Datasets:

roseking
/

openAI-math-dataset

import OAI.Geometry.Relativity.CKS.LogRadialIdentity
import OAI.Geometry.Relativity.CKS.CKSScaledBounds

namespace OAI

noncomputable section
namespace CKSMixedGeometry
noncomputable section
open CKSCalculus Set Filter
open scoped Topology ContDiff NNReal Matrix.Norms.Elementwise

structure SourceMassFields where
  sigma : Angle → Mat
  mg : Angle → Mat
  mK : Angle → Mat
  mr : Angle → ℝ
  eg : Point → Mat
  ek : Point → Mat
  b : Point → Angle
  err : Point → ℝ

def SourceMassFields.logFields (f : SourceMassFields) : MassFields where
  sigma := angularLift f.sigma
  mg := angularLift f.mg
  mK := angularLift f.mK
  mr := angularLift f.mr
  eg := fun y => f.eg (logRadiusChart y)
  ek := fun y => f.ek (logRadiusChart y)
  b := fun y => f.b (logRadiusChart y)
  err := fun y => f.err (logRadiusChart y)
  er := 0

lemma angularProjection_logRadiusChart (x : Point) :
    angularProjection (logRadiusChart x) = angularProjection x := by
  ext a
  simp [angularProjection,logRadiusChart]

structure SourceMassFields.RegularAt (f : SourceMassFields) (y : Point) : Prop where
  sigma : ContDiffAt ℝ 3 f.sigma (angularProjection y)
  mg : ContDiffAt ℝ 3 f.mg (angularProjection y)
  mK : ContDiffAt ℝ 2 f.mK (angularProjection y)
  mr : ContDiffAt ℝ 2 f.mr (angularProjection y)
  eg : ContDiffAt ℝ 3 f.eg y
  ek : ContDiffAt ℝ 2 f.ek y
  b : ContDiffAt ℝ 3 f.b y
  err : ContDiffAt ℝ 2 f.err y

lemma source_logFields_regular {f : SourceMassFields} {x : Point}
    (hf : f.RegularAt (logRadiusChart x)) : f.logFields.RegularAt x := by
  have hσ := hf.sigma
  have hmg := hf.mg
  have hmk := hf.mK
  have hmr := hf.mr
  rw [angularProjection_logRadiusChart] at hσ hmg hmk hmr
  have hc3 := ((logRadiusChart_smooth.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 3)).contDiffAt (x := x))
  have hc2 := ((logRadiusChart_smooth.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)).contDiffAt (x := x))
  exact ⟨angularLift_diff hσ,angularLift_diff hmg,hf.eg.comp x hc3,contDiffAt_const,
    angularLift_diff hmk,hf.ek.comp x hc2,hf.b.comp x hc3,angularLift_diff hmr,hf.err.comp x hc2⟩

lemma source_logGammaRadial_actual {f : SourceMassFields} {x : Point}
    (hf : f.RegularAt (logRadiusChart x)) :
    logGammaRadial f.logFields x = (1/Real.exp (x 0)) • radialMatrixD (logGamma f.logFields) x := by
  have hσ := hf.sigma
  have hmg := hf.mg
  rw [angularProjection_logRadiusChart] at hσ hmg
  exact logGammaRadial_actual (source_logFields_regular hf)
    (radialMatrixD_angularLift (hσ.of_le (by norm_num)))
    (radialMatrixD_angularLift (hmg.of_le (by norm_num)))

def sourceGamma (f : SourceMassFields) : Point → Mat := fun y =>
  (y 0)^2 • f.sigma (angularProjection y)+(1/(y 0)) • f.mg (angularProjection y)+f.eg y

def sourceTangentialK (f : SourceMassFields) : Point → Mat := fun y =>
  (y 0)^2 • f.sigma (angularProjection y)+(1/(y 0)) • f.mK (angularProjection y)+f.ek y

def sourceRadialMetric (f : SourceMassFields) : Point → ℝ := fun y =>
  1/(1+(y 0)^2)+f.mr (angularProjection y)/(y 0)^5+f.err y

lemma sourceGamma_pullback (f : SourceMassFields) :
    (fun y => sourceGamma f (logRadiusChart y)) = logGamma f.logFields := by
  funext y
  simp [sourceGamma,logGamma,SourceMassFields.logFields,angularLift,
    angularProjection_logRadiusChart,logRadiusChart]

