Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.MinimalDescriptorLength
import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
import OAI.MeasureTheory.FalconerCompletion.Recursion.PrefixDescriptors

namespace OAI

/-! Dyadic shells and prefix decompositions. -/

open _root_.OAI.PlanarFalconer

section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace CurrentFalconer.TreeND
open Set MeasureTheory PlanarFalconer
open scoped ContDiff

section Transport
variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
  [NormedAddCommGroup W] [InnerProductSpace ℝ W]

noncomputable def transportedAngularField (e : V ≃ₗᵢ[ℝ] W) (A : V →L[ℝ] V) : W →L[ℝ] W :=
  e.toContinuousLinearEquiv.toContinuousLinearMap.comp
    (A.comp e.symm.toContinuousLinearEquiv.toContinuousLinearMap)

lemma angularWord_pull_exact (e : V ≃ₗᵢ[ℝ] W) (L : List (V →L[ℝ] V))
    (b : SmoothFunctionND W) (u : V) :
    (Attempt4PrefixShell.angularWordOperatorND L (Attempt4PrefixShell.smoothPullND e b)).1 u=
      (angularWordOperatorND (L.map (transportedAngularField e)) b).1 (e u) := by
  induction L generalizing u with
  | nil => rfl
  | cons A L ih =>
    have heq : (Attempt4PrefixShell.angularWordOperatorND L
        (Attempt4PrefixShell.smoothPullND e b)).1=
        (angularWordOperatorND (L.map (transportedAngularField e)) b).1 ∘ e := by
      funext z
      exact ih (u:=z)
    change fderiv ℝ (Attempt4PrefixShell.angularWordOperatorND L
      (Attempt4PrefixShell.smoothPullND e b)).1 u (A u)=
      fderiv ℝ (angularWordOperatorND (L.map (transportedAngularField e)) b).1 (e u)
        (transportedAngularField e A (e u))
    rw [heq,fderiv_comp u
      ((smoothFunctionND_contDiff _).differentiable (by simp) (e u))
      (e.toContinuousLinearEquiv.differentiableAt)]
    simp only [ContinuousLinearMap.comp_apply,LinearIsometryEquiv.fderiv,
      transportedAngularField,ContinuousLinearEquiv.coe_coe]
    congr 2
    exact congrArg A (e.symm_apply_apply u).symm

lemma normalizedAngularWord_pull_exact (e : V ≃ₗᵢ[ℝ] W) (D : ℝ)
    (L : List (V →L[ℝ] V)) (b : SmoothFunctionND W) (u : V) :
    (Attempt4PrefixShell.normalizedAngularWordND D L
      (Attempt4PrefixShell.smoothPullND e b)).1 u=
      (normalizedAngularWordND D (L.map (transportedAngularField e)) b).1 (e u) := by
  change ((D:ℂ)^L.length)⁻¹*(Attempt4PrefixShell.angularWordOperatorND L
    (Attempt4PrefixShell.smoothPullND e b)).1 u=
    ((D:ℂ)^(L.map (transportedAngularField e)).length)⁻¹*
      (angularWordOperatorND (L.map (transportedAngularField e)) b).1 (e u)
  rw [List.length_map,angularWord_pull_exact]
end Transport

variable {n : ℕ}
namespace PositiveDistanceState
variable (e : PositiveDistanceState (n+1))

lemma prefixEndpoint_word_exact (x : EuclideanSpace ℝ (Fin (n+1)))
    (b : PolicyDescriptor)
    (L : List (Attempt4PrefixShell.SphereAmbientND n →L[ℝ] Attempt4PrefixShell.SphereAmbientND n))
    (w : EuclideanSpace ℝ (Fin (n+1))) :
    (Attempt4PrefixShell.euclideanAngularWordND
      (sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig) L (e.prefixEndpointSmooth x b)).1 w=
      (actualEndpointAngular
        (L.map (transportedAngularField (Attempt4PrefixShell.sphereEuclideanIsoND n)))
        e.ρ e.mass e.N e.sig e.C e.J e.i b x).1 w := by
  change (Attempt4PrefixShell.normalizedAngularWordND _ L
    (Attempt4PrefixShell.smoothPullND (Attempt4PrefixShell.sphereEuclideanIsoND n)
      (e.prefixEndpointSmooth x b))).1 ((Attempt4PrefixShell.sphereEuclideanIsoND n).symm w)=_
  simpa only [LinearIsometryEquiv.apply_symm_apply,actualEndpointAngular,
    actualEndpointSmooth,prefixEndpointSmooth,CurrentFalconer.smoothFunctionSpaceND,
    Attempt4PrefixShell.smoothFunctionSpaceND] using
    normalizedAngularWord_pull_exact (Attempt4PrefixShell.sphereEuclideanIsoND n)
      (sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig) L
      (actualEndpointSmooth e.ρ e.mass e.N e.sig e.C e.J e.i b x)
      ((Attempt4PrefixShell.sphereEuclideanIsoND n).symm w)

