| 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 |
|
|
