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695bd70 | 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 | import OAI.Geometry.Relativity.CKS.CKSSourceMass
namespace OAI
noncomputable section
namespace CKSAngularSlice
noncomputable section
open CKSCalculus Set Filter
open scoped Topology ContDiff NNReal Matrix.Norms.Elementwise
abbrev AP := CKSAngularGeometry.Point
abbrev MP := CKSMixedGeometry.Point
abbrev A := Fin 2
def injection : AP →L[ℝ] MP :=
ContinuousLinearMap.pi (fun i => Fin.cases 0 (fun a => ContinuousLinearMap.proj a) i)
def slice (r : ℝ) (x : AP) : MP := Pi.single 0 r+injection x
@[simp] lemma injection_zero (x : AP) : injection x 0=0 := rfl
@[simp] lemma injection_succ (x : AP) (a : A) : injection x a.succ=x a := rfl
@[simp] lemma slice_zero (r : ℝ) (x : AP) : slice r x 0=r := by simp [slice]
@[simp] lemma slice_succ (r : ℝ) (x : AP) (a : A) : slice r x a.succ=x a := by simp [slice]
lemma injection_basis (a : A) : injection (CKSAngularGeometry.basis a)=CKSMixedGeometry.basis a.succ := by
funext i
refine Fin.cases ?_ (fun k => ?_) i
· simp [CKSMixedGeometry.basis]
· simp [CKSAngularGeometry.basis,CKSMixedGeometry.basis,Pi.single_apply]
lemma slice_hasFDerivAt (r : ℝ) (x : AP) : HasFDerivAt (slice r) injection x :=
injection.hasFDerivAt.const_add _
lemma slice_smooth (r : ℝ) : ContDiff ℝ ∞ (slice r) :=
contDiff_const.add injection.contDiff
lemma D_slice (a : A) (r : ℝ) {f : MP → ℝ} {x : AP}
(hf : DifferentiableAt ℝ f (slice r x)) :
D (CKSAngularGeometry.basis a) (fun y => f (slice r y)) x=
D (CKSMixedGeometry.basis a.succ) f (slice r x) := by
unfold D
erw [fderiv_comp x hf (slice_hasFDerivAt r x).differentiableAt,
(slice_hasFDerivAt r x).fderiv,ContinuousLinearMap.comp_apply,injection_basis]
def restrictJet (p : CKSMixedGeometry.ScalarJet) : CKSAngularGeometry.ScalarJet :=
(p.1,fun a => p.2.1 a.succ,fun a b => p.2.2 a.succ b.succ)
def restrictThree (p : CKSMixedGeometry.ScalarThreeJet) : CKSAngularGeometry.ScalarThreeJet :=
(restrictJet p.1,fun a => restrictJet (p.2 a.succ))
lemma actualScalarJet_slice (r : ℝ) {f : MP → ℝ} {x : AP}
(hf : ContDiffAt ℝ 2 f (slice r x)) :
CKSAngularGeometry.actualScalarJet (fun y => f (slice r y)) x=
restrictJet (CKSMixedGeometry.actualScalarJet f (slice r x)) := by
apply Prod.ext
· rfl
apply Prod.ext
· funext a; exact D_slice a r (hf.differentiableAt (by norm_num))
· funext a b
have he : D (CKSAngularGeometry.basis b) (fun y => f (slice r y)) =ᶠ[𝓝 x]
(fun y => D (CKSMixedGeometry.basis b.succ) f (slice r y)) := by
have hn := (slice_smooth r).continuous.continuousAt.tendsto.eventually (hf.eventually (by simp))
filter_upwards [hn] with y hy
exact D_slice b r (hy.differentiableAt (by norm_num))
change D (CKSAngularGeometry.basis a) (D (CKSAngularGeometry.basis b) (fun y => f (slice r y))) x=_
rw [D_congr he]
exact D_slice a r ((contDiffAt_D hf (m:=1) (by norm_num) _).differentiableAt (by norm_num))
lemma actualThreeJet_slice (r : ℝ) {f : MP → ℝ} {x : AP}
(hf : ContDiffAt ℝ 3 f (slice r x)) :
CKSAngularGeometry.actualThreeJet (fun y => f (slice r y)) x=
restrictThree (CKSMixedGeometry.actualThreeJet f (slice r x)) := by
apply Prod.ext (actualScalarJet_slice r (hf.of_le (by norm_num)))
funext a
have he : D (CKSAngularGeometry.basis a) (fun y => f (slice r y)) =ᶠ[𝓝 x]
(fun y => D (CKSMixedGeometry.basis a.succ) f (slice r y)) := by
have hn := (slice_smooth r).continuous.continuousAt.tendsto.eventually (hf.eventually (by simp))
filter_upwards [hn] with y hy
exact D_slice a r (hy.differentiableAt (by norm_num))
change CKSAngularGeometry.actualScalarJet (D (CKSAngularGeometry.basis a) (fun y => f (slice r y))) x=_
rw [CKSAngularGeometry.actualScalarJet_congr he]
exact actualScalarJet_slice r (contDiffAt_D hf (m:=2) (by norm_num) _)
lemma restrictJet_norm (p : CKSMixedGeometry.ScalarJet) : ‖restrictJet p‖ ≤ ‖p‖ := by
apply norm_prod_le_iff.mpr
refine ⟨norm_fst_le p, norm_prod_le_iff.mpr ⟨?_,?_⟩⟩
· apply (pi_norm_le_iff_of_nonneg (norm_nonneg p)).mpr
intro a
exact (norm_le_pi_norm p.2.1 a.succ).trans ((norm_fst_le p.2).trans (norm_snd_le p))
· apply (pi_norm_le_iff_of_nonneg (norm_nonneg p)).mpr
intro a
apply (pi_norm_le_iff_of_nonneg (norm_nonneg p)).mpr
intro b
exact ((norm_le_pi_norm (p.2.2 a.succ) b.succ).trans (norm_le_pi_norm p.2.2 a.succ)).trans
((norm_snd_le p.2).trans (norm_snd_le p))
lemma restrictThree_norm (p : CKSMixedGeometry.ScalarThreeJet) : ‖restrictThree p‖ ≤ ‖p‖ := by
apply norm_prod_le_iff.mpr
refine ⟨(restrictJet_norm p.1).trans (norm_fst_le p),?_⟩
apply (pi_norm_le_iff_of_nonneg (norm_nonneg p)).mpr
intro a
exact (restrictJet_norm (p.2 a.succ)).trans ((norm_le_pi_norm p.2 a.succ).trans (norm_snd_le p))
lemma angular_twojet_le_mixed (r : ℝ) {f : MP → ℝ} {x : AP}
(hf : ContDiffAt ℝ 2 f (slice r x)) :
‖CKSAngularGeometry.actualScalarJet (fun y => f (slice r y)) x‖ ≤
‖CKSMixedGeometry.actualScalarJet f (slice r x)‖ := by
rw [actualScalarJet_slice r hf]; exact restrictJet_norm _
lemma angular_threejet_le_mixed (r : ℝ) {f : MP → ℝ} {x : AP}
(hf : ContDiffAt ℝ 3 f (slice r x)) :
‖CKSAngularGeometry.actualThreeJet (fun y => f (slice r y)) x‖ ≤
‖CKSMixedGeometry.actualThreeJet f (slice r x)‖ := by
rw [actualThreeJet_slice r hf]; exact restrictThree_norm _
end
end CKSAngularSlice
end
end OAI
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