end PositiveDistanceState

end CurrentFalconer.TreeND
end GraphLiteralSupport
end
end

section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace CurrentFalconer.TreeND
open Set MeasureTheory PlanarFalconer
open scoped ContDiff
variable {n : ℕ}
abbrev PrefixAssignment (n : ℕ) := ℂ × ((PolicyDescriptor ⊕ PolicyDescriptor) →
  List (Attempt4PrefixShell.SphereAmbientND n →L[ℝ] Attempt4PrefixShell.SphereAmbientND n))

noncomputable def prefixLeftWords (p : PrefixAssignment n) (b : PolicyDescriptor) :
    List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))) :=
  (p.2 (Sum.inl b)).map (transportedAngularField (Attempt4PrefixShell.sphereEuclideanIsoND n))
noncomputable def prefixRightWords (p : PrefixAssignment n) (b : PolicyDescriptor) :
    List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))) :=
  (p.2 (Sum.inr b)).map (transportedAngularField (Attempt4PrefixShell.sphereEuclideanIsoND n))

namespace PositiveDistanceState
variable (e : PositiveDistanceState (n+1))

noncomputable def prefixInverseLeft (p : PrefixAssignment n) :
    EuclideanSpace ℝ (Fin (n+1)) → EuclideanSpace ℝ (Fin (n+1)) → ℂ :=
  Attempt4PrefixShell.inverseLeftEndpointND e.prefixDescriptors
    (fun b => sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig)
    e.prefixEndpointSmooth (fun _ => 1) p
noncomputable def prefixInverseRight (p : PrefixAssignment n) :
    EuclideanSpace ℝ (Fin (n+1)) → EuclideanSpace ℝ (Fin (n+1)) → ℂ :=
  Attempt4PrefixShell.inverseRightEndpointND e.prefixDescriptors
    (fun b => sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig)
    e.prefixEndpointSmooth (fun _ => 1) p

lemma prefixInverseLeft_eq (p : PrefixAssignment n) (x w : EuclideanSpace ℝ (Fin (n+1))) :
    e.prefixInverseLeft p x w=
      (actualAngularProduct e.prefixDescriptors id (prefixLeftWords p)
        e.ρ e.mass e.N e.sig e.C e.J e.i x).1 w := by
  classical
  simp only [prefixInverseLeft,Attempt4PrefixShell.inverseLeftEndpointND,one_mul,
    actualAngularProduct,smoothFinsetProductND]
  apply Finset.prod_congr rfl
  intro b hb
  exact e.prefixEndpoint_word_exact x b (p.2 (Sum.inl b)) w

lemma prefixInverseRight_eq (p : PrefixAssignment n) (x w : EuclideanSpace ℝ (Fin (n+1))) :
    e.prefixInverseRight p x w=
      (actualAngularProduct e.prefixDescriptors id (prefixRightWords p)
        e.ρ e.mass e.N e.sig e.C e.J e.i x).1 w := by
  classical
  simp only [prefixInverseRight,Attempt4PrefixShell.inverseRightEndpointND,one_mul,
    actualAngularProduct,smoothFinsetProductND]
  apply Finset.prod_congr rfl
  intro b hb
  exact e.prefixEndpoint_word_exact x b (p.2 (Sum.inr b)) w

lemma prefixInverseLeft_measurable (p : PrefixAssignment n) :
    Measurable (Function.uncurry (e.prefixInverseLeft p)) := by
  simpa only [Function.uncurry_def,e.prefixInverseLeft_eq] using
    actualAngularProduct_measurable e.prefixDescriptors id (prefixLeftWords p)
      e.ρ e.mass e.N e.sig e.C e.J e.i

lemma prefixInverseRight_measurable (p : PrefixAssignment n) :
    Measurable (Function.uncurry (e.prefixInverseRight p)) := by
  simpa only [Function.uncurry_def,e.prefixInverseRight_eq] using
    actualAngularProduct_measurable e.prefixDescriptors id (prefixRightWords p)
      e.ρ e.mass e.N e.sig e.C e.J e.i

