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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 | import Mathlib
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.FalconerCompletion.Graphs.PairAmbientGoodRadialIffDistanceGood
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 ENNReal Topology Classical
variable {n : ℕ}
theorem occupiedFar_count_terminal
(ρ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ]
(m : ℝ → ℝ) (N q b t : ℕ) (sig C : ℝ) (J i : ℕ)
{B : ℕ} {s lam : ℝ} (hd : RegularizedProfileData ρ N B s lam m)
(hN : 0<N) (hb : b≤N) (ht : t≤b+100*q)
(P : CellPath (n+1)) (hab : P.length≤b)
{K : ℝ} (hK : 32*(n+2:ℝ)≤K)
(p : (Fin (n+1) → ℤ) × (Fin (n+1) → ℤ))
(hp : p∈occupiedFarPairsND ρ P.length b K (fun j => (pathIndex P j:ℤ))) :
(positiveDistanceCountND
((gridPairMeasureND ρ b p).restrict {z | euclideanPairDirectionND z ∈ actualPairAmbientGood
(adjustedProfile (d := n+1) m N) N q b t ρ m sig C J i z})
(n+1) t b (((2:ℝ)^N)^(-m b))).toReal ≤
((2:ℝ)^N)^(100*(q:ℝ)/N+4*(n+2:ℝ)*lam) := by
obtain ⟨X,Y,hX,hY,hXa,hYa,hXp,hYp,hμX,hμY,he⟩ :=
occupiedFarPair_actual_state ρ m N q b t sig C J i P hab hK p hp
rw [he]
have hh := PositiveDistanceState.actual_distance_terminal_rpow
(actualChildState ρ m N q b t sig C J i X Y) hd hN hb hX hY ht
simpa only [actualChildState,Nat.cast_add,Nat.cast_one,add_assoc,one_add_one_eq_two] using hh
theorem actualFarChildMaximum_terminal
(ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ]
(m : ℝ → ℝ) (N q a b t : ℕ) (sig C : ℝ) (J i : ℕ)
{Bρ Bκ : ℕ} {sρ sκ lam : ℝ}
(hdρ : RegularizedProfileData ρ N Bρ sρ lam m)
(hdκ : RegularizedProfileData κ N Bκ sκ lam m)
(hN : 0<N) (hab : a≤b) (hb : b≤N) (ht : t≤b+100*q)
(X Y : CellPath (n+1)) (hX : X.length=a) (hY : Y.length=a)
{K : ℝ} (hK : 32*(n+2:ℝ)≤K) :
(farChildMaximumND ρ κ b
(occupiedFarPairsND ρ a b K (fun j => (pathIndex X j:ℤ)))
(occupiedFarPairsND κ a b K (fun j => (pathIndex Y j:ℤ)))
t (((2:ℝ)^N)^(-m b))
(actualPairAmbientGood (adjustedProfile (d := n+1) m N) N q b t ρ m sig C J i)
(actualPairAmbientGood (adjustedProfile (d := n+1) m N) N q b t κ m sig C J i)).toReal ≤
((2:ℝ)^N)^(100*(q:ℝ)/N+4*(n+2:ℝ)*lam) := by
let U := ((2:ℝ)^N)^(100*(q:ℝ)/N+4*(n+2:ℝ)*lam)
have hU : 0≤U := by positivity
have one (μ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure μ]
{B : ℕ} {s : ℝ} (hd : RegularizedProfileData μ N B s lam m)
(P : CellPath (n+1)) (hP : P.length=a) :
(occupiedFarPairsND μ a b K (fun j => (pathIndex P j:ℤ))).sup
(fun p => positiveDistanceCountND
((gridPairMeasureND μ b p).restrict {z | euclideanPairDirectionND z ∈ actualPairAmbientGood
(adjustedProfile (d := n+1) m N) N q b t μ m sig C J i z})
(n+1) t b (((2:ℝ)^N)^(-m b))) ≤ ENNReal.ofReal U := by
apply Finset.sup_le
intro p hp
have ht' := occupiedFar_count_terminal μ m N q b t sig C J i hd hN hb ht P
(hP ▸ hab) hK p (by rwa [hP])
apply (ENNReal.toReal_le_toReal (positiveDistanceCountND_ne_top _ _ _ _ _)
ENNReal.ofReal_ne_top).mp
simpa only [ENNReal.toReal_ofReal hU] using ht'
have hh : farChildMaximumND ρ κ b
(occupiedFarPairsND ρ a b K (fun j => (pathIndex X j:ℤ)))
(occupiedFarPairsND κ a b K (fun j => (pathIndex Y j:ℤ)))
t (((2:ℝ)^N)^(-m b))
(actualPairAmbientGood (adjustedProfile (d := n+1) m N) N q b t ρ m sig C J i)
(actualPairAmbientGood (adjustedProfile (d := n+1) m N) N q b t κ m sig C J i) ≤
ENNReal.ofReal U := sup_le (one ρ hdρ X hX) (one κ hdκ Y hY)
simpa only [ENNReal.toReal_ofReal hU] using ENNReal.toReal_mono ENNReal.ofReal_ne_top hh
end CurrentFalconer.TreeND
end GraphLiteralSupport
end
end
end OAI
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