Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Analysis.BiholderTransport.Geodesics.MinimizingSubinterval
import OAI.Analysis.BiholderTransport.Geodesics.GlobalGauss

namespace OAI

noncomputable section
open Set Filter Manifold Bundle Metric
open scoped Topology ContDiff NNReal

namespace WeakMTWTransport
variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
  [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
  [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
  [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
    (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]

attribute [local instance] normedAddCommGroupTangentSpaceVectorSpace
  normedSpaceTangentSpaceVectorSpace

lemma chart_spray_hasDerivAt (a : M) (z : TangentBundle 𝓘(ℝ,Model n) M) {t : ℝ}
    (ht : (sprayFlow t z).1∈(extChartAt 𝓘(ℝ,Model n) a).source) :
    HasDerivAt (fun s => extChartAt 𝓘(ℝ,Model n) a (sprayFlow s z).1)
      (mfderiv 𝓘(ℝ,Model n) 𝓘(ℝ,Model n) (extChartAt 𝓘(ℝ,Model n) a)
        (sprayFlow t z).1 (sprayFlow t z).2) t := by
  have hsrc : (sprayFlow t z).1∈(chartAt (Model n) a).source := by
    simpa only [extChartAt_source] using ht
  have projectedDeriv :=
    (ContinuousLinearMap.fst ℝ (Model n) (Model n)).hasFDerivAt.comp_hasDerivAt t
      (sprayFlow_chart_hasDerivAt (⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M) z ht)
  change HasDerivAt (fun s => extChartAt 𝓘(ℝ,Model n) a (sprayFlow s z).1)
    (tangentCoordChange 𝓘(ℝ,Model n) (sprayFlow t z).1 a
      (sprayFlow t z).1 (sprayFlow t z).2) t at projectedDeriv
  apply projectedDeriv.congr_deriv
  symm
  change (mfderiv 𝓘(ℝ,Model n) 𝓘(ℝ,Model n) (chartAt (Model n) a)
    (sprayFlow t z).1) (sprayFlow t z).2 = _
  rw [mfderiv_chartAt_eq_tangentCoordChange hsrc]
  rfl

variable [IsRiemannianManifold 𝓘(ℝ,Model n) M]

lemma exists_lipschitzOn_chart (a : M) :
    ∃ C : ℝ≥0, ∃ r>0, ball a r⊆(extChartAt 𝓘(ℝ,Model n) a).source ∧
      LipschitzOnWith C (extChartAt 𝓘(ℝ,Model n) a) (ball a r) := by
  have : IsContinuousRiemannianBundle (Model n)
      (fun x : M => TangentSpace 𝓘(ℝ,Model n) x) :=
    continuousRiemannianBundle_of_smooth (IB := 𝓘(ℝ,Model n))
  obtain ⟨C,hC,H⟩ := eventually_norm_mfderiv_extChartAt_lt 𝓘(ℝ,Model n) a
  obtain ⟨R,hR,hsub⟩ := Metric.mem_nhds_iff.mp
    (inter_mem (extChartAt_source_mem_nhds (I := 𝓘(ℝ,Model n)) a) H)
  let Cnn : ℝ≥0 := ⟨C,hC.le⟩
  refine ⟨Cnn,R/3,by positivity,?_,?_⟩
  · exact fun x hx => (hsub ((ball_subset_ball (by linarith : R/3≤R)) hx)).1
  apply LipschitzOnWith.of_dist_le_mul
  intro x hx y hy
  obtain ⟨p,hp,he⟩ := exists_minimizing_vector (n := n) x y
  let z : TangentBundle 𝓘(ℝ,Model n) M := ⟨x,p⟩
  have hpN : ‖p‖=dist x y := by rw [←he]; exact hp.symm
  have hpath : ∀ t∈Icc (0:ℝ) 1, (sprayFlow t z).1∈ball a R := by
    intro t ht
    have hD := dist_sprayFlow_le z 0 t
    simp only [sprayFlow_zero,zero_sub,abs_neg,abs_of_nonneg ht.1] at hD
    change dist x (sprayFlow t z).1≤t*‖p‖ at hD
    have hD' : dist x (sprayFlow t z).1≤dist x y :=
      hD.trans (by rw [←hpN]; exact mul_le_of_le_one_left (norm_nonneg _) ht.2)
    have hxy := dist_triangle x a y
    have hax : dist x a<R/3 := hx
    have hay : dist y a<R/3 := hy
    have htri := dist_triangle (sprayFlow t z).1 x a
    rw [dist_comm (sprayFlow t z).1 x] at htri
    rw [dist_comm a y] at hxy
    change dist (sprayFlow t z).1 a<R
    linarith
  let γ := fun t : ℝ => extChartAt 𝓘(ℝ,Model n) a (sprayFlow t z).1
  let dγ : ℝ → Model n := fun t => mfderiv 𝓘(ℝ,Model n) 𝓘(ℝ,Model n)
    (extChartAt 𝓘(ℝ,Model n) a) (sprayFlow t z).1 (sprayFlow t z).2
  have hd : ∀ t∈Icc (0:ℝ) 1, HasDerivWithinAt γ (dγ t) (Icc 0 1) t := by
    intro t ht
    exact (chart_spray_hasDerivAt a z (hsub (hpath t ht)).1).hasDerivWithinAt
  have hb : ∀ t∈Icc (0:ℝ) 1, ‖dγ t‖≤C*‖p‖ := by
    intro t ht
    apply (ContinuousLinearMap.le_opNorm _ _).trans
    rw [sprayFlow_speed]
    exact mul_le_mul_of_nonneg_right (hsub (hpath t ht)).2.le (norm_nonneg _)
  have hfin := norm_image_sub_le_of_norm_deriv_le_segment_01' hd (fun t ht => hb t ⟨ht.1,ht.2.le⟩)
  have he1 : (sprayFlow 1 z).1=y := by
    simpa only [z,←riemannianExp_eq_sprayFlow] using he
  have he0 : (sprayFlow 0 z).1=x := by rw [sprayFlow_zero]
  change dist (extChartAt 𝓘(ℝ,Model n) a x) (extChartAt 𝓘(ℝ,Model n) a y)≤C*dist x y
  simpa only [γ,he1,he0,←dist_eq_norm,dist_comm,hpN] using hfin

