| import OAI.NumberTheory.Jacobsthal.Estimates.AdmittedHighRemoval |
| import OAI.NumberTheory.Jacobsthal.Probability.SubMarkovFlagPowers |
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|
| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| section |
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| namespace NumberTheoryLean.ContinuousHighPartition |
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|
| open _root_.Set _root_.MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open FinitePathMeasures RegeneratingInverseBands ArrivalKernelGeometry |
| open AdmittedHarmonicPaths LowStateHorizon SubMarkovFlagPowers |
| open PersistentFailureFlag FlaggedOccupationBound |
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|
| def high (S : ℝ) (z : CostState) : Prop := S < stateRatio z.1 |
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| theorem high_measurable (S : ℝ) : MeasurableSet {z | high S z} := |
| measurableSet_lt measurable_const (stateRatio_measurable.comp measurable_fst) |
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| 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] |
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| noncomputable def tiltedFlagged (v ell S : ℝ) : Kernel (CostState×Bool) (CostState×Bool) := |
| marked (admittedTilted v ell) (high_measurable S) |
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| 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] |
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| end NumberTheoryLean.ContinuousHighPartition |
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| end |
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| end Erdos970 |
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| end OAI |
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