| import OAI.Geometry.IsometricImmersion.Darboux.QActualSegmentTransfer |
| import OAI.Geometry.IsometricImmersion.Energy.MultiplierIntegral |
| import Mathlib.Analysis.Calculus.MeanValue |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.HighEquation |
| open SmoothLocal.Geometry SmoothLocal.Weighted |
|
|
| theorem horizontal_partial_eq_of_slice_eq {f f0 : Coord → ℝ} {U : Set Coord} |
| (hf : ContDiffOn ℝ ∞ f U) (hf0 : ContDiffOn ℝ ∞ f0 U) (hU : IsOpen U) |
| {left right a : ℝ} |
| (hline : ∀ x ∈ Ioo left right, boxPoint x a ∈ U) |
| (heq : ∀ x ∈ Ioo left right, f0 (boxPoint x a) = f (boxPoint x a)) |
| {x : ℝ} (hx : x ∈ Ioo left right) : |
| coordPartial 0 f0 (boxPoint x a) = coordPartial 0 f (boxPoint x a) := by |
| have hd : HasDerivAt (fun t => f (boxPoint t a)) (coordPartial 0 f (boxPoint x a)) x := |
| ((hf.contDiffAt (hU.mem_nhds (hline x hx))).differentiableAt (by simp)).hasFDerivAt.comp_hasDerivAt |
| x (boxPoint_hasDerivAt_t x a) |
| have hd0 : HasDerivAt (fun t => f0 (boxPoint t a)) (coordPartial 0 f0 (boxPoint x a)) x := |
| ((hf0.contDiffAt (hU.mem_nhds (hline x hx))).differentiableAt (by simp)).hasFDerivAt.comp_hasDerivAt |
| x (boxPoint_hasDerivAt_t x a) |
| have hev : (fun t => f0 (boxPoint t a)) =ᶠ[𝓝 x] (fun t => f (boxPoint t a)) := by |
| filter_upwards [isOpen_Ioo.mem_nhds hx] with t ht |
| exact heq t ht |
| rw [← hd0.deriv, ← hd.deriv] |
| exact hev.deriv_eq |
|
|
| theorem qSolutionJet_eq_of_cauchy_data {z z0 : Coord → ℝ} {U : Set Coord} |
| (hz : ContDiffOn ℝ ∞ z U) (hz0 : ContDiffOn ℝ ∞ z0 U) (hU : IsOpen U) |
| {left right a : ℝ} |
| (hline : ∀ x ∈ Ioo left right, boxPoint x a ∈ U) |
| (hvalue : ∀ x ∈ Ioo left right, z0 (boxPoint x a) = z (boxPoint x a)) |
| (hvelocity : ∀ x ∈ Ioo left right, |
| coordPartial 1 z0 (boxPoint x a) = coordPartial 1 z (boxPoint x a)) |
| {x : ℝ} (hx : x ∈ Ioo left right) : |
| qSolutionJet z0 (boxPoint x a) = qSolutionJet z (boxPoint x a) := by |
| have hxfirst : ∀ x ∈ Ioo left right, |
| coordPartial 0 z0 (boxPoint x a) = coordPartial 0 z (boxPoint x a) := |
| fun x hx => horizontal_partial_eq_of_slice_eq hz hz0 hU hline hvalue hx |
| have hxx := horizontal_partial_eq_of_slice_eq (partial_contDiffOn hz hU 0) |
| (partial_contDiffOn hz0 hU 0) hU hline hxfirst hx |
| have hxy := horizontal_partial_eq_of_slice_eq (partial_contDiffOn hz hU 1) |
| (partial_contDiffOn hz0 hU 1) hU hline hvelocity hx |
| ext i |
| fin_cases i |
| · rfl |
| · rfl |
| · exact hxfirst x hx |
| · exact hvelocity x hx |
| · exact hxy |
| · exact hxx |
|
|
| theorem coordinate_word_vertical_variation {f : Coord → ℝ} {U : Set Coord} |
| (hf : ContDiffOn ℝ ∞ f U) (hU : IsOpen U) |
| {left right a b x t : ℝ} |
| (hbox : closedRectangle left right a b ⊆ U) |
| (hx : x ∈ Icc left right) (hab : a ≤ b) (ht : t ∈ Icc a b) |
| {M : ℝ} (hB : CoordinateBound f (closedRectangle left right a b) 3 M) |
| (ds : List (Fin 2)) (hds : ds.length ≤ 2) : |
| |iteratedCoordPartial ds f (boxPoint x t) - |
| iteratedCoordPartial ds f (boxPoint x a)| ≤ M * |t - a| := by |
| have hd (s : ℝ) (hs : s ∈ Icc a b) : |
| HasDerivAt (fun y => iteratedCoordPartial ds f (boxPoint x y)) |
| (coordPartial 1 (iteratedCoordPartial ds f) (boxPoint x s)) s := |
| (LowQuotient.iterated_differentiableAt hf hU (hbox (boxPoint_mem hx hs)) ds).hasFDerivAt.comp_hasDerivAt |
| s (boxPoint_hasDerivAt_s x s) |
| have hb (s : ℝ) (hs : s ∈ Icc a b) : |
| ‖deriv (fun y => iteratedCoordPartial ds f (boxPoint x y)) s‖ ≤ M := by |
| rw [(hd s hs).deriv, Real.norm_eq_abs] |
| exact hB (1 :: ds) (Nat.succ_le_succ hds) _ (boxPoint_mem hx hs) |
| simpa only [Real.norm_eq_abs] using |
