File size: 15,976 Bytes
b224933 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
import OAI.MeasureTheory.FalconerCompletion.Graphs.ResidualPairSourceSymbolEqScalarPair
namespace OAI
/-! Recursive policies and quantitative depth estimates. -/
open _root_.OAI.PlanarFalconer
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace CurrentFalconer.TreeND
open Set MeasureTheory Filter PlanarFalconer
open scoped ContDiff ENNReal Topology Classical
variable {n : ℕ}
lemma treeCell_membership_constant_on_finer_grid
{P : CellPath (n+1)} {h : ℕ} (hP : P.length≤h) (X : Fin (n+1) → ℤ)
{x y : EuclideanSpace ℝ (Fin (n+1))}
(hx : x∈CurrentFalconer.ambientGridCell h X)
(hy : y∈CurrentFalconer.ambientGridCell h X) : x∈treeCell P ↔ y∈treeCell P := by
have hs := CurrentFalconer.ambientGridCell_subset_ancestor hP X
have one {u v : EuclideanSpace ℝ (Fin (n+1))}
(hu : u∈CurrentFalconer.ambientGridCell h X)
(hv : v∈CurrentFalconer.ambientGridCell h X) (huP : u∈treeCell P) : v∈treeCell P := by
have he : (fun i => X i/(2:ℤ)^(h-P.length))=(fun i => (pathIndex P i:ℤ)) := by
by_contra hne
exact Set.disjoint_left.mp (CurrentFalconer.ambientGridCell_disjoint hne) (hs hu) huP
have hh := hs hv
rw [he] at hh
exact hh
exact ⟨one hx hy,one hy hx⟩
lemma actualEndpointSmooth_grid_reached
(ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (N : ℕ)
(sig C : ℝ) (J i : ℕ) (b : PolicyDescriptor) {h : ℕ} (ha : descriptorAnchor b≤h)
(X : Fin (n+1) → ℤ) {x y : EuclideanSpace ℝ (Fin (n+1))}
(hx : x∈CurrentFalconer.ambientGridCell h X) (hy : y∈CurrentFalconer.ambientGridCell h X) :
actualEndpointSmooth ρ m N sig C J i b x=actualEndpointSmooth ρ m N sig C J i b y := by
have he (P : CellPath (n+1)) (hP : P∈dyadicPaths (descriptorAnchor b)) :
x∈treeCell P ↔ y∈treeCell P :=
treeCell_membership_constant_on_finer_grid ((mem_dyadicPaths _ _).mp hP ▸ ha) X hx hy
by_cases ex : ∃ P∈dyadicPaths (descriptorAnchor b), x∈treeCell P
· obtain ⟨P,hP,hxP⟩ := ex
rw [actualEndpointSmooth_of_mem ρ m N sig C J i b hP hxP, actualEndpointSmooth_of_mem ρ m N sig C J i b hP ((he P hP).mp hxP)]
· have hx0 : ∀ P∈dyadicPaths (descriptorAnchor b), x∉treeCell P := by
intro P hP hxP
exact ex ⟨P,hP,hxP⟩
rw [actualEndpointSmooth_of_absent ρ m N sig C J i b hx0, actualEndpointSmooth_of_absent ρ m N sig C J i b (fun P hP hyP => hx0 P hP ((he P hP).mpr hyP))]
lemma actualAngularProduct_grid_reached
(D : Finset PolicyDescriptor)
(L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))
(ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (N : ℕ)
(sig C : ℝ) (J i : ℕ) {h : ℕ} (ha : ∀ b∈D, descriptorAnchor b≤h)
(X : Fin (n+1) → ℤ) {x y : EuclideanSpace ℝ (Fin (n+1))}
(hx : x∈CurrentFalconer.ambientGridCell h X) (hy : y∈CurrentFalconer.ambientGridCell h X) :
actualAngularProduct D id L ρ m N sig C J i x=actualAngularProduct D id L ρ m N sig C J i y := by
apply Subtype.ext
funext w
apply Finset.prod_congr rfl
intro b hb
have he : actualEndpointAngular (L b) ρ m N sig C J i b x=actualEndpointAngular (L b) ρ m N sig C J i b y := by
unfold actualEndpointAngular
rw [actualEndpointSmooth_grid_reached ρ m N sig C J i b (ha b hb) X hx hy]
exact congrArg (fun f : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1))) => f.1 w) he
noncomputable def newlyReachedDescriptorList (A : ℕ → ℝ) (N q a t p h : ℕ) : Finset PolicyDescriptor :=
(unreachedDescriptorList A N q a t p).filter (fun b => descriptorAnchor b≤h)
