import Erdos1135.Tao.Renewal.Lemma79Case3EventAdapter
import Erdos1135.Tao.Renewal.QEndpointFreshDomain

/-!
# Canonical Endpoint/Fresh Pair FSlack

This leaf places the endpoint-dependent canonical FSlack event on the exact
endpoint/fresh carrier and applies native Markov to the checked all-R moment.
-/

namespace Erdos1135
namespace Tao

noncomputable section

open TaoSection7Lemma77

namespace TaoSection7Case3SourceStoppingRun
namespace Lemma79TailExpectation

/-- Endpoint-dependent canonical FSlack event on an endpoint/fresh atom. -/
noncomputable def lemma79CanonicalEndpointFreshFSlackEvent
    (entry : TaoSection7RenewalPoint)
    (family : Set TaoSection7Triangle)
    (n : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ)
    (R Amarkov : ℕ) :
    Set ((ℕ × ℤ) × List TaoSection7RenewalPoint) :=
  {atom |
    atom.2 ∈ lemma79CanonicalFSlackEvent
      (lemma79EndpointFreshOrigin entry atom)
      family n xi epsilon (n / 2) R Amarkov}

/-- Native form of the endpoint-dependent FSlack event. -/
theorem lemma79CanonicalEndpointFreshFSlackEvent_eq_native
    (entry : TaoSection7RenewalPoint)
    (family : Set TaoSection7Triangle)
    (n : ℕ) (xi : ZMod (3 ^ n)) (epsilon : ℝ)
    (R Amarkov : ℕ) :
    lemma79CanonicalEndpointFreshFSlackEvent
        entry family n xi epsilon R Amarkov =
      {atom |
        (10 : ENNReal) ^ (Amarkov + 2) *
              ENNReal.ofReal (Real.exp epsilon) <
          lemma79HoldPathCutoffTailMomentENN
            (lemma79EndpointFreshOrigin entry atom)
            family n xi epsilon (n / 2) R atom.2} := by
  ext atom
  change
    atom.2 ∈ lemma79CanonicalFSlackEvent
        (lemma79EndpointFreshOrigin entry atom)
        family n xi epsilon (n / 2) R Amarkov ↔ _
  rw [lemma79CanonicalFSlackEvent_eq_native]
  rfl

/-- Countable endpoint/fresh FSlack mass bound.  The endpoint-dependent
origins are already integrated by the canonical all-R expectation theorem. -/
theorem lemma79CanonicalEndpointFreshPMF_fSlackEvent_outerMeasure_le
    (L : ℕ)
    (hlocalizedMass : ∀ start s,
      (1 / 2 : ℝ) ≤
        ((lemma77CanonicalFirstPassageEndpointPMF start s).toOuterMeasure
          (lemma77CanonicalLocalizedEndpointEvent s L)).toReal)
    {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ}
    (hxi : zmodThreePrimitive n xi)
    (hscalar : TaoSection7ClaimStarScalarPacket epsilon)
    (hcollar :
      taoSection7Case2HorizontalCollar L ^ 2 + (L : ℝ) ^ 2 ≤
        taoSection7TriangleSeparation epsilon ^ 2)
    (entry : TaoSection7RenewalPoint) (gap R Amarkov Aweight : ℕ)
    (hAmarkov : Amarkov = Aweight + 1) :
    (lemma79CanonicalEndpointFreshPMF (n / 2) entry gap).toOuterMeasure
        (lemma79CanonicalEndpointFreshFSlackEvent entry
          (taoSection7CanonicalTriangleFamily hxi hscalar)
          n xi epsilon R Amarkov) ≤
      ENNReal.ofReal (1 / ((10 : ℝ) ^ (Aweight + 3))) := by
  subst Amarkov
  let μ := lemma79CanonicalEndpointFreshPMF (n / 2) entry gap
  let family := taoSection7CanonicalTriangleFamily hxi hscalar
  let moment :
      ((ℕ × ℤ) × List TaoSection7RenewalPoint) → ENNReal :=
    fun atom =>
      lemma79HoldPathCutoffTailMomentENN
        (lemma79EndpointFreshOrigin entry atom)
        family n xi epsilon (n / 2) R atom.2
  let base := ENNReal.ofReal (Real.exp epsilon)
  let scale := (10 : ENNReal) ^ (Aweight + 1 + 2)
  have hbase0 : base ≠ 0 :=
    (ENNReal.ofReal_pos.mpr (Real.exp_pos epsilon)).ne'
  have hbaseTop : base ≠ ⊤ := ENNReal.ofReal_ne_top
  have hmean : lemma79PMFENNExpectation μ moment ≤ base := by
    simpa [μ, moment, base, family, lemma79EndpointFreshOrigin] using
      (lemma79CanonicalEndpointFreshPMF_all_R
        L hlocalizedMass hxi hscalar hcollar entry gap R)
  rw [lemma79CanonicalEndpointFreshFSlackEvent_eq_native]
  calc
    μ.toOuterMeasure {atom | scale * base < moment atom} ≤ scale⁻¹ := by
      simpa [mul_comm] using
        (lemma79_pmfToOuterMeasure_base_mul_scale_gt_le_inv
          μ moment base scale hbase0 hbaseTop hmean)
    _ = ENNReal.ofReal (1 / ((10 : ℝ) ^ (Aweight + 3))) := by
      dsimp [scale]
      rw [show Aweight + 1 + 2 = Aweight + 3 by omega]
      rw [one_div, ENNReal.ofReal_inv_of_pos (by positivity)]
      rw [ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 10)]
      norm_num

/-- Safe real projection of the endpoint/fresh FSlack mass bound. -/
theorem lemma79CanonicalEndpointFreshPMF_fSlackEvent_toReal_le
    (L : ℕ)
    (hlocalizedMass : ∀ start s,
      (1 / 2 : ℝ) ≤
        ((lemma77CanonicalFirstPassageEndpointPMF start s).toOuterMeasure
          (lemma77CanonicalLocalizedEndpointEvent s L)).toReal)
    {n : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ}
    (hxi : zmodThreePrimitive n xi)
    (hscalar : TaoSection7ClaimStarScalarPacket epsilon)
    (hcollar :
      taoSection7Case2HorizontalCollar L ^ 2 + (L : ℝ) ^ 2 ≤
        taoSection7TriangleSeparation epsilon ^ 2)
    (entry : TaoSection7RenewalPoint) (gap R Amarkov Aweight : ℕ)
    (hAmarkov : Amarkov = Aweight + 1) :
    ((lemma79CanonicalEndpointFreshPMF (n / 2) entry gap).toOuterMeasure
        (lemma79CanonicalEndpointFreshFSlackEvent entry
          (taoSection7CanonicalTriangleFamily hxi hscalar)
          n xi epsilon R Amarkov)).toReal ≤
      1 / ((10 : ℝ) ^ (Aweight + 3)) := by
  exact ENNReal.toReal_le_of_le_ofReal (by positivity)
    (lemma79CanonicalEndpointFreshPMF_fSlackEvent_outerMeasure_le
      L hlocalizedMass hxi hscalar hcollar
      entry gap R Amarkov Aweight hAmarkov)

end Lemma79TailExpectation
end TaoSection7Case3SourceStoppingRun

end

end Tao
end Erdos1135