end WeakMTWTransport

end

end OAI

	import Mathlib
	import OAI.Analysis.BiholderTransport.Geodesics.MinimizingSubinterval
	import OAI.Analysis.BiholderTransport.Geodesics.GlobalGauss

	namespace OAI

	noncomputable section
	open Set Filter Manifold Bundle Metric
	open scoped Topology ContDiff NNReal

	namespace WeakMTWTransport
	variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M]
	[ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M]
	[RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]
	[IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n)
	(fun x : M => TangentSpace 𝓘(ℝ,Model n) x)]

	attribute [local instance] normedAddCommGroupTangentSpaceVectorSpace
	normedSpaceTangentSpaceVectorSpace

	lemma chart_spray_hasDerivAt (a : M) (z : TangentBundle 𝓘(ℝ,Model n) M) {t : ℝ}
	(ht : (sprayFlow t z).1∈(extChartAt 𝓘(ℝ,Model n) a).source) :
	HasDerivAt (fun s => extChartAt 𝓘(ℝ,Model n) a (sprayFlow s z).1)
	(mfderiv 𝓘(ℝ,Model n) 𝓘(ℝ,Model n) (extChartAt 𝓘(ℝ,Model n) a)
	(sprayFlow t z).1 (sprayFlow t z).2) t := by
	have hsrc : (sprayFlow t z).1∈(chartAt (Model n) a).source := by
	simpa only [extChartAt_source] using ht
	have projectedDeriv :=
	(ContinuousLinearMap.fst ℝ (Model n) (Model n)).hasFDerivAt.comp_hasDerivAt t
	(sprayFlow_chart_hasDerivAt (⟨a,0⟩ : TangentBundle 𝓘(ℝ,Model n) M) z ht)
	change HasDerivAt (fun s => extChartAt 𝓘(ℝ,Model n) a (sprayFlow s z).1)
	(tangentCoordChange 𝓘(ℝ,Model n) (sprayFlow t z).1 a
	(sprayFlow t z).1 (sprayFlow t z).2) t at projectedDeriv
	apply projectedDeriv.congr_deriv
	symm
	change (mfderiv 𝓘(ℝ,Model n) 𝓘(ℝ,Model n) (chartAt (Model n) a)
	(sprayFlow t z).1) (sprayFlow t z).2 = _
	rw [mfderiv_chartAt_eq_tangentCoordChange hsrc]
	rfl