lemma actualResidualAngular_step_factorization
(A : ℕ → ℝ) (N q a t p h : ℕ) (hph : p≤h)
(L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))
(ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ)
(x : EuclideanSpace ℝ (Fin (n+1))) :
actualResidualAngular A N q a t p L ρ m sig C J i x =
smoothMulND (actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i x)
(actualResidualAngular A N q a t h L ρ m sig C J i x) := by
have hcomp : (unreachedDescriptorList A N q a t p).filter (fun b => ¬descriptorAnchor b≤h)=
unreachedDescriptorList A N q a t h := by
ext b
simp only [unreachedDescriptorList,Finset.mem_filter,not_le]
constructor
· rintro ⟨⟨hb, _⟩, hbh⟩
exact ⟨hb,hbh⟩
· rintro ⟨hb,hbh⟩
exact ⟨⟨hb,lt_of_le_of_lt hph hbh⟩,hbh⟩
apply Subtype.ext
funext w
have he := (Finset.prod_filter_mul_prod_filter_not
(unreachedDescriptorList A N q a t p) (fun b => descriptorAnchor b≤h)
(fun b => (actualEndpointAngular (L b) ρ m N sig C J i b x).1 w)).symm
rw [hcomp] at he
exact he
lemma actualResidualSourceSymbol_grid_factorization
(A : ℕ → ℝ) (N q a t p h : ℕ) (hph : p≤h)
(L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))
(ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ)
(f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1))))
(X : Fin (n+1) → ℤ) {x : EuclideanSpace ℝ (Fin (n+1))}
(hx : x∈CurrentFalconer.ambientGridCell h X) (w : EuclideanSpace ℝ (Fin (n+1))) :
(actualResidualSourceSymbol A N q a t p L ρ m sig C J i f Q x).1 w =
(actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i
(CurrentFalconer.ambientGridCenter h X)).1 w *
(actualResidualSourceSymbol A N q a t h L ρ m sig C J i f Q x).1 w := by
have he := actualResidualAngular_step_factorization A N q a t p h hph L ρ m sig C J i x
have hc := actualAngularProduct_grid_reached (newlyReachedDescriptorList A N q a t p h)
L ρ m N sig C J i (fun b hb => (Finset.mem_filter.mp hb).2) X hx
(CurrentFalconer.ambientGridCenter_mem h X)
change f x*(Q.1 w*(actualResidualAngular A N q a t p L ρ m sig C J i x).1 w)=_
rw [he]
change f x*(Q.1 w*((actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i x).1 w * _))=_
rw [hc]
change f x*(Q.1 w*(_*_))=_*(f x*(Q.1 w*_))
dsimp only [actualResidualAngular]
ring
lemma actualResidualTransform_grid_factorization
(A : ℕ → ℝ) (N q a t p h : ℕ) (hph : p≤h)
(L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))
(ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (sig C : ℝ) (J i : ℕ)
(f : EuclideanSpace ℝ (Fin (n+1)) → ℂ) (Q : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1))))
(X : Fin (n+1) → ℤ) (M r : ℝ) (w : EuclideanSpace ℝ (Fin (n+1))) :
sourceAngularTransformND ρ (CurrentFalconer.ambientGridCell h X) M
(fun x => (actualResidualSourceSymbol A N q a t p L ρ m sig C J i f Q x).1) r w =
(actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i
(CurrentFalconer.ambientGridCenter h X)).1 w *
sourceAngularTransformND ρ (CurrentFalconer.ambientGridCell h X) M
(fun x => (actualResidualSourceSymbol A N q a t h L ρ m sig C J i f Q x).1) r w := by
unfold sourceAngularTransformND
rw [show (∫ x in CurrentFalconer.ambientGridCell h X, spherePlaneWaveND r x w*(actualResidualSourceSymbol A N q a t p L ρ m sig C J i f Q x).1 w ∂ρ) = ∫ x in CurrentFalconer.ambientGridCell h X, (actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i (CurrentFalconer.ambientGridCenter h X)).1 w * (spherePlaneWaveND r x w*(actualResidualSourceSymbol A N q a t h L ρ m sig C J i f Q x).1 w) ∂ρ by apply setIntegral_congr_fun (CurrentFalconer.measurableSet_ambientGridCell h X) intro x hx dsimp only rw [actualResidualSourceSymbol_grid_factorization A N q a t p h hph L ρ m sig C J i f Q X hx w]