lemma prefixInverseWord_total (j : ℕ) {p : PrefixAssignment n}
    (hp : p ∈ Attempt4PrefixShell.inverseAngularTermsND j (e.prefixDescriptors.disjSum e.prefixDescriptors)) :
    (∑ b ∈ e.prefixDescriptors, (prefixLeftWords p b).length)+
      (∑ b ∈ e.prefixDescriptors, (prefixRightWords p b).length) ≤ 2*j := by
  have h := Attempt4PrefixShell.inverseAngularTermsND_budget j _ hp
  simpa only [Finset.sum_disjSum,prefixLeftWords,prefixRightWords,List.length_map] using h

end PositiveDistanceState

lemma uniform_prefixInverse_support {sig : ℝ} (hsig : 0 < sig) {J : ℕ} (hJ : 0 < J) :
    ∃ N₀ : ℕ, ∀ (e : PositiveDistanceState (n+1)), N₀ ≤ e.N → e.sig=sig → e.J=J →
      IsFiniteMeasure e.ρ → 0 < e.q → e.t ≤ e.N → ∀ (p : PrefixAssignment n) (k : ℕ)
        (x w : EuclideanSpace ℝ (Fin (n+1))), ‖w‖=1 →
      (w∈tsupport (iteratedFDeriv ℝ k (e.prefixInverseLeft p x)) →
        ∀ b∈e.prefixDescriptors, (x,w)∈endpointGood e.ρ e.mass e.N e.sig e.C e.J (e.i+1) b) ∧
      (w∈tsupport (iteratedFDeriv ℝ k (e.prefixInverseRight p x)) →
        ∀ b∈e.prefixDescriptors, (x,w)∈endpointGood e.ρ e.mass e.N e.sig e.C e.J (e.i+1) b) := by
  obtain ⟨N₀,h⟩ := uniform_actualAngularProduct_support (d:=n+1) (ι:=PolicyDescriptor) hsig hJ
  refine ⟨N₀,?_⟩
  intro e he hse hJe hf hq ht p k x w hw
  have hv : ∀ b∈e.prefixDescriptors, ValidDescriptor e.N (id b) :=
    fun b hb => policyDescriptor_valid _ hq ht hb
  have hleft := h e.N he e.ρ hf e.mass e.C e.i e.prefixDescriptors id hv
    (prefixLeftWords p) k x w hw
  have hright := h e.N he e.ρ hf e.mass e.C e.i e.prefixDescriptors id hv
    (prefixRightWords p) k x w hw
  rw [← hse,← hJe] at hleft hright
  have hl : e.prefixInverseLeft p x=
      (actualAngularProduct e.prefixDescriptors id (prefixLeftWords p)
        e.ρ e.mass e.N e.sig e.C e.J e.i x).1 := funext (e.prefixInverseLeft_eq p x)
  have hr : e.prefixInverseRight p x=
      (actualAngularProduct e.prefixDescriptors id (prefixRightWords p)
        e.ρ e.mass e.N e.sig e.C e.J e.i x).1 := funext (e.prefixInverseRight_eq p x)
  rw [hl,hr]
  exact ⟨hleft,hright⟩

end CurrentFalconer.TreeND
end GraphLiteralSupport
end
end

section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace CurrentFalconer.TreeND
open Set MeasureTheory PlanarFalconer Filter
open scoped ContDiff
variable {n : ℕ}

noncomputable def prefixInverseVocabulary (n J : ℕ) :
    Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) := by
  classical
  exact ((Finset.range (J+1)).biUnion (fun j =>
    Attempt4PrefixShell.polynomialAngularSubwordsND (Attempt4PrefixShell.sphereBasisND n)
      (Attempt4PrefixShell.sphereBPolynomialND n j))).image
    (List.map (transportedAngularField (Attempt4PrefixShell.sphereEuclideanIsoND n)))