| (convex_Icc a b).norm_image_sub_le_of_norm_deriv_le |
| (fun s hs => (hd s hs).differentiableAt) hb (show a ∈ Icc a b from ⟨le_rfl, hab⟩) ht |
|
|
| theorem coordinate_word_difference_on_strip {z z0 : Coord → ℝ} {U : Set Coord} |
| (hz : ContDiffOn ℝ ∞ z U) (hz0 : ContDiffOn ℝ ∞ z0 U) (hU : IsOpen U) |
| {left right a b x t : ℝ} |
| (hbox : closedRectangle left right a b ⊆ U) |
| (hx : x ∈ Icc left right) (hab : a ≤ b) (ht : t ∈ Icc a b) |
| {M M0 : ℝ} (hB : CoordinateBound z (closedRectangle left right a b) 3 M) |
| (hB0 : CoordinateBound z0 (closedRectangle left right a b) 3 M0) |
| (ds : List (Fin 2)) (hds : ds.length ≤ 2) |
| (hinitial : iteratedCoordPartial ds z0 (boxPoint x a) = |
| iteratedCoordPartial ds z (boxPoint x a)) : |
| |iteratedCoordPartial ds z0 (boxPoint x t) - iteratedCoordPartial ds z (boxPoint x t)| ≤ |
| (M + M0) * |t - a| := by |
| have h0 := coordinate_word_vertical_variation hz0 hU hbox hx hab ht hB0 ds hds |
| have h1 := coordinate_word_vertical_variation hz hU hbox hx hab ht hB ds hds |
| calc |
| _ = |(iteratedCoordPartial ds z0 (boxPoint x t) - iteratedCoordPartial ds z0 (boxPoint x a)) - |
| (iteratedCoordPartial ds z (boxPoint x t) - iteratedCoordPartial ds z (boxPoint x a))| := by |
| rw [hinitial] |
| congr 1 |
| ring |
| _ ≤ |iteratedCoordPartial ds z0 (boxPoint x t) - iteratedCoordPartial ds z0 (boxPoint x a)| + |
| |iteratedCoordPartial ds z (boxPoint x t) - iteratedCoordPartial ds z (boxPoint x a)| := abs_sub _ _ |
| _ ≤ M0 * |t - a| + M * |t - a| := add_le_add h0 h1 |
| _ = (M + M0) * |t - a| := by ring |
|
|
| theorem qSolutionJet_distance_on_same_strip {z z0 : Coord → ℝ} {U : Set Coord} |
| (hz : ContDiffOn ℝ ∞ z U) (hz0 : ContDiffOn ℝ ∞ z0 U) (hU : IsOpen U) |
| {left right a b x t : ℝ} |
| (hbox : closedRectangle left right a b ⊆ U) |
| (hx : x ∈ Ioo left right) (hab : a ≤ b) (ht : t ∈ Icc a b) |
| {M M0 : ℝ} (hM : 0 ≤ M) (hM0 : 0 ≤ M0) |
| (hB : CoordinateBound z (closedRectangle left right a b) 3 M) |
| (hB0 : CoordinateBound z0 (closedRectangle left right a b) 3 M0) |
| (hvalue : ∀ x ∈ Ioo left right, z0 (boxPoint x a) = z (boxPoint x a)) |
| (hvelocity : ∀ x ∈ Ioo left right, |
| coordPartial 1 z0 (boxPoint x a) = coordPartial 1 z (boxPoint x a)) : |
| ‖qSolutionJet z0 (boxPoint x t) - qSolutionJet z (boxPoint x t)‖ ≤ |
| (M + M0) * (b - a) := by |
| have hx' : x ∈ Icc left right := ⟨hx.1.le, hx.2.le⟩ |
| have hinitial := qSolutionJet_eq_of_cauchy_data hz hz0 hU |
| (fun y hy => hbox (boxPoint_mem ⟨hy.1.le, hy.2.le⟩ ⟨le_rfl, hab⟩)) hvalue hvelocity hx |
| have hword (ds : List (Fin 2)) (hds : ds.length ≤ 2) |
| (he : iteratedCoordPartial ds z0 (boxPoint x a) = iteratedCoordPartial ds z (boxPoint x a)) : |
| |iteratedCoordPartial ds z0 (boxPoint x t) - iteratedCoordPartial ds z (boxPoint x t)| ≤ |
| (M + M0) * (b - a) := by |
| apply (coordinate_word_difference_on_strip hz hz0 hU hbox hx' hab ht hB hB0 ds hds he).trans |
| apply mul_le_mul_of_nonneg_left _ (add_nonneg hM hM0) |
| rw [abs_of_nonneg (sub_nonneg.mpr ht.1)] |
| exact sub_le_sub_right ht.2 a |
| apply state_distance_le_of_coordinate_error (mul_nonneg (add_nonneg hM hM0) (sub_nonneg.mpr hab)) |
| apply qSolutionJet_coordinate_error z z0 (boxPoint x t) |
| (mul_nonneg (add_nonneg hM hM0) (sub_nonneg.mpr hab)) |
| · intro i |
| fin_cases i |
| · exact hword [0] (by norm_num) (congrFun hinitial 2) |
| · exact hword [1] (by norm_num) (congrFun hinitial 3) |
| · exact hword [0, 1] (by norm_num) (congrFun hinitial 4) |
| · exact hword [0, 0] (by norm_num) (congrFun hinitial 5) |
|
|
| end SmoothLocal.HighEquation |
|
|
| end |
|
|
| end OAI |
|
|
