| import OAI.Geometry.IsometricImmersion.Caps.PulseReflection |
| import OAI.Geometry.IsometricImmersion.Taylor.TaylorCauchyData |
| import OAI.Geometry.IsometricImmersion.Caps.UpperCapWords |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function |
| open scoped ContDiff Topology |
|
|
| namespace SmoothLocal.Taylor |
| open SmoothLocal.Geometry SmoothLocal.Flow.Reflection SmoothLocal.Pulse |
|
|
| theorem reflectPoint_initialPoint (x a : ℝ) : |
| reflectPoint (![x,a] : Coord) = ![x,-a] := by |
| ext i |
| fin_cases i <;> simp |
|
|
| theorem coordPartial_one_reflectedScalar (f : Coord → ℝ) (p : Coord) : |
| coordPartial 1 (reflectedScalar f) p = -coordPartial 1 f (reflectPoint p) := by |
| have he : fderiv ℝ (f ∘ reflectPoint) p = |
| (fderiv ℝ f (reflectPoint p)).comp |
| (reflectionIsometry.toContinuousLinearEquiv : Coord →L[ℝ] Coord) := |
| reflectionIsometry.toContinuousLinearEquiv.comp_right_fderiv (f := f) (x := p) |
| change fderiv ℝ (f ∘ reflectPoint) p (Pi.single 1 (1 : ℝ)) = |
| -fderiv ℝ f (reflectPoint p) (Pi.single 1 (1 : ℝ)) |
| rw [he, ContinuousLinearMap.comp_apply] |
| have hb : reflectionIsometry.toContinuousLinearEquiv (Pi.single 1 (1 : ℝ)) = |
| -(Pi.single 1 (1 : ℝ) : Coord) := by |
| ext i |
| fin_cases i <;> simp [reflectionIsometry,reflectPoint] |
| change (fderiv ℝ f (reflectPoint p)) (reflectionIsometry (Pi.single 1 1)) = |
| -(fderiv ℝ f (reflectPoint p)) (Pi.single 1 1) |
| change reflectionIsometry (Pi.single 1 (1 : ℝ)) = -(Pi.single 1 (1 : ℝ) : Coord) at hb |
| rw [hb,map_neg] |
|
|
| theorem heightCauchyValue_reflected (z : Coord → ℝ) (a : ℝ) : |
| heightCauchyValue (reflectedScalar z) a = heightCauchyValue z (-a) := by |
| funext x |
| change z (reflectPoint ![x,a]) = z ![x,-a] |
| rw [reflectPoint_initialPoint] |
|
|
| theorem heightCauchyVelocity_reflected (z : Coord → ℝ) (a : ℝ) : |
| heightCauchyVelocity (reflectedScalar z) a = -heightCauchyVelocity z (-a) := by |
| funext x |
| change coordPartial 1 (reflectedScalar z) ![x,a] = -coordPartial 1 z ![x,-a] |
| rw [coordPartial_one_reflectedScalar,reflectPoint_initialPoint] |
|
|
| theorem reflected_Cauchy_jet_norms (z : Coord → ℝ) (a x : ℝ) (n : ℕ) : |
| ‖iteratedFDeriv ℝ n (heightCauchyValue (reflectedScalar z) a) x‖ = |
| ‖iteratedFDeriv ℝ n (heightCauchyValue z (-a)) x‖ ∧ |
| ‖iteratedFDeriv ℝ n (heightCauchyVelocity (reflectedScalar z) a) x‖ = |
| ‖iteratedFDeriv ℝ n (heightCauchyVelocity z (-a)) x‖ := by |
| simp [heightCauchyValue_reflected,heightCauchyVelocity_reflected, |
| iteratedFDeriv_neg_apply] |
|
|
| theorem sheared_reflected_Cauchy_jet_norms (z : Coord → ℝ) (q0 a x : ℝ) (n : ℕ) : |
| ‖iteratedFDeriv ℝ n (heightCauchyValue |
| (heightInShearCoordinates (reflectedScalar z) (-q0)) a) x‖ = |
| ‖iteratedFDeriv ℝ n (heightCauchyValue (heightInShearCoordinates z q0) (-a)) x‖ ∧ |
| ‖iteratedFDeriv ℝ n (heightCauchyVelocity |
| (heightInShearCoordinates (reflectedScalar z) (-q0)) a) x‖ = |
| ‖iteratedFDeriv ℝ n (heightCauchyVelocity (heightInShearCoordinates z q0) (-a)) x‖ := by |
| rw [height_shear_reflection] |
| exact reflected_Cauchy_jet_norms _ a x n |
|
|
| end SmoothLocal.Taylor |
|
|
| end |
|
|
| end OAI |
|
|
