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f89a930 | 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 | import OAI.NumberTheory.Jacobsthal.Estimates.AdmittedHighRemoval
import OAI.NumberTheory.Jacobsthal.Probability.SubMarkovFlagPowers
namespace OAI
namespace Erdos970
open scoped _root_.Erdos970
section
namespace NumberTheoryLean.ContinuousHighPartition
open _root_.Set _root_.MeasureTheory ProbabilityTheory
open scoped ENNReal
open FinitePathMeasures RegeneratingInverseBands ArrivalKernelGeometry
open AdmittedHarmonicPaths LowStateHorizon SubMarkovFlagPowers
open PersistentFailureFlag FlaggedOccupationBound
def high (S : ℝ) (z : CostState) : Prop := S < stateRatio z.1
theorem high_measurable (S : ℝ) : MeasurableSet {z | high S z} :=
measurableSet_lt measurable_const (stateRatio_measurable.comp measurable_fst)
theorem good_admitted_eq_low (K : Kernel CostState CostState) (v ell S : ℝ) :
goodKernel (K.restrict (arrivalSet_measurable v ell)) (high_measurable S) = lowKernel K v ell S := by
ext z : 1
simp only [goodKernel,lowKernel,Kernel.restrict_apply,Measure.restrict_restrict (high_measurable S).compl]
congr 1
ext y
simp only [lowDomain,mem_inter_iff,mem_compl_iff,mem_ofPred_eq,high,not_lt]
exact and_comm
theorem marked_harmonic_eq (v ell S : ℝ) :
marked (admittedHarmonic v ell) (high_measurable S) = FlaggedHarmonicPaths.kernel v ell S := by
ext z : 1
rw [marked_eq_map_sfinite,FlaggedHarmonicPaths.kernel_eq_map]
congr 1
funext y
by_cases hy : S < stateRatio y.1 <;>
cases z.2 <;> simp [FlaggedHarmonicPaths.flagUpdate,badBit,high,hy]
noncomputable def tiltedFlagged (v ell S : ℝ) : Kernel (CostState×Bool) (CostState×Bool) :=
marked (admittedTilted v ell) (high_measurable S)
theorem harmonic_finite_partition (v ell S : ℝ) {F : CostState → ℝ≥0∞} (hF : Measurable F)
(n : ℕ) (z : CostState) (hz : stateRatio z.1 ≤ S) :
(∫⁻ y,F y ∂((admittedHarmonic v ell)^n) z) =
(∫⁻ y,F y ∂((lowKernel HarmonicWeightKernel.harmonicKernel v ell S)^n) z)+
(∫⁻ y,selected F y ∂((FlaggedHarmonicPaths.kernel v ell S)^n) (FlaggedHarmonicPaths.initial S z)) := by
have h := finite_integral_partition (admittedHarmonic v ell) (high_measurable S) hF n z
have he : goodKernel (admittedHarmonic v ell) (high_measurable S) =
lowKernel HarmonicWeightKernel.harmonicKernel v ell S :=
good_admitted_eq_low HarmonicWeightKernel.harmonicKernel v ell S
rw [he,marked_harmonic_eq] at h
simpa only [FlaggedHarmonicPaths.initial,ite_eq_right (not_lt.mpr hz)] using h
theorem tilted_finite_partition (v ell S : ℝ) {F : CostState → ℝ≥0∞} (hF : Measurable F)
(n : ℕ) (z : CostState) :
(∫⁻ y,F y ∂((admittedTilted v ell)^n) z) =
(∫⁻ y,F y ∂((lowKernel costKernel v ell S)^n) z)+
(∫⁻ y,selected F y ∂((tiltedFlagged v ell S)^n) (z,false)) := by
have h := finite_integral_partition (admittedTilted v ell) (high_measurable S) hF n z
have he : goodKernel (admittedTilted v ell) (high_measurable S) = lowKernel costKernel v ell S :=
good_admitted_eq_low costKernel v ell S
rw [he] at h
exact h
theorem harmonic_high_compact_bound (v ell S : ℝ) (z : CostState) {F : CostState → ℝ≥0∞}
{D : ℝ≥0∞} (hD : D ≠ ∞) {K : ℝ} (hbound : ∀ y,F y ≤ D)
(hsupport : ∀ y,K < gapValue v y → F y=0) :
(∑' n : ℕ,∫⁻ y,selected F y ∂((FlaggedHarmonicPaths.kernel v ell S)^n) (FlaggedHarmonicPaths.initial S z)) ≤
D*AdmittedHighRemoval.lowHighMass v ell S z K := by
have hA : MeasurableSet {y : CostState×Bool | y.2=true ∧ gapValue v y.1 ≤ K} :=
(measurable_snd (measurableSet_singleton true)).inter
(measurableSet_le ((gapValue_measurable v).comp measurable_fst) measurable_const)
have hpoint : ∀ y : CostState×Bool,selected F y ≤
D*{y : CostState×Bool | y.2=true ∧ gapValue v y.1 ≤ K}.indicator (fun _ => (1:ℝ≥0∞)) y := by
rintro ⟨y,b⟩
cases b with
| false => simp [selected,failed]
| true =>
by_cases hy : gapValue v y ≤ K
· simp [selected,failed,hy,hbound y]
· simp [selected,failed,hy,hsupport y (lt_of_not_ge hy)]
unfold AdmittedHighRemoval.lowHighMass
rw [← ENNReal.tsum_mul_left]
apply ENNReal.tsum_le_tsum
intro n
calc
_ ≤ ∫⁻ y,D*{y : CostState×Bool | y.2=true ∧ gapValue v y.1 ≤ K}.indicator (fun _ => (1:ℝ≥0∞)) y
∂((FlaggedHarmonicPaths.kernel v ell S)^n) (FlaggedHarmonicPaths.initial S z) := lintegral_mono hpoint
_ = _ := by rw [lintegral_const_mul' _ _ hD,lintegral_indicator hA,setLIntegral_const,one_mul]
end NumberTheoryLean.ContinuousHighPartition
end
end Erdos970
end OAI
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