ring]
rw [integral_const_mul]
exact (mul_smul_comm _ _ _).symm
lemma uniform_newlyReachedSymbol_bound
(W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))))
(B total : ℕ) (hW : ∀ L∈W, L.length≤B) (sig : ℝ) (hsig : 0<sig) :
∃ D : ℝ, 1≤D ∧ ∀ᶠ N : ℕ in Filter.atTop, ∀ (A : ℕ → ℝ) (q a t p h : ℕ), 0<q →
∀ (L : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))
(ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) (m : ℝ → ℝ) (C : ℝ) (J i : ℕ),
(∀ b ∈ policyDescriptorList A N q a t, L b∈W) →
(∑ b∈policyDescriptorList A N q a t, (L b).length ≤ total) →
∀ (x w : EuclideanSpace ℝ (Fin (n+1))), ‖w‖=1 →
‖(actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L ρ m N sig C J i x).1 w‖ ≤ D^total := by
obtain ⟨D,hD,hDb⟩ := uniform_actualAngularProduct_derivatives
(ι:=PolicyDescriptor) W 0 B total 3 hW 8 sig hsig
refine ⟨D,hD,?_⟩
filter_upwards [hDb,Filter.eventually_ge_atTop 1] with N hDb hN
intro A q a t p h hq L ρ m C J i hL htotal x w hw
have hsub : newlyReachedDescriptorList A N q a t p h ⊆ policyDescriptorList A N q a t := (Finset.filter_subset _ _).trans (Finset.filter_subset _ _)
have hcard : ((newlyReachedDescriptorList A N q a t p h).card:ℝ)≤8*(N:ℝ)^3 := by
calc
_ ≤ ((policyDescriptorList A N q a t).card:ℝ) := by exact_mod_cast Finset.card_le_card hsub
_ ≤ _ := actualPolicy_card_bound A hN hq
have hsum : ∑ b∈newlyReachedDescriptorList A N q a t p h, (L b).length ≤ total :=
(Finset.sum_le_sum_of_subset_of_nonneg hsub (by intros; exact Nat.zero_le _)).trans htotal
have hh := hDb (newlyReachedDescriptorList A N q a t p h) id L ρ m C J i
((policyDescriptorList A N q a t).sup descriptorLength) hcard
(fun b hb => hL b (hsub hb)) hsum (fun b hb => Finset.le_sup (f:=descriptorLength) (hsub hb)) x 0 le_rfl w hw
simpa only [norm_iteratedFDeriv_zero,pow_zero,mul_one] using hh
theorem actualResidual_graph_reached_removalND
(W : Finset (List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1)))))
(B total : ℕ) (hW : ∀ L∈W, L.length≤B) {sig : ℝ} (hsig : 0<sig)
(Cq : ℝ) (hCq : 0≤Cq) :
∃ D : ℝ, 1≤D ∧ ∀ᶠ N : ℕ in atTop,
∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ],
∀ (A : ℕ → ℝ) (q a t p h : ℕ), 0<q → p≤h →
∀ (L₁ L₂ : PolicyDescriptor → List (EuclideanSpace ℝ (Fin (n+1)) →L[ℝ] EuclideanSpace ℝ (Fin (n+1))))
(m : ℝ → ℝ) (C : ℝ) (J i : ℕ)
(f g : EuclideanSpace ℝ (Fin (n+1)) → ℂ)
(Q S : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1)))),
Measurable f → Measurable g → (∀ x, ‖f x‖≤1) → (∀ x, ‖g x‖≤1) →
(∀ b∈policyDescriptorList A N q a t, L₁ b∈W) →
(∀ b∈policyDescriptorList A N q a t, L₂ b∈W) →
(∑ b∈policyDescriptorList A N q a t, (L₁ b).length≤total) →
(∑ b∈policyDescriptorList A N q a t, (L₂ b).length≤total) →
(∀ w, ‖w‖=1 → ‖Q.1 w‖≤Cq) → (∀ w, ‖w‖=1 → ‖S.1 w‖≤Cq) →
∀ (X Y : Finset (Fin (n+1) → ℤ)) (hXY : (X.product Y).Nonempty) (M K v : ℝ) (Ψ : ℝ → ℝ),
Integrable (fun r => Ψ (r/(2:ℝ)^v)) → (∀ r, 0≤Ψ r) →
Attempt4Near.graphChildMaxND (n+1) (Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) h v Ψ X Y hXY
(originalCellTransformND ρ h M (fun x => (actualResidualSourceSymbol A N q a t p L₁ ρ m sig C J i f Q x).1))
(originalCellTransformND κ h K (fun x => (actualResidualSourceSymbol A N q a t p L₂ κ m sig C J i g S x).1)) ≤