lemma prefixInverseVocabulary_budget (J : ℕ) {L : List
    (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))}
    (hL : L ∈ prefixInverseVocabulary n J) : L.length ≤ 2*J := by
  classical
  obtain ⟨L',hL',rfl⟩ := Finset.mem_image.mp hL
  obtain ⟨j,hj,hL'⟩ := Finset.mem_biUnion.mp hL'
  rw [List.length_map]
  exact (Attempt4PrefixShell.polynomialAngularSubwordsND_budget _ _ hL').trans
    ((Nat.mul_le_mul_left 2 (Attempt4PrefixShell.sphereBPolynomialND_degree n j)).trans
      (Nat.mul_le_mul_left 2 (Nat.le_of_lt_succ (Finset.mem_range.mp hj))))
lemma prefixInverseVocabulary_mem (J j : ℕ) (hj : j ≤ J)
    (S : Finset (PolicyDescriptor ⊕ PolicyDescriptor)) {p : PrefixAssignment n}
    (hp : p ∈ Attempt4PrefixShell.inverseAngularTermsND j S)
    (b : PolicyDescriptor ⊕ PolicyDescriptor) :
    (p.2 b).map (transportedAngularField (Attempt4PrefixShell.sphereEuclideanIsoND n)) ∈
      prefixInverseVocabulary n J := by
  classical
  apply Finset.mem_image.mpr
  refine ⟨p.2 b,Finset.mem_biUnion.mpr ⟨j,Finset.mem_range.mpr (Nat.lt_succ_of_le hj),?_⟩,rfl⟩
  exact Attempt4PrefixShell.polynomialLeibnizTermsND_subword _ _ S hp b
lemma uniform_prefixInverse_derivatives (M J : ℕ) {sig : ℝ} (hsig : 0 < sig) :
    ∃ D : ℝ, 1 ≤ D ∧ ∀ᶠ N : ℕ in atTop, ∀ (e : PositiveDistanceState (n+1)),
      e.N=N → e.sig=sig → 0 < e.q → ∀ j ≤ J, ∀ p ∈
        Attempt4PrefixShell.inverseAngularTermsND j (e.prefixDescriptors.disjSum e.prefixDescriptors),
      ∀ (x : EuclideanSpace ℝ (Fin (n+1))), ∀ r ≤ M, ∀ w, ‖w‖=1 →
      ‖iteratedFDeriv ℝ r (e.prefixInverseLeft p x) w‖ ≤
        D^(2*J)*((((2:ℝ)^N)^sig)*sourceMaskScaleND ((e.t-e.a)/2) ((2:ℝ)^N) sig)^r ∧
      ‖iteratedFDeriv ℝ r (e.prefixInverseRight p x) w‖ ≤
        D^(2*J)*((((2:ℝ)^N)^sig)*sourceMaskScaleND ((e.t-e.a)/2) ((2:ℝ)^N) sig)^r := by
  classical
  obtain ⟨D,hD,h⟩ := uniform_actualAngularProduct_derivatives
    (ι:=PolicyDescriptor) (prefixInverseVocabulary n J) M (2*J) (2*J) 3
    (fun _ hL => prefixInverseVocabulary_budget J hL) 8 sig hsig
  refine ⟨D,hD,?_⟩
  filter_upwards [h,eventually_ge_atTop 1] with N hN hN₁
  intro e he hse hq j hj p hp x r hr w hw
  subst N
  rw [← hse] at hN ⊢
  have hc : (e.prefixDescriptors.card:ℝ) ≤ 8*(e.N:ℝ)^3 := by
    exact actualPolicy_card_bound (adjustedProfile (d:=n+1) e.mass e.N) hN₁ hq
  have hl : ∀ b ∈ e.prefixDescriptors, descriptorLength (id b) ≤ (e.t-e.a)/2 :=
    fun b hb => policyDescriptor_length_le_half _ hq hb
  have ht := (e.prefixInverseWord_total j hp).trans (Nat.mul_le_mul_left 2 hj)
  have hleft := hN e.prefixDescriptors id (prefixLeftWords p) e.ρ e.mass e.C e.J e.i
    ((e.t-e.a)/2) hc
    (fun b _ => prefixInverseVocabulary_mem J j hj _ hp (Sum.inl b))
    ((Nat.le_add_right _ _).trans ht) hl x r hr w hw
  have hright := hN e.prefixDescriptors id (prefixRightWords p) e.ρ e.mass e.C e.J e.i
    ((e.t-e.a)/2) hc
    (fun b _ => prefixInverseVocabulary_mem J j hj _ hp (Sum.inr b))
    ((Nat.le_add_left _ _).trans ht) hl x r hr w hw
  have hleftEq : e.prefixInverseLeft p x =
      (actualAngularProduct e.prefixDescriptors id (prefixLeftWords p)
        e.ρ e.mass e.N e.sig e.C e.J e.i x).1 := funext (e.prefixInverseLeft_eq p x)
  have hrightEq : e.prefixInverseRight p x =
      (actualAngularProduct e.prefixDescriptors id (prefixRightWords p)
        e.ρ e.mass e.N e.sig e.C e.J e.i x).1 := funext (e.prefixInverseRight_eq p x)
  rw [hleftEq,hrightEq]
  exact ⟨hleft,hright⟩
end CurrentFalconer.TreeND
end GraphLiteralSupport
end
end
end OAI

	import Mathlib
	import OAI.MeasureTheory.Falconer.Model
	import OAI.MeasureTheory.Falconer.Recursion.MinimalDescriptorLength
	import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
	import OAI.MeasureTheory.FalconerCompletion.Recursion.PrefixDescriptors

