import Erdos1135.Tao.Renewal.CanonicalFirstPassageEndpoint
import Erdos1135.Tao.Renewal.HoldListHorizontalMoment
import Erdos1135.Tao.Renewal.Lemma77TerminalHoldMoment

/-!
# Canonical First-Passage Horizontal Exponential Moment

The canonical stopped endpoint is pulled back to the original iid Hold master
of length `s+1`.  Stopping removes only positive horizontal increments, so a
nonnegative exponential rate is dominated by the exact full-list MGF.
-/

open scoped BigOperators

namespace Erdos1135
namespace Tao

noncomputable section

namespace TaoSection7Lemma77

open TaoSection7Case3SourceStoppingRun.Lemma79TailExpectation

/-- Endpoint selected from a full Hold master by the canonical vertical cut. -/
def lemma77CanonicalStoppedEndpoint
    (start : TaoSection7RenewalPoint) (s : ℕ)
    (full : List TaoSection7RenewalPoint) : ℕ × ℤ :=
  lemma77EndpointOfPrefix start
    (full.take (lemma79VerticalFirstPassageCut start s full))

/-- The canonical endpoint law is a direct map of the original length-`s+1`
iid Hold master.  No terminal or fresh Hold is appended. -/
theorem lemma77CanonicalFirstPassageEndpointPMF_eq_holdList_map
    (start : TaoSection7RenewalPoint) (s : ℕ) :
    lemma77CanonicalFirstPassageEndpointPMF start s =
      (taoSection7HoldListPMF (s + 1)).map
        (lemma77CanonicalStoppedEndpoint start s) := by
  unfold lemma77CanonicalFirstPassageEndpointPMF
    lemma77CanonicalFirstPassagePrefixPMF
    lemma77CanonicalStoppedEndpoint
  rw [PMF.map_comp]
  rw [← taoSection7HoldSourcePrefixListPMF_map_holdPoint_eq (s + 1)]
  rw [PMF.map_comp]
  rfl

/-- Taking an initial segment cannot increase the total horizontal increment. -/
theorem lemma77HoldPrefixHorizontalDelta_take_le
    (K : ℕ) (full : List TaoSection7RenewalPoint) :
    lemma77HoldPrefixHorizontalDelta (full.take K).length (full.take K) ≤
      lemma77HoldPrefixHorizontalDelta full.length full := by
  induction full generalizing K with
  | nil => simp [lemma77HoldPrefixHorizontalDelta]
  | cons h hs ih =>
      cases K with
      | zero => simp [lemma77HoldPrefixHorizontalDelta]
      | succ K =>
          have htail := ih K
          simp [lemma77HoldPrefixHorizontalDelta] at htail ⊢
          omega

/-- The stopped endpoint's horizontal coordinate is bounded by the complete
master-list horizontal displacement. -/
theorem lemma77CanonicalStoppedEndpoint_fst_le_fullHorizontalDelta
    (start : TaoSection7RenewalPoint) (s : ℕ)
    (full : List TaoSection7RenewalPoint) :
    (lemma77CanonicalStoppedEndpoint start s full).1 ≤
      lemma77HoldPrefixHorizontalDelta full.length full := by
  unfold lemma77CanonicalStoppedEndpoint lemma77EndpointOfPrefix
  rw [lemma77HoldPrefixIncrement_eq_horizontalDelta]
  exact lemma77HoldPrefixHorizontalDelta_take_le _ _

/-- Unrestricted countable horizontal MGF of the canonical endpoint law.

The right side has exactly `s+1` Geom(4) factors, inherited from the original
master used to define the first-passage prefix.
-/
theorem lemma77CanonicalFirstPassageEndpointPMF_horizontalExpMoment_le
    (start : TaoSection7RenewalPoint) (s : ℕ) {t : ℝ}
    (ht : 0 ≤ t) (hadm : 3 * Real.exp t < 4) :
    Summable (fun x : ℕ × ℤ =>
      (lemma77CanonicalFirstPassageEndpointPMF start s x).toReal *
        Real.exp (t * (x.1 : ℝ))) ∧
    (∑' x : ℕ × ℤ,
      (lemma77CanonicalFirstPassageEndpointPMF start s x).toReal *
        Real.exp (t * (x.1 : ℝ))) ≤
      taoSection7Geom4ExpMoment t ^ (s + 1) := by
  let p := taoSection7HoldListPMF (s + 1)
  let f := lemma77CanonicalStoppedEndpoint start s
  let W : ℕ × ℤ → ℝ := fun x => Real.exp (t * (x.1 : ℝ))
  let fullWeight : List TaoSection7RenewalPoint → ℝ := fun full =>
    Real.exp
      (t * (lemma77HoldPrefixHorizontalDelta (s + 1) full : ℝ))
  have hmaster :=
    taoSection7HoldListPMF_horizontalExpMoment (s + 1) hadm
  have hpoint : ∀ full : List TaoSection7RenewalPoint,
      (p full).toReal * W (f full) ≤
        (p full).toReal * fullWeight full := by
    intro full
    by_cases hlen : full.length = s + 1
    · apply mul_le_mul_of_nonneg_left _ ENNReal.toReal_nonneg
      apply Real.exp_le_exp.mpr
      apply mul_le_mul_of_nonneg_left _ ht
      have hdelta :=
        lemma77CanonicalStoppedEndpoint_fst_le_fullHorizontalDelta
          start s full
      rw [hlen] at hdelta
      exact_mod_cast hdelta
    · have hzero : p full = 0 := by
        exact taoSection7HoldListPMF_apply_eq_zero_of_length_ne
          (s + 1) full hlen
      simp [hzero]
  have hmaster' : Summable fun full =>
      (p full).toReal * fullWeight full := by
    simpa [p, fullWeight] using hmaster.1
  have hstopped : Summable fun full =>
      (p full).toReal * W (f full) :=
    Summable.of_nonneg_of_le
      (fun full => mul_nonneg ENNReal.toReal_nonneg (Real.exp_pos _).le)
      hpoint hmaster'
  have hstopped_le :
      (∑' full, (p full).toReal * W (f full)) ≤
        taoSection7Geom4ExpMoment t ^ (s + 1) := by
    calc
      (∑' full, (p full).toReal * W (f full)) ≤
          ∑' full, (p full).toReal * fullWeight full :=
        hstopped.tsum_le_tsum hpoint hmaster'
      _ = taoSection7Geom4ExpMoment t ^ (s + 1) := by
        simpa [p, fullWeight] using hmaster.2
  have hfinite :
      ((∑' full, p full * ENNReal.ofReal (W (f full))) : ENNReal) ≠ ⊤ := by
    have heq :
        (∑' full, p full * ENNReal.ofReal (W (f full))) =
          ∑' full, ENNReal.ofReal ((p full).toReal * W (f full)) := by
      apply tsum_congr
      intro full
      rw [ENNReal.ofReal_mul ENNReal.toReal_nonneg,
        ENNReal.ofReal_toReal (PMF.apply_ne_top p full)]
    rw [heq]
    exact hstopped.tsum_ofReal_ne_top
  have htransport := pmf_map_weighted_tsum_toReal_eq_of_finite
    p f W (fun _ => (Real.exp_pos _).le) hfinite
  rw [lemma77CanonicalFirstPassageEndpointPMF_eq_holdList_map start s]
  have hsumEq :
      (∑' x, ((p.map f) x).toReal * W x) =
        ∑' full, (p full).toReal * W (f full) := htransport.2.2
  exact ⟨by simpa [p, f, W] using htransport.1,
    hsumEq.trans_le hstopped_le⟩

end TaoSection7Lemma77

end

end Tao
end Erdos1135