lean/OAI/NumberTheory/Jacobsthal/Probability/HighPrefixKernel.lean

import Mathlib.Probability.Kernel.Composition.Prod
import OAI.NumberTheory.Jacobsthal.Paths.HighPrefixState

namespace OAI

namespace Erdos970

section

namespace ErdosPrimeInputs.HighPrefixKernel

open Set Filter MeasureTheory ProbabilityTheory
open scoped ENNReal ProbabilityTheory
open WeightedKernelMass HarmonicPrefixKernel HarmonicPrefixTilt HarmonicLongPrefixes HighPrefixState

noncomputable def kernel (ell S : ℝ) : Kernel State State :=
  ((Kernel.id : Kernel State State).prod
    ((HarmonicPrefixKernel.kernel ell).comap (fun z : State => z.1) measurable_fst)).map
      (Function.uncurry (update S))

instance kernel_isSFinite (ell S : ℝ) : IsSFiniteKernel (kernel ell S) := by
  unfold kernel
  infer_instance

lemma update_at_measurable (S : ℝ) (z : State) : Measurable (update S z) :=
  (update_measurable S).of_uncurry_left

theorem kernel_eq_map (ell S : ℝ) (z : State) :
    kernel ell S z = (HarmonicPrefixKernel.kernel ell z.1).map (update S z) := by
  apply Measure.ext_of_lintegral
  intro f hf
  rw [kernel,Kernel.map_apply _ (update_measurable S),lintegral_map hf (update_measurable S)]
  have hg : Measurable (fun p : State × ℝ => f (update S p.1 p.2)) := hf.comp (update_measurable S)
  calc
    _ = ∫⁻ t, f (update S z t) ∂HarmonicPrefixKernel.kernel ell z.1 :=
      Kernel.lintegral_id_prod (f := fun p : State × ℝ => f (update S p.1 p.2)) hg _ z
    _ = _ := (lintegral_map hf (update_at_measurable S z)).symm

lemma harmonic_ae_interval (ell B : ℝ) :
    ∀ᵐ t ∂HarmonicPrefixKernel.kernel ell B, t ∈ Ioc ell B := by
  rw [HarmonicPrefixKernel.kernel_eq_restrict,
    ae_withDensity_iff (by fun_prop : Measurable (fun t : ℝ => ENNReal.ofReal (t⁻¹)))]
  filter_upwards [self_mem_ae_restrict measurableSet_Ioc] with t ht
  intro _
  exact ht

theorem ae_update (ell S : ℝ) (z : State) {P : State → Prop}
    (hP : MeasurableSet {w | P w}) (h : ∀ t ∈ Ioc ell z.1, P (update S z t)) :
    ∀ᵐ w ∂kernel ell S z, P w := by
  rw [kernel_eq_map]
  apply (ae_map_iff (update_at_measurable S z).aemeasurable hP).mpr
  exact (harmonic_ae_interval ell z.1).mono (fun t ht => h t ht)

lemma state_schur_one {ell : ℝ} (hell : 0<ell) (S : ℝ) (z : State) :
    1+(∫⁻ w, boundE ell w.1 ∂kernel ell S z) = boundE ell z.1 := by
  rw [kernel_eq_map]
  have hf : Measurable (fun w : State => boundE ell w.1) := (boundE_measurable ell).comp measurable_fst
  rw [lintegral_map (f := fun w : State => boundE ell w.1) hf (update_at_measurable S z)]
  exact schur_one hell z.1

lemma state_schur_two {ell : ℝ} (hell : 0<ell) (S : ℝ) (z : State) :
    1+2*(∫⁻ w, squareBoundE ell w.1 ∂kernel ell S z) = squareBoundE ell z.1 := by
  rw [kernel_eq_map]
  have hf : Measurable (fun w : State => squareBoundE ell w.1) := (squareBoundE_measurable ell).comp measurable_fst
  rw [lintegral_map (f := fun w : State => squareBoundE ell w.1) hf (update_at_measurable S z)]
  exact schur_two hell z.1

theorem total_mass {ell B : ℝ} (hell : 0<ell) (hB : ell≤B) (S R : ℝ) :
    (∑' n : ℕ, mass (kernel ell S) n (initial S B R)) ≤ ENNReal.ofReal (B/ell) := by
  have h := weighted_total_bound (kernel ell S) 1 (fun z : State => boundE ell z.1)
    (fun z => by simpa only [one_mul] using (state_schur_one hell S z).le) (initial S B R)
  simpa only [one_pow,one_mul,initial,boundE,bound_on hell hB] using h

theorem weighted_mass {ell B : ℝ} (hell : 0<ell) (hB : ell≤B) (S R : ℝ) :
    (∑' n : ℕ, (2:ℝ≥0∞)^n*mass (kernel ell S) n (initial S B R)) ≤ ENNReal.ofReal ((B/ell)^2) := by
  have h := weighted_total_bound (kernel ell S) 2 (fun z : State => squareBoundE ell z.1)
    (fun z => (state_schur_two hell S z).le) (initial S B R)
  simpa only [initial,squareBoundE,bound_on hell hB] using h

theorem ae_invariant {ell B : ℝ} (hell : 0<ell) (hB : ell≤B) (S R : ℝ) :
    ∀ n : ℕ, (n:ℝ)≤S →
      ∀ᵐ z ∂(kernel ell S ^ n) (initial S B R), Invariant ell S n z := by
  intro n
  induction n with
  | zero =>
    intro _
    change ∀ᵐ z ∂Measure.dirac (initial S B R), Invariant ell S 0 z
    exact (ae_dirac_iff (invariant_measurable ell S 0)).mpr (initial_invariant hell hB)
  | succ n ih =>
    intro hn
    have hn' : (n:ℝ)≤S := by push_cast at hn; linarith
    rw [pow_succ']
    apply Kernel.ae_comp_of_ae_ae (invariant_measurable ell S (n+1))
    filter_upwards [ih hn'] with z hz
    exact ae_update ell S z (invariant_measurable ell S (n+1))
      (fun t ht => update_invariant hell n z t hz ht hn')

end ErdosPrimeInputs.HighPrefixKernel

end

end Erdos970

end OAI