	namespace OAI

	/-! Dyadic shells and prefix decompositions. -/

	open _root_.OAI.PlanarFalconer

	section
	section
	open Set MeasureTheory
	open scoped ComplexConjugate RealInnerProductSpace
	namespace GraphLiteralSupport
	namespace CurrentFalconer.TreeND
	open Set MeasureTheory PlanarFalconer
	open scoped ContDiff

	section Transport
	variable {V W : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
	[NormedAddCommGroup W] [InnerProductSpace ℝ W]

	noncomputable def transportedAngularField (e : V ≃ₗᵢ[ℝ] W) (A : V →L[ℝ] V) : W →L[ℝ] W :=
	e.toContinuousLinearEquiv.toContinuousLinearMap.comp
	(A.comp e.symm.toContinuousLinearEquiv.toContinuousLinearMap)

	lemma angularWord_pull_exact (e : V ≃ₗᵢ[ℝ] W) (L : List (V →L[ℝ] V))
	(b : SmoothFunctionND W) (u : V) :
	(Attempt4PrefixShell.angularWordOperatorND L (Attempt4PrefixShell.smoothPullND e b)).1 u=
	(angularWordOperatorND (L.map (transportedAngularField e)) b).1 (e u) := by
	induction L generalizing u with
	\| nil => rfl
	\| cons A L ih =>
	have heq : (Attempt4PrefixShell.angularWordOperatorND L
	(Attempt4PrefixShell.smoothPullND e b)).1=
	(angularWordOperatorND (L.map (transportedAngularField e)) b).1 ∘ e := by
	funext z
	exact ih (u:=z)
	change fderiv ℝ (Attempt4PrefixShell.angularWordOperatorND L
	(Attempt4PrefixShell.smoothPullND e b)).1 u (A u)=
	fderiv ℝ (angularWordOperatorND (L.map (transportedAngularField e)) b).1 (e u)
	(transportedAngularField e A (e u))
	rw [heq,fderiv_comp u
	((smoothFunctionND_contDiff _).differentiable (by simp) (e u))
	(e.toContinuousLinearEquiv.differentiableAt)]
	simp only [ContinuousLinearMap.comp_apply,LinearIsometryEquiv.fderiv,
	transportedAngularField,ContinuousLinearEquiv.coe_coe]
	congr 2
	exact congrArg A (e.symm_apply_apply u).symm

	lemma normalizedAngularWord_pull_exact (e : V ≃ₗᵢ[ℝ] W) (D : ℝ)
	(L : List (V →L[ℝ] V)) (b : SmoothFunctionND W) (u : V) :
	(Attempt4PrefixShell.normalizedAngularWordND D L
	(Attempt4PrefixShell.smoothPullND e b)).1 u=
	(normalizedAngularWordND D (L.map (transportedAngularField e)) b).1 (e u) := by
	change ((D:ℂ)^L.length)⁻¹*(Attempt4PrefixShell.angularWordOperatorND L
	(Attempt4PrefixShell.smoothPullND e b)).1 u=
	((D:ℂ)^(L.map (transportedAngularField e)).length)⁻¹*
	(angularWordOperatorND (L.map (transportedAngularField e)) b).1 (e u)
	rw [List.length_map,angularWord_pull_exact]
	end Transport

	variable {n : ℕ}
	namespace PositiveDistanceState
	variable (e : PositiveDistanceState (n+1))

	lemma prefixEndpoint_word_exact (x : EuclideanSpace ℝ (Fin (n+1)))
	(b : PolicyDescriptor)
	(L : List (Attempt4PrefixShell.SphereAmbientND n →L[ℝ] Attempt4PrefixShell.SphereAmbientND n))
	(w : EuclideanSpace ℝ (Fin (n+1))) :
	(Attempt4PrefixShell.euclideanAngularWordND
	(sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig) L (e.prefixEndpointSmooth x b)).1 w=
	(actualEndpointAngular
	(L.map (transportedAngularField (Attempt4PrefixShell.sphereEuclideanIsoND n)))
	e.ρ e.mass e.N e.sig e.C e.J e.i b x).1 w := by
	change (Attempt4PrefixShell.normalizedAngularWordND _ L
	(Attempt4PrefixShell.smoothPullND (Attempt4PrefixShell.sphereEuclideanIsoND n)
	(e.prefixEndpointSmooth x b))).1 ((Attempt4PrefixShell.sphereEuclideanIsoND n).symm w)=_
	simpa only [LinearIsometryEquiv.apply_symm_apply,actualEndpointAngular,
	actualEndpointSmooth,prefixEndpointSmooth,CurrentFalconer.smoothFunctionSpaceND,
	Attempt4PrefixShell.smoothFunctionSpaceND] using
	normalizedAngularWord_pull_exact (Attempt4PrefixShell.sphereEuclideanIsoND n)
	(sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig) L
	(actualEndpointSmooth e.ρ e.mass e.N e.sig e.C e.J e.i b x)
	((Attempt4PrefixShell.sphereEuclideanIsoND n).symm w)

