openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Recursion /OccupiedFarCountTerminal.lean
| 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 | |