	variable [IsRiemannianManifold 𝓘(ℝ,Model n) M]

	lemma exists_lipschitzOn_chart (a : M) :
	∃ C : ℝ≥0, ∃ r>0, ball a r⊆(extChartAt 𝓘(ℝ,Model n) a).source ∧
	LipschitzOnWith C (extChartAt 𝓘(ℝ,Model n) a) (ball a r) := by
	have : IsContinuousRiemannianBundle (Model n)
	(fun x : M => TangentSpace 𝓘(ℝ,Model n) x) :=
	continuousRiemannianBundle_of_smooth (IB := 𝓘(ℝ,Model n))
	obtain ⟨C,hC,H⟩ := eventually_norm_mfderiv_extChartAt_lt 𝓘(ℝ,Model n) a
	obtain ⟨R,hR,hsub⟩ := Metric.mem_nhds_iff.mp
	(inter_mem (extChartAt_source_mem_nhds (I := 𝓘(ℝ,Model n)) a) H)
	let Cnn : ℝ≥0 := ⟨C,hC.le⟩
	refine ⟨Cnn,R/3,by positivity,?_,?_⟩
	· exact fun x hx => (hsub ((ball_subset_ball (by linarith : R/3≤R)) hx)).1
	apply LipschitzOnWith.of_dist_le_mul
	intro x hx y hy
	obtain ⟨p,hp,he⟩ := exists_minimizing_vector (n := n) x y
	let z : TangentBundle 𝓘(ℝ,Model n) M := ⟨x,p⟩
	have hpN : ‖p‖=dist x y := by rw [←he]; exact hp.symm
	have hpath : ∀ t∈Icc (0:ℝ) 1, (sprayFlow t z).1∈ball a R := by
	intro t ht
	have hD := dist_sprayFlow_le z 0 t
	simp only [sprayFlow_zero,zero_sub,abs_neg,abs_of_nonneg ht.1] at hD
	change dist x (sprayFlow t z).1≤t*‖p‖ at hD
	have hD' : dist x (sprayFlow t z).1≤dist x y :=
	hD.trans (by rw [←hpN]; exact mul_le_of_le_one_left (norm_nonneg _) ht.2)
	have hxy := dist_triangle x a y
	have hax : dist x a<R/3 := hx
	have hay : dist y a<R/3 := hy
	have htri := dist_triangle (sprayFlow t z).1 x a
	rw [dist_comm (sprayFlow t z).1 x] at htri
	rw [dist_comm a y] at hxy
	change dist (sprayFlow t z).1 a<R
	linarith
	let γ := fun t : ℝ => extChartAt 𝓘(ℝ,Model n) a (sprayFlow t z).1
	let dγ : ℝ → Model n := fun t => mfderiv 𝓘(ℝ,Model n) 𝓘(ℝ,Model n)
	(extChartAt 𝓘(ℝ,Model n) a) (sprayFlow t z).1 (sprayFlow t z).2
	have hd : ∀ t∈Icc (0:ℝ) 1, HasDerivWithinAt γ (dγ t) (Icc 0 1) t := by
	intro t ht
	exact (chart_spray_hasDerivAt a z (hsub (hpath t ht)).1).hasDerivWithinAt
	have hb : ∀ t∈Icc (0:ℝ) 1, ‖dγ t‖≤C*‖p‖ := by
	intro t ht
	apply (ContinuousLinearMap.le_opNorm _ _).trans
	rw [sprayFlow_speed]
	exact mul_le_mul_of_nonneg_right (hsub (hpath t ht)).2.le (norm_nonneg _)
	have hfin := norm_image_sub_le_of_norm_deriv_le_segment_01' hd (fun t ht => hb t ⟨ht.1,ht.2.le⟩)
	have he1 : (sprayFlow 1 z).1=y := by
	simpa only [z,←riemannianExp_eq_sprayFlow] using he
	have he0 : (sprayFlow 0 z).1=x := by rw [sprayFlow_zero]
	change dist (extChartAt 𝓘(ℝ,Model n) a x) (extChartAt 𝓘(ℝ,Model n) a y)≤C*dist x y
	simpa only [γ,he1,he0,←dist_eq_norm,dist_comm,hpN] using hfin

	end WeakMTWTransport

	end

	end OAI