File size: 3,187 Bytes
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 | 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
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