lemma sourceTangentialK_pullback (f : SourceMassFields) :
    (fun y => sourceTangentialK f (logRadiusChart y)) = logTangentialK f.logFields := by
  funext y
  simp [sourceTangentialK,logTangentialK,SourceMassFields.logFields,angularLift,
    angularProjection_logRadiusChart,logRadiusChart]

lemma sourceRadialMetric_pullback (f : SourceMassFields) :
    (fun y => sourceRadialMetric f (logRadiusChart y)) = logRadialMetric f.logFields := by
  funext y
  simp [sourceRadialMetric,logRadialMetric,SourceMassFields.logFields,angularLift,
    angularProjection_logRadiusChart,logRadiusChart]

structure SourceMassFields.ComponentBounds (f : SourceMassFields) (B : ℝ) (y : Point) : Prop where
  eg : ∀ i k, ScaledComponentBound (fun z => f.eg z i k) 3 2 B y
  ek : ∀ i k, ScaledComponentBound (fun z => f.ek z i k) 2 2 B y
  b : ∀ a, ScaledComponentBound (fun z => f.b z a) 3 3 B y
  err : ScaledComponentBound f.err 2 6 B y

lemma source_log_remainder_bounds {f : SourceMassFields} {x : Point} {B : ℝ} (hB : 0 ≤ B)
    (hf : f.RegularAt (logRadiusChart x)) (hb : f.ComponentBounds B (logRadiusChart x)) :
    ‖matrixThreeJets f.logFields.eg x‖ ≤ B/Real.exp (x 0)^2 ∧
    ‖matrixScalarJets f.logFields.ek x‖ ≤ B/Real.exp (x 0)^2 ∧
    ‖fun a => actualThreeJet (fun y => f.logFields.b y a) x‖ ≤ B/Real.exp (x 0)^3 ∧
    ‖actualScalarJet f.logFields.err x‖ ≤ B/Real.exp (x 0)^6 := by
  refine ⟨source_matrix_three_bound hB hf.eg hb.eg,source_matrix_two_bound hB hf.ek hb.ek,?_,
    source_two_jet_bound hB hf.err hb.err⟩
  apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr
  intro a
  exact source_three_jet_bound hB (contDiffAt_pi.mp hf.b a) (hb.b a)

end
end CKSMixedGeometry

end

end OAI

	import OAI.Geometry.Relativity.CKS.LogRadialIdentity
	import OAI.Geometry.Relativity.CKS.CKSScaledBounds

	namespace OAI

	noncomputable section
	namespace CKSMixedGeometry
	noncomputable section
	open CKSCalculus Set Filter
	open scoped Topology ContDiff NNReal Matrix.Norms.Elementwise

	structure SourceMassFields where
	sigma : Angle → Mat
	mg : Angle → Mat
	mK : Angle → Mat
	mr : Angle → ℝ
	eg : Point → Mat
	ek : Point → Mat
	b : Point → Angle
	err : Point → ℝ

	def SourceMassFields.logFields (f : SourceMassFields) : MassFields where
	sigma := angularLift f.sigma
	mg := angularLift f.mg
	mK := angularLift f.mK
	mr := angularLift f.mr
	eg := fun y => f.eg (logRadiusChart y)
	ek := fun y => f.ek (logRadiusChart y)
	b := fun y => f.b (logRadiusChart y)
	err := fun y => f.err (logRadiusChart y)
	er := 0

	lemma angularProjection_logRadiusChart (x : Point) :
	angularProjection (logRadiusChart x) = angularProjection x := by
	ext a
	simp [angularProjection,logRadiusChart]

	structure SourceMassFields.RegularAt (f : SourceMassFields) (y : Point) : Prop where
	sigma : ContDiffAt ℝ 3 f.sigma (angularProjection y)
	mg : ContDiffAt ℝ 3 f.mg (angularProjection y)
	mK : ContDiffAt ℝ 2 f.mK (angularProjection y)
	mr : ContDiffAt ℝ 2 f.mr (angularProjection y)
	eg : ContDiffAt ℝ 3 f.eg y
	ek : ContDiffAt ℝ 2 f.ek y
	b : ContDiffAt ℝ 3 f.b y
	err : ContDiffAt ℝ 2 f.err y