	end PositiveDistanceState

	end CurrentFalconer.TreeND
	end GraphLiteralSupport
	end
	end

	section
	section
	open Set MeasureTheory
	open scoped ComplexConjugate RealInnerProductSpace
	namespace GraphLiteralSupport
	namespace CurrentFalconer.TreeND
	open Set MeasureTheory PlanarFalconer
	open scoped ContDiff
	variable {n : ℕ}
	abbrev PrefixAssignment (n : ℕ) := ℂ × ((PolicyDescriptor ⊕ PolicyDescriptor) →
	List (Attempt4PrefixShell.SphereAmbientND n →L[ℝ] Attempt4PrefixShell.SphereAmbientND n))

	noncomputable def prefixLeftWords (p : PrefixAssignment n) (b : PolicyDescriptor) :
	List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))) :=
	(p.2 (Sum.inl b)).map (transportedAngularField (Attempt4PrefixShell.sphereEuclideanIsoND n))
	noncomputable def prefixRightWords (p : PrefixAssignment n) (b : PolicyDescriptor) :
	List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))) :=
	(p.2 (Sum.inr b)).map (transportedAngularField (Attempt4PrefixShell.sphereEuclideanIsoND n))

	namespace PositiveDistanceState
	variable (e : PositiveDistanceState (n+1))

	noncomputable def prefixInverseLeft (p : PrefixAssignment n) :
	EuclideanSpace ℝ (Fin (n+1)) → EuclideanSpace ℝ (Fin (n+1)) → ℂ :=
	Attempt4PrefixShell.inverseLeftEndpointND e.prefixDescriptors
	(fun b => sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig)
	e.prefixEndpointSmooth (fun _ => 1) p
	noncomputable def prefixInverseRight (p : PrefixAssignment n) :
	EuclideanSpace ℝ (Fin (n+1)) → EuclideanSpace ℝ (Fin (n+1)) → ℂ :=
	Attempt4PrefixShell.inverseRightEndpointND e.prefixDescriptors
	(fun b => sourceMaskScaleND (descriptorLength b) ((2:ℝ)^e.N) e.sig)
	e.prefixEndpointSmooth (fun _ => 1) p

	lemma prefixInverseLeft_eq (p : PrefixAssignment n) (x w : EuclideanSpace ℝ (Fin (n+1))) :
	e.prefixInverseLeft p x w=
	(actualAngularProduct e.prefixDescriptors id (prefixLeftWords p)
	e.ρ e.mass e.N e.sig e.C e.J e.i x).1 w := by
	classical
	simp only [prefixInverseLeft,Attempt4PrefixShell.inverseLeftEndpointND,one_mul,
	actualAngularProduct,smoothFinsetProductND]
	apply Finset.prod_congr rfl
	intro b hb
	exact e.prefixEndpoint_word_exact x b (p.2 (Sum.inl b)) w

	lemma prefixInverseRight_eq (p : PrefixAssignment n) (x w : EuclideanSpace ℝ (Fin (n+1))) :
	e.prefixInverseRight p x w=
	(actualAngularProduct e.prefixDescriptors id (prefixRightWords p)
	e.ρ e.mass e.N e.sig e.C e.J e.i x).1 w := by
	classical
	simp only [prefixInverseRight,Attempt4PrefixShell.inverseRightEndpointND,one_mul,
	actualAngularProduct,smoothFinsetProductND]
	apply Finset.prod_congr rfl
	intro b hb
	exact e.prefixEndpoint_word_exact x b (p.2 (Sum.inr b)) w

	lemma prefixInverseLeft_measurable (p : PrefixAssignment n) :
	Measurable (Function.uncurry (e.prefixInverseLeft p)) := by
	simpa only [Function.uncurry_def,e.prefixInverseLeft_eq] using
	actualAngularProduct_measurable e.prefixDescriptors id (prefixLeftWords p)
	e.ρ e.mass e.N e.sig e.C e.J e.i

	lemma prefixInverseRight_measurable (p : PrefixAssignment n) :
	Measurable (Function.uncurry (e.prefixInverseRight p)) := by
	simpa only [Function.uncurry_def,e.prefixInverseRight_eq] using
	actualAngularProduct_measurable e.prefixDescriptors id (prefixRightWords p)
	e.ρ e.mass e.N e.sig e.C e.J e.i

