp : p ∈ modelSquare) : |q p| ≤ 1 :=
(hq0 p hp).trans (by norm_num)
have h00 : |iteratedCoordPartial [0, 0] (capFlowHeight Y) p| ≤ flowCoordinateBound M := by
rw [capFlowHeight_word_00 hs hp]
exact (cap_flow_time_second_bound hq hU hSU hY hrange hstart hode hM hqweak hq1 hp.2 hp.1).trans htt
have h10 : |iteratedCoordPartial [1, 0] (capFlowHeight Y) p| ≤ flowCoordinateBound M := by
rw [capFlowHeight_word_10 hs hp]
exact (cap_flow_mixed_bound hq hU hSU hY hrange hstart hode hM hq1 hp.2 hp.1).trans hts
have h11 : |iteratedCoordPartial [1, 1] (capFlowHeight Y) p| ≤ flowCoordinateBound M := by
rw [capFlowHeight_word_11 hs hp]
exact (cap_flow_initial_second_bound hq hU hSU hY hrange hstart hode hM (hq1 1) (hq2 1 1) hp.2 hp.1).trans hss
fin_cases i <;> fin_cases j <;> simp only [Fin.mk_zero, Fin.mk_one]
· exact h00
· have he : iteratedCoordPartial [0, 1] (capFlowHeight Y) p = iteratedCoordPartial [1, 0] (capFlowHeight Y) p :=
coordPartial_comm hf capChartDomain_isOpen hp 0 1
rw [he]; exact h10
· exact h10
· exact h11
theorem capFlowHeight_word_three_bound {p : Coord} (hp : p ∈ capChartDomain) (i j k : Fin 2) :
|iteratedCoordPartial [i, j, k] (capFlowHeight Y) p| ≤ flowCoordinateBound M := by
have hs := cap_flow_joint_contDiffOn hq hU hSU hY hrange hstart hode
have hf := capFlowHeight_contDiffOn hs
obtain ⟨_, _, _, _, _, _, httt, htts, htss, hsss⟩ := flowCoordinateBound_dominates hM
have hqweak (p : Coord) (hp : p ∈ modelSquare) : |q p| ≤ 1 :=
(hq0 p hp).trans (by norm_num)
have h000 : |iteratedCoordPartial [0, 0, 0] (capFlowHeight Y) p| ≤ flowCoordinateBound M := by
rw [capFlowHeight_word_000 hs hp]
exact (cap_flow_time_third_bound hq hU hSU hY hrange hstart hode hM hqweak hq1 hq2 hp.2 hp.1).trans httt
have h100 : |iteratedCoordPartial [1, 0, 0] (capFlowHeight Y) p| ≤ flowCoordinateBound M := by
rw [capFlowHeight_word_100 hs hp]
exact (cap_flow_time_second_initial_bound hq hU hSU hY hrange hstart hode hM hqweak hq1 hq2 hp.2 hp.1).trans htts
have h110 : |iteratedCoordPartial [1, 1, 0] (capFlowHeight Y) p| ≤ flowCoordinateBound M := by
rw [capFlowHeight_word_110 hs hp]
exact (cap_flow_mixed_initial_bound hq hU hSU hY hrange hstart hode hM hq1 hq2 hp.2 hp.1).trans htss
have h111 : |iteratedCoordPartial [1, 1, 1] (capFlowHeight Y) p| ≤ flowCoordinateBound M := by
rw [capFlowHeight_word_111 hs hp]
exact (cap_flow_initial_third_bound hq hU hSU hY hrange hstart hode hM (hq1 1) (hq2 1 1) (hq3 1 1 1) hp.2 hp.1).trans hsss
fin_cases i <;> fin_cases j <;> fin_cases k <;> simp only [Fin.mk_zero, Fin.mk_one]
· exact h000
· rw [iterated_three_swap_inner hf capChartDomain_isOpen hp 0 0 1,
iterated_three_swap_outer hf capChartDomain_isOpen hp 0 1 0]
exact h100
· rw [iterated_three_swap_outer hf capChartDomain_isOpen hp 0 1 0]
exact h100
· rw [iterated_three_swap_outer hf capChartDomain_isOpen hp 0 1 1,
iterated_three_swap_inner hf capChartDomain_isOpen hp 1 0 1]
exact h110
· exact h100
· rw [iterated_three_swap_inner hf capChartDomain_isOpen hp 1 0 1]
exact h110
· exact h110
· exact h111
theorem capFlowHeight_coordinate_bound :
CoordinateBound (capFlowHeight Y) capChartDomain 3 (flowCoordinateBound M) := by
intro ds hds p hp
cases ds with
| nil =>
have hh : |capFlowHeight Y p| ≤ 3 := abs_le.mpr
(hrange (p 1) ⟨hp.2.1.le, hp.2.2.le⟩ (p 0) ⟨hp.1.1.le, hp.1.2.le⟩)
exact hh.trans (flowCoordinateBound_dominates hM).1
| cons i ds =>
cases ds with
| nil => exact capFlowHeight_word_one_bound hq hU hSU hY hrange hstart hode hM hq0 hq1 hq2 hq3 hp i
| cons j ds =>
cases ds with
| nil => exact capFlowHeight_word_two_bound hq hU hSU hY hrange hstart hode hM hq0 hq1 hq2 hq3 hp i j
| cons k ds =>
cases ds with
| nil => exact capFlowHeight_word_three_bound hq hU hSU hY hrange hstart hode hM hq0 hq1 hq2 hq3 hp i j k
| cons l ds => simp only [List.length_cons] at hds; omega
theorem capChart_coefficient_coordinate_bound (j : Fin 2) :
CoordinateBound (fun p => capChart Y p j) capChartDomain 3 (flowCoordinateBound M) := by
fin_cases j <;> simp only [Fin.mk_zero, Fin.mk_one]
· have he : (fun p => capChart Y p 0) = (fun p : Coord => p 0) := funext (capChart_zero Y)
rw [he]
exact (coordinateProjection_bound 0).mono le_rfl (flowCoordinateBound_dominates hM).1
· have he : (fun p => capChart Y p 1) = capFlowHeight Y := funext (capChart_one Y)
rw [he]
exact capFlowHeight_coordinate_bound hq hU hSU hY hrange hstart hode hM hq0 hq1 hq2 hq3
end ActualFlow
end SmoothLocal.Flow
end
end OAI
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