D^(4*total)*
Attempt4Near.graphChildMaxND (n+1) (Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) h v Ψ X Y hXY
(originalCellTransformND ρ h M (fun x => (actualResidualSourceSymbol A N q a t h L₁ ρ m sig C J i f Q x).1))
(originalCellTransformND κ h K (fun x => (actualResidualSourceSymbol A N q a t h L₂ κ m sig C J i g S x).1)) := by
obtain ⟨D,hD,hDb⟩ := uniform_newlyReachedSymbol_bound W B total hW sig hsig
obtain ⟨E,hE,hEb⟩ := actual_residual_transform_uniform_package W B total hW hsig Cq hCq
refine ⟨D,hD,?_⟩
filter_upwards [hDb,hEb] with N hDb hEb
intro ρ κ hρ hκ A q a t p h hq hph L₁ L₂ m C J i f g Q S hf hg hfb hgb hL₁ hL₂ htotal₁ htotal₂ hQ hS X Y hXY M K v Ψ hΨ hΨpos
obtain ⟨hbm,hbn,hbt⟩ := hEb ρ A q a t h hq L₁ m C J i f Q hf hfb hL₁ htotal₁ hQ
obtain ⟨hcm,hcn,hct⟩ := hEb κ A q a t h hq L₂ m C J i g S hg hgb hL₂ htotal₂ hS
let ψ := fun X w => (actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L₁ ρ m N sig C J i (CurrentFalconer.ambientGridCenter h X)).1 w
let φ := fun Y w => (actualAngularProduct (newlyReachedDescriptorList A N q a t p h) id L₂ κ m N sig C J i (CurrentFalconer.ambientGridCenter h Y)).1 w
have hψ (X : Fin (n+1) → ℤ) : ∀ᵐ w ∂Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1))), ‖ψ X w‖≤D^total := by
filter_upwards [Attempt4FullFar.canonicalSphereMeasureND_ae_norm] with w hw
exact hDb A q a t p h hq L₁ ρ m C J i hL₁ htotal₁ _ w hw
have hφ (Y : Fin (n+1) → ℤ) : ∀ᵐ w ∂Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1))), ‖φ Y w‖≤D^total := by
filter_upwards [Attempt4FullFar.canonicalSphereMeasureND_ae_norm] with w hw
exact hDb A q a t p h hq L₂ κ m C J i hL₂ htotal₂ _ w hw
have hh := Attempt4Near.graphChildMaxND_bounded_factors (n+1)
(Attempt4FullFar.canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) h v X Y hXY
(originalCellTransformND ρ h M (fun x => (actualResidualSourceSymbol A N q a t h L₁ ρ m sig C J i f Q x).1))
(originalCellTransformND κ h K (fun x => (actualResidualSourceSymbol A N q a t h L₂ κ m sig C J i g S x).1))
(fun X _ => (hbt (Attempt4FullFar.ambientGridCell h X) M).1)
(fun Y _ => (hct (Attempt4FullFar.ambientGridCell h Y) K).1)
(fun X => |M⁻¹| * (ρ (Attempt4FullFar.ambientGridCell h X)).toReal*E)
(fun Y => |K⁻¹| * (κ (Attempt4FullFar.ambientGridCell h Y)).toReal*E)
(fun X _ => (hbt (Attempt4FullFar.ambientGridCell h X) M).2)
(fun Y _ => (hct (Attempt4FullFar.ambientGridCell h Y) K).2)
ψ φ (fun X _ => hψ X) (fun Y _ => hφ Y) hΨ hΨpos
have hpow : (D^total)^2*(D^total)^2=D^(4*total) := by
rw [← pow_add,← pow_mul]
congr 1
omega
rw [hpow] at hh
have hUeq : originalCellTransformND ρ h M (fun x => (actualResidualSourceSymbol A N q a t p L₁ ρ m sig C J i f Q x).1) =
fun X z => ψ X z.2 * originalCellTransformND ρ h M (fun x => (actualResidualSourceSymbol A N q a t h L₁ ρ m sig C J i f Q x).1) X z := by
funext X z
exact actualResidualTransform_grid_factorization A N q a t p h hph L₁ ρ m sig C J i f Q X M z.1 z.2
have hWeq : originalCellTransformND κ h K (fun x => (actualResidualSourceSymbol A N q a t p L₂ κ m sig C J i g S x).1) =
fun Y z => φ Y z.2 * originalCellTransformND κ h K (fun x => (actualResidualSourceSymbol A N q a t h L₂ κ m sig C J i g S x).1) Y z := by
funext Y z
exact actualResidualTransform_grid_factorization A N q a t p h hph L₂ κ m sig C J i g S Y K z.1 z.2
rw [hUeq,hWeq]
exact hh
end CurrentFalconer.TreeND
end GraphLiteralSupport
end
end
end OAI
|