	lemma prefixInverseWord_total (j : ℕ) {p : PrefixAssignment n}
	(hp : p ∈ Attempt4PrefixShell.inverseAngularTermsND j (e.prefixDescriptors.disjSum e.prefixDescriptors)) :
	(∑ b ∈ e.prefixDescriptors, (prefixLeftWords p b).length)+
	(∑ b ∈ e.prefixDescriptors, (prefixRightWords p b).length) ≤ 2*j := by
	have h := Attempt4PrefixShell.inverseAngularTermsND_budget j _ hp
	simpa only [Finset.sum_disjSum,prefixLeftWords,prefixRightWords,List.length_map] using h

	end PositiveDistanceState

	lemma uniform_prefixInverse_support {sig : ℝ} (hsig : 0 < sig) {J : ℕ} (hJ : 0 < J) :
	∃ N₀ : ℕ, ∀ (e : PositiveDistanceState (n+1)), N₀ ≤ e.N → e.sig=sig → e.J=J →
	IsFiniteMeasure e.ρ → 0 < e.q → e.t ≤ e.N → ∀ (p : PrefixAssignment n) (k : ℕ)
	(x w : EuclideanSpace ℝ (Fin (n+1))), ‖w‖=1 →
	(w∈tsupport (iteratedFDeriv ℝ k (e.prefixInverseLeft p x)) →
	∀ b∈e.prefixDescriptors, (x,w)∈endpointGood e.ρ e.mass e.N e.sig e.C e.J (e.i+1) b) ∧
	(w∈tsupport (iteratedFDeriv ℝ k (e.prefixInverseRight p x)) →
	∀ b∈e.prefixDescriptors, (x,w)∈endpointGood e.ρ e.mass e.N e.sig e.C e.J (e.i+1) b) := by
	obtain ⟨N₀,h⟩ := uniform_actualAngularProduct_support (d:=n+1) (ι:=PolicyDescriptor) hsig hJ
	refine ⟨N₀,?_⟩
	intro e he hse hJe hf hq ht p k x w hw
	have hv : ∀ b∈e.prefixDescriptors, ValidDescriptor e.N (id b) :=
	fun b hb => policyDescriptor_valid _ hq ht hb
	have hleft := h e.N he e.ρ hf e.mass e.C e.i e.prefixDescriptors id hv
	(prefixLeftWords p) k x w hw
	have hright := h e.N he e.ρ hf e.mass e.C e.i e.prefixDescriptors id hv
	(prefixRightWords p) k x w hw
	rw [← hse,← hJe] at hleft hright
	have hl : e.prefixInverseLeft p x=
	(actualAngularProduct e.prefixDescriptors id (prefixLeftWords p)
	e.ρ e.mass e.N e.sig e.C e.J e.i x).1 := funext (e.prefixInverseLeft_eq p x)
	have hr : e.prefixInverseRight p x=
	(actualAngularProduct e.prefixDescriptors id (prefixRightWords p)
	e.ρ e.mass e.N e.sig e.C e.J e.i x).1 := funext (e.prefixInverseRight_eq p x)
	rw [hl,hr]
	exact ⟨hleft,hright⟩

	end CurrentFalconer.TreeND
	end GraphLiteralSupport
	end
	end

	section
	section
	open Set MeasureTheory
	open scoped ComplexConjugate RealInnerProductSpace
	namespace GraphLiteralSupport
	namespace CurrentFalconer.TreeND
	open Set MeasureTheory PlanarFalconer Filter
	open scoped ContDiff
	variable {n : ℕ}

	noncomputable def prefixInverseVocabulary (n J : ℕ) :
	Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))) := by
	classical
	exact ((Finset.range (J+1)).biUnion (fun j =>
	Attempt4PrefixShell.polynomialAngularSubwordsND (Attempt4PrefixShell.sphereBasisND n)
	(Attempt4PrefixShell.sphereBPolynomialND n j))).image
	(List.map (transportedAngularField (Attempt4PrefixShell.sphereEuclideanIsoND n)))

	lemma prefixInverseVocabulary_budget (J : ℕ) {L : List
	(EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))}
	(hL : L ∈ prefixInverseVocabulary n J) : L.length ≤ 2*J := by
	classical
	obtain ⟨L',hL',rfl⟩ := Finset.mem_image.mp hL
	obtain ⟨j,hj,hL'⟩ := Finset.mem_biUnion.mp hL'
	rw [List.length_map]
	exact (Attempt4PrefixShell.polynomialAngularSubwordsND_budget _ _ hL').trans
	((Nat.mul_le_mul_left 2 (Attempt4PrefixShell.sphereBPolynomialND_degree n j)).trans
	(Nat.mul_le_mul_left 2 (Nat.le_of_lt_succ (Finset.mem_range.mp hj))))

