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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 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Recursion.PolicyDescriptors
import OAI.MeasureTheory.FalconerCompletion.Angular.AngularHTrivialBound
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 _root_.OAI.GraphLiteralSupport.Attempt4FullFar
open scoped ContDiff ENNReal Topology
variable {n : ℕ}
theorem actual_residual_angular_terminal
{Ψ : ℝ → ℝ} (hΨ : Integrable Ψ)
(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) :
∃ H : ℝ, 0≤H ∧ ∀ᶠ N : ℕ in atTop,
∀ (ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ],
∀ (A : ℕ → ℝ) (q a₀ t₀ a : ℕ), 0<q → 0<N → a≤N →
∀ (words₁ words₂ : 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 P : SmoothFunctionND (EuclideanSpace ℝ (Fin (n+1))))
(X Y : CellPath (n+1)) (v lam sρ sκ : ℝ) (Bρ Bκ : ℕ),
RegularizedProfileData ρ N Bρ sρ lam m → RegularizedProfileData κ N Bκ sκ lam m →
X.length=a → Y.length=a → v-a≤100*(q:ℝ) →
Measurable f → Measurable g → (∀ x, ‖f x‖≤1) → (∀ x, ‖g x‖≤1) →
(∀ b∈policyDescriptorList A N q a₀ t₀, words₁ b∈W) →
(∀ b∈policyDescriptorList A N q a₀ t₀, words₂ b∈W) →
(∑ b∈policyDescriptorList A N q a₀ t₀, (words₁ b).length≤total) →
(∑ b∈policyDescriptorList A N q a₀ t₀, (words₂ b).length≤total) →
(∀ w, ‖w‖=1 → ‖Q.1 w‖≤Cq) → (∀ w, ‖w‖=1 → ‖P.1 w‖≤Cq) →
let b := fun x w => (actualResidualSourceSymbol A N q a₀ t₀ a words₁ ρ m sig C J i f Q x).1 w
let c := fun x w => (actualResidualSourceSymbol A N q a₀ t₀ a words₂ κ m sig C J i g P x).1 w
Attempt4Near.sourceAngularHND (n+1) (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) a v Ψ
(fun p => sourceAngularTransformND ρ (treeCell X) (((2:ℝ)^N)^(-m a)) b p.1 p.2)
(fun p => sourceAngularTransformND κ (treeCell Y) (((2:ℝ)^N)^(-m a)) c p.1 p.2) ≤
H*((2:ℝ)^N)^(100*(n+1:ℝ)*q/N+4*(n+2:ℝ)*lam) := by
obtain ⟨E,hE,he⟩ := actual_residual_transform_uniform_package W B total hW hsig Cq hCq
let μ := canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))
let I : ℝ := ∫ r, |Ψ r|
let H := I*E^4*(μ.real univ)^2
have hI : 0≤I := integral_nonneg (fun _ => abs_nonneg _)
refine ⟨H,by dsimp [H]; positivity,?_⟩
filter_upwards [he] with N he
intro ρ κ hρ hκ A q a₀ t₀ a hq hN ha words₁ words₂ m C J i f g Q P X Y v lam sρ sκ Bρ Bκ
hdρ hdκ hX hY hv hfm hgm hfb hgb hw₁ hw₂ ht₁ ht₂ hQ hP
let R : ℝ := (2:ℝ)^N
let S : ℝ := R^((n+2:ℝ)*lam)
have hS : 0≤S := by dsimp [S]; positivity
have hu := (he ρ A q a₀ t₀ a hq words₁ m C J i f Q hfm hfb hw₁ ht₁ hQ).2.2 (treeCell X) (R^(-m a))
have hw := (he κ A q a₀ t₀ a hq words₂ m C J i g P hgm hgb hw₂ ht₂ hP).2.2 (treeCell Y) (R^(-m a))
have hmass₁ : |(R^(-m a))⁻¹| *(ρ (treeCell X)).toReal*E ≤ S*E := by
rw [abs_of_pos (inv_pos.mpr (Real.rpow_pos_of_pos (by dsimp [R]; positivity) _))]
apply mul_le_mul_of_nonneg_right _ hE
rw [mul_comm,← div_eq_mul_inv]
simpa only [R,S,Nat.cast_add,Nat.cast_one,add_assoc,one_add_one_eq_two] using hdρ.normalized_cell_mass_le ha hX
have hmass₂ : |(R^(-m a))⁻¹| *(κ (treeCell Y)).toReal*E ≤ S*E := by
rw [abs_of_pos (inv_pos.mpr (Real.rpow_pos_of_pos (by dsimp [R]; positivity) _))]
apply mul_le_mul_of_nonneg_right _ hE
rw [mul_comm,← div_eq_mul_inv]
simpa only [R,S,Nat.cast_add,Nat.cast_one,add_assoc,one_add_one_eq_two] using hdκ.normalized_cell_mass_le ha hY
have hb₁ r : ∀ᵐ w ∂μ, ‖sourceAngularTransformND ρ (treeCell X) (R^(-m a))
(fun x w => (actualResidualSourceSymbol A N q a₀ t₀ a words₁ ρ m sig C J i f Q x).1 w) r w‖≤S*E :=
(hu.2 r).mono (fun _ hw => hw.trans hmass₁)
have hb₂ r : ∀ᵐ w ∂μ, ‖sourceAngularTransformND κ (treeCell Y) (R^(-m a))
(fun x w => (actualResidualSourceSymbol A N q a₀ t₀ a words₂ κ m sig C J i g P x).1 w) r w‖≤S*E :=
(hw.2 r).mono (fun _ hw => hw.trans hmass₂)
have hh := Attempt4Near.sourceAngularHND_trivial_bound (n+1) μ a v hΨ hu.1 hw.1 hb₁ hb₂
have hS4 : S^4=R^(4*(n+2:ℝ)*lam) := by
rw [← Real.rpow_mul_natCast (by dsimp [R]; positivity : 0≤R)]
congr 1
norm_num
ring
have hcoeff : (2:ℝ)^((↑(n+1):ℝ)*(v-a))*I*((S*E)^2*μ.real univ)*((S*E)^2*μ.real univ) =
H*((2:ℝ)^((n+1:ℝ)*(v-a))*R^(4*(n+2:ℝ)*lam)) := by
rw [← hS4]
dsimp [H]
push_cast
ring
change _ ≤ _ at hh
rw [hcoeff] at hh
apply hh.trans
apply mul_le_mul_of_nonneg_left _ (by dsimp [H]; positivity)
have hexp : (2:ℝ)^((n+1:ℝ)*(v-a)) ≤ R^(100*(n+1:ℝ)*q/N) := by
dsimp [R]
rw [← Real.rpow_natCast,← Real.rpow_mul (by norm_num : (0:ℝ)≤2)]
have hn : (N:ℝ)≠0 := by exact_mod_cast (Nat.ne_of_gt hN)
have heq : (N:ℝ)*(100*(n+1:ℝ)*q/N)=100*(n+1:ℝ)*q := by field_simp
rw [heq]
apply Real.rpow_le_rpow_of_exponent_le (by norm_num)
nlinarith [mul_le_mul_of_nonneg_left hv (show 0≤(n+1:ℝ) by positivity)]
rw [Real.rpow_add (by dsimp [R]; positivity : 0<R)]
exact mul_le_mul_of_nonneg_right hexp (by positivity)
end CurrentFalconer.TreeND
end GraphLiteralSupport
end
end
end OAI
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