	lemma source_logFields_regular {f : SourceMassFields} {x : Point}
	(hf : f.RegularAt (logRadiusChart x)) : f.logFields.RegularAt x := by
	have hσ := hf.sigma
	have hmg := hf.mg
	have hmk := hf.mK
	have hmr := hf.mr
	rw [angularProjection_logRadiusChart] at hσ hmg hmk hmr
	have hc3 := ((logRadiusChart_smooth.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 3)).contDiffAt (x := x))
	have hc2 := ((logRadiusChart_smooth.of_le (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)).contDiffAt (x := x))
	exact ⟨angularLift_diff hσ,angularLift_diff hmg,hf.eg.comp x hc3,contDiffAt_const,
	angularLift_diff hmk,hf.ek.comp x hc2,hf.b.comp x hc3,angularLift_diff hmr,hf.err.comp x hc2⟩

	lemma source_logGammaRadial_actual {f : SourceMassFields} {x : Point}
	(hf : f.RegularAt (logRadiusChart x)) :
	logGammaRadial f.logFields x = (1/Real.exp (x 0)) • radialMatrixD (logGamma f.logFields) x := by
	have hσ := hf.sigma
	have hmg := hf.mg
	rw [angularProjection_logRadiusChart] at hσ hmg
	exact logGammaRadial_actual (source_logFields_regular hf)
	(radialMatrixD_angularLift (hσ.of_le (by norm_num)))
	(radialMatrixD_angularLift (hmg.of_le (by norm_num)))

	def sourceGamma (f : SourceMassFields) : Point → Mat := fun y =>
	(y 0)^2 • f.sigma (angularProjection y)+(1/(y 0)) • f.mg (angularProjection y)+f.eg y

	def sourceTangentialK (f : SourceMassFields) : Point → Mat := fun y =>
	(y 0)^2 • f.sigma (angularProjection y)+(1/(y 0)) • f.mK (angularProjection y)+f.ek y

	def sourceRadialMetric (f : SourceMassFields) : Point → ℝ := fun y =>
	1/(1+(y 0)^2)+f.mr (angularProjection y)/(y 0)^5+f.err y

	lemma sourceGamma_pullback (f : SourceMassFields) :
	(fun y => sourceGamma f (logRadiusChart y)) = logGamma f.logFields := by
	funext y
	simp [sourceGamma,logGamma,SourceMassFields.logFields,angularLift,
	angularProjection_logRadiusChart,logRadiusChart]

	lemma sourceTangentialK_pullback (f : SourceMassFields) :
	(fun y => sourceTangentialK f (logRadiusChart y)) = logTangentialK f.logFields := by
	funext y
	simp [sourceTangentialK,logTangentialK,SourceMassFields.logFields,angularLift,
	angularProjection_logRadiusChart,logRadiusChart]

	lemma sourceRadialMetric_pullback (f : SourceMassFields) :
	(fun y => sourceRadialMetric f (logRadiusChart y)) = logRadialMetric f.logFields := by
	funext y
	simp [sourceRadialMetric,logRadialMetric,SourceMassFields.logFields,angularLift,
	angularProjection_logRadiusChart,logRadiusChart]

	structure SourceMassFields.ComponentBounds (f : SourceMassFields) (B : ℝ) (y : Point) : Prop where
	eg : ∀ i k, ScaledComponentBound (fun z => f.eg z i k) 3 2 B y
	ek : ∀ i k, ScaledComponentBound (fun z => f.ek z i k) 2 2 B y
	b : ∀ a, ScaledComponentBound (fun z => f.b z a) 3 3 B y
	err : ScaledComponentBound f.err 2 6 B y

	lemma source_log_remainder_bounds {f : SourceMassFields} {x : Point} {B : ℝ} (hB : 0 ≤ B)
	(hf : f.RegularAt (logRadiusChart x)) (hb : f.ComponentBounds B (logRadiusChart x)) :
	‖matrixThreeJets f.logFields.eg x‖ ≤ B/Real.exp (x 0)^2 ∧
	‖matrixScalarJets f.logFields.ek x‖ ≤ B/Real.exp (x 0)^2 ∧
	‖fun a => actualThreeJet (fun y => f.logFields.b y a) x‖ ≤ B/Real.exp (x 0)^3 ∧
	‖actualScalarJet f.logFields.err x‖ ≤ B/Real.exp (x 0)^6 := by
	refine ⟨source_matrix_three_bound hB hf.eg hb.eg,source_matrix_two_bound hB hf.ek hb.ek,?_,
	source_two_jet_bound hB hf.err hb.err⟩
	apply (pi_norm_le_iff_of_nonneg (by positivity)).mpr
	intro a
	exact source_three_jet_bound hB (contDiffAt_pi.mp hf.b a) (hb.b a)

	end
	end CKSMixedGeometry

	end

	end OAI