	lemma prefixInverseVocabulary_mem (J j : ℕ) (hj : j ≤ J)
	(S : Finset (PolicyDescriptor ⊕ PolicyDescriptor)) {p : PrefixAssignment n}
	(hp : p ∈ Attempt4PrefixShell.inverseAngularTermsND j S)
	(b : PolicyDescriptor ⊕ PolicyDescriptor) :
	(p.2 b).map (transportedAngularField (Attempt4PrefixShell.sphereEuclideanIsoND n)) ∈
	prefixInverseVocabulary n J := by
	classical
	apply Finset.mem_image.mpr
	refine ⟨p.2 b,Finset.mem_biUnion.mpr ⟨j,Finset.mem_range.mpr (Nat.lt_succ_of_le hj),?_⟩,rfl⟩
	exact Attempt4PrefixShell.polynomialLeibnizTermsND_subword _ _ S hp b

	lemma uniform_prefixInverse_derivatives (M J : ℕ) {sig : ℝ} (hsig : 0 < sig) :
	∃ D : ℝ, 1 ≤ D ∧ ∀ᶠ N : ℕ in atTop, ∀ (e : PositiveDistanceState (n+1)),
	e.N=N → e.sig=sig → 0 < e.q → ∀ j ≤ J, ∀ p ∈
	Attempt4PrefixShell.inverseAngularTermsND j (e.prefixDescriptors.disjSum e.prefixDescriptors),
	∀ (x : EuclideanSpace ℝ (Fin (n+1))), ∀ r ≤ M, ∀ w, ‖w‖=1 →
	‖iteratedFDeriv ℝ r (e.prefixInverseLeft p x) w‖ ≤
	D^(2J)((((2:ℝ)^N)^sig)*sourceMaskScaleND ((e.t-e.a)/2) ((2:ℝ)^N) sig)^r ∧
	‖iteratedFDeriv ℝ r (e.prefixInverseRight p x) w‖ ≤
	D^(2J)((((2:ℝ)^N)^sig)*sourceMaskScaleND ((e.t-e.a)/2) ((2:ℝ)^N) sig)^r := by
	classical
	obtain ⟨D,hD,h⟩ := uniform_actualAngularProduct_derivatives
	(ι:=PolicyDescriptor) (prefixInverseVocabulary n J) M (2J) (2J) 3
	(fun _ hL => prefixInverseVocabulary_budget J hL) 8 sig hsig
	refine ⟨D,hD,?_⟩
	filter_upwards [h,eventually_ge_atTop 1] with N hN hN₁
	intro e he hse hq j hj p hp x r hr w hw
	subst N
	rw [← hse] at hN ⊢
	have hc : (e.prefixDescriptors.card:ℝ) ≤ 8*(e.N:ℝ)^3 := by
	exact actualPolicy_card_bound (adjustedProfile (d:=n+1) e.mass e.N) hN₁ hq
	have hl : ∀ b ∈ e.prefixDescriptors, descriptorLength (id b) ≤ (e.t-e.a)/2 :=
	fun b hb => policyDescriptor_length_le_half _ hq hb
	have ht := (e.prefixInverseWord_total j hp).trans (Nat.mul_le_mul_left 2 hj)
	have hleft := hN e.prefixDescriptors id (prefixLeftWords p) e.ρ e.mass e.C e.J e.i
	((e.t-e.a)/2) hc
	(fun b _ => prefixInverseVocabulary_mem J j hj _ hp (Sum.inl b))
	((Nat.le_add_right _ _).trans ht) hl x r hr w hw
	have hright := hN e.prefixDescriptors id (prefixRightWords p) e.ρ e.mass e.C e.J e.i
	((e.t-e.a)/2) hc
	(fun b _ => prefixInverseVocabulary_mem J j hj _ hp (Sum.inr b))
	((Nat.le_add_left _ _).trans ht) hl x r hr w hw
	have hleftEq : e.prefixInverseLeft p x =
	(actualAngularProduct e.prefixDescriptors id (prefixLeftWords p)
	e.ρ e.mass e.N e.sig e.C e.J e.i x).1 := funext (e.prefixInverseLeft_eq p x)
	have hrightEq : e.prefixInverseRight p x =
	(actualAngularProduct e.prefixDescriptors id (prefixRightWords p)
	e.ρ e.mass e.N e.sig e.C e.J e.i x).1 := funext (e.prefixInverseRight_eq p x)
	rw [hleftEq,hrightEq]
	exact ⟨hleft,hright⟩

	end CurrentFalconer.TreeND
	end GraphLiteralSupport
	end
	end

	end OAI