| import Mathlib.MeasureTheory.Integral.Bochner.Basic |
| import OAI.NumberTheory.Jacobsthal.Partitions.CompactTestGeometry |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
| open scoped _root_.Erdos970 |
|
|
| section |
|
|
| namespace NumberTheoryLean.CompactTestMeasurable |
|
|
| open _root_.Set _root_.Finset _root_.MeasureTheory ProbabilityTheory |
| open scoped ENNReal |
| open FinitePathGeometry FinitePathMeasures PrimeHistories PrimeKilledChain |
| open CompactTestGeometry RegeneratingInverseBands ActualCouplingUpdates ArrivalKernelGeometry |
|
|
| variable {w ell S : ℝ} {start : Node} |
|
|
| theorem primeTest_measurable (F : Side → ℝ×ℝ → ℝ) : |
| Measurable (primeTest (w:=w) (ell:=ell) (S:=S) (start:=start) F) := measurable_of_countable _ |
|
|
| theorem continuousTest_measurable (v : ℝ) (F : Side → ℝ×ℝ → ℝ) (hF : ∀ i,Measurable (F i)) : |
| Measurable (continuousTest v F) := by |
| have hcoord : Measurable (fun z : CostState => (gapValue v z,stateRatio z.1)) := |
| (gapValue_measurable v).prodMk (stateRatio_measurable.comp measurable_fst) |
| have hside : MeasurableSet {z : CostState | stateSide z.1 = .even} := |
| measurable_fst (side_set_measurable .even) |
| have heq : (fun z : CostState => F (stateSide z.1) (gapValue v z,stateRatio z.1)) = |
| (fun z => if stateSide z.1 = .even then F .even (gapValue v z,stateRatio z.1) |
| else F .odd (gapValue v z,stateRatio z.1)) := by |
| funext z |
| rcases z with ⟨s,t⟩ |
| cases s <;> rfl |
| have hb : Measurable (fun z : CostState => F (stateSide z.1) (gapValue v z,stateRatio z.1)) := by |
| rw [heq] |
| exact Measurable.ite hside ((hF .even).comp hcoord) ((hF .odd).comp hcoord) |
| exact hb.sumElim measurable_const |
|
|
| theorem primeTest_bound {F : Side → ℝ×ℝ → ℝ} {A : ℝ} (hA : 0 ≤ A) |
| (hF : ∀ i x,|F i x| ≤ A) (z : ChainState w ell S start) : |primeTest F z| ≤ A := by |
| cases z with |
| | none => simpa only [primeTest,abs_zero] using hA |
| | some h => exact hF _ _ |
|
|
| theorem continuousTest_bound {F : Side → ℝ×ℝ → ℝ} {A : ℝ} (hA : 0 ≤ A) |
| (hF : ∀ i x,|F i x| ≤ A) (v : ℝ) (z : CemeteryKernel.Space CostState) : |
| |continuousTest v F z| ≤ A := by |
| cases z with |
| | inr u => simpa only [continuousTest,abs_zero] using hA |
| | inl y => exact hF _ _ |
|
|
| theorem test_difference_bound {F : Side → ℝ×ℝ → ℝ} {A : ℝ} (hA : 0 ≤ A) |
| (hF : ∀ i x,|F i x| ≤ A) (v : ℝ) (p : ChainState w ell S start) (z : CemeteryKernel.Space CostState) : |
| |primeTest F p-continuousTest v F z| ≤ 2*A := by |
| exact (abs_sub _ _).trans (by linarith [primeTest_bound hA hF p,continuousTest_bound hA hF v z]) |
|
|
| section Generic |
| variable {α : Type*} [MeasurableSpace α] {μ : Measure α} |
|
|
| theorem bounded_integrable [IsFiniteMeasure μ] {f : α → ℝ} (hf : Measurable f) |
| {A : ℝ} (hb : ∀ x,|f x| ≤ A) : Integrable f μ := by |
| refine (integrable_const A).mono' hf.aestronglyMeasurable ?_ |
| exact Filter.Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using hb x) |
|
|
| theorem integral_difference_enorm {f g : α → ℝ} (hf : Integrable f μ) (hg : Integrable g μ) : |
| ENNReal.ofReal |(∫ x,f x ∂μ)-(∫ x,g x ∂μ)| ≤ ∫⁻ x,ENNReal.ofReal |f x-g x| ∂μ := by |
| rw [← integral_sub hf hg] |
| simpa only [Real.enorm_eq_ofReal_abs] using enorm_integral_le_lintegral_enorm (fun x => f x-g x) |
|
|
| theorem finite_integral_difference (μ : ℕ → Measure α) (f g : α → ℝ) (N : ℕ) |
| (hf : ∀ n ∈ range N,Integrable f (μ n)) (hg : ∀ n ∈ range N,Integrable g (μ n)) : |
| ENNReal.ofReal |(∑ n ∈ range N,∫ x,f x ∂μ n)-(∑ n ∈ range N,∫ x,g x ∂μ n)| ≤ |
| ∑ n ∈ range N,∫⁻ x,ENNReal.ofReal |f x-g x| ∂μ n := by |
| rw [← Finset.sum_sub_distrib] |
| refine (ENNReal.ofReal_le_ofReal (Finset.abs_sum_le_sum_abs _ _)).trans ?_ |
| rw [ENNReal.ofReal_sum_of_nonneg (fun _ _ => abs_nonneg _)] |
| exact Finset.sum_le_sum (fun n hn => integral_difference_enorm (hf n hn) (hg n hn)) |
| end Generic |
|
|
| end NumberTheoryLean.CompactTestMeasurable |
|
|
| end |
|
|
| end Erdos970 |
|
|
| end OAI |
|
|
