import Erdos1135.Tao.Renewal.CanonicalFirstPassageEndpoint
import Mathlib.Tactic

/-!
# Proposition 7.8 Case 2 Discount Algebra

This proof leaf checks the scalar and expectation algebra in Tao's `(7.47)`.
It is specialized to the canonical first-passage endpoint law at its public
consumer, but it does not prove the exponential-moment estimate `(7.49)`, the
white-exit probability `(7.51)`, or the near-top case of Proposition 7.8.
-/

open scoped BigOperators

namespace Erdos1135
namespace Tao

noncomputable section

/-- Tao's elementary scalar discount used before `(7.47)`. -/
theorem taoSection7Prop78_exp_neg_epsilon_cube_half_le
    {epsilon : ℝ} (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1) :
    Real.exp (-(epsilon ^ 3 / 2)) ≤ 1 - epsilon ^ 3 / 4 := by
  let x : ℝ := epsilon ^ 3 / 2
  have hx0 : 0 ≤ x := by
    dsimp [x]
    positivity
  have hepsilonCube : epsilon ^ 3 ≤ 1 := by
    exact pow_le_one₀ hepsilon0 hepsilon1
  have hx1 : x ≤ 1 := by
    dsimp [x]
    linarith
  have hden : 0 < 1 + x := by linarith
  have hexp : Real.exp (-x) ≤ (1 + x)⁻¹ := by
    rw [Real.exp_neg]
    exact (inv_le_inv₀ (Real.exp_pos x) hden).2 (by
      simpa [add_comm] using Real.add_one_le_exp x)
  have hinv : (1 + x)⁻¹ ≤ 1 - x / 2 := by
    rw [inv_eq_one_div, div_le_iff₀ hden]
    nlinarith [mul_nonneg hx0 (sub_nonneg.mpr hx1)]
  calc
    Real.exp (-(epsilon ^ 3 / 2)) = Real.exp (-x) := by rfl
    _ ≤ (1 + x)⁻¹ := hexp
    _ ≤ 1 - x / 2 := hinv
    _ = 1 - epsilon ^ 3 / 4 := by
      dsimp [x]
      ring

/-- Real event mass of a PMF as the sum of its indicated point masses. -/
theorem pmf_eventMass_toReal_eq_tsum_indicator
    {alpha : Type*} (p : PMF alpha) (E : Set alpha) :
    (p.toOuterMeasure E).toReal =
      ∑' x : alpha, E.indicator (fun x => (p x).toReal) x := by
  classical
  rw [PMF.toOuterMeasure_apply, ENNReal.tsum_toReal_eq]
  · apply tsum_congr
    intro x
    by_cases hx : x ∈ E <;> simp [Set.indicator, hx]
  · intro x
    by_cases hx : x ∈ E <;>
      simp [Set.indicator, hx, PMF.apply_ne_top p x]

/-- Abstract expectation algebra behind `(7.47)`.

The exponential weight is allowed to be unbounded, but its PMF-weighted sum
must be summable. Nonnegativity of `g` ensures that a discounted event atom
loses at least `c` times its underlying PMF mass.
-/
theorem pmf_discounted_exp_expectation_le_sub_eventMass
    {alpha : Type*} (p : PMF alpha) (E : Set alpha)
    (g : alpha → ℝ) {d c : ℝ}
    (hc0 : 0 ≤ c) (hdiscount : Real.exp (-d) ≤ 1 - c)
    (hg0 : ∀ x, 0 ≤ g x)
    (hsum : Summable fun x : alpha => (p x).toReal * Real.exp (g x)) :
    (∑' x : alpha,
        (p x).toReal *
          (Real.exp (-(d * E.indicator (fun _ => (1 : ℝ)) x)) *
            Real.exp (g x))) ≤
      (∑' x : alpha, (p x).toReal * Real.exp (g x)) -
        c * (p.toOuterMeasure E).toReal := by
  classical
  let base : alpha → ℝ := fun x => (p x).toReal * Real.exp (g x)
  let discounted : alpha → ℝ := fun x =>
    (p x).toReal *
      (Real.exp (-(d * E.indicator (fun _ => (1 : ℝ)) x)) *
        Real.exp (g x))
  let eventTerm : alpha → ℝ := fun x =>
    c * E.indicator (fun x => (p x).toReal) x
  have hdiscounted0 : ∀ x, 0 ≤ discounted x := by
    intro x
    dsimp [discounted]
    positivity
  have hdiscounted_le : ∀ x, discounted x ≤ base x := by
    intro x
    dsimp [discounted, base]
    by_cases hx : x ∈ E
    · simp only [Set.indicator_of_mem hx, mul_one]
      have hfactor : Real.exp (-d) ≤ 1 :=
        hdiscount.trans (sub_le_self 1 hc0)
      have hinner :
          Real.exp (-d) * Real.exp (g x) ≤ Real.exp (g x) := by
        calc
          Real.exp (-d) * Real.exp (g x) ≤ 1 * Real.exp (g x) :=
            mul_le_mul_of_nonneg_right hfactor (Real.exp_nonneg _)
          _ = Real.exp (g x) := one_mul _
      exact mul_le_mul_of_nonneg_left hinner ENNReal.toReal_nonneg
    · simp [Set.indicator, hx]
  have hdiscountedSum : Summable discounted :=
    Summable.of_nonneg_of_le hdiscounted0 hdiscounted_le (by
      simpa [base] using hsum)
  have hpSum : Summable fun x : alpha => (p x).toReal :=
    ENNReal.summable_toReal p.tsum_coe_ne_top
  have heventSum : Summable eventTerm := by
    have hi := hpSum.indicator E
    refine (hi.mul_left c).congr ?_
    intro x
    simp only [eventTerm]
  have hpoint : ∀ x, discounted x + eventTerm x ≤ base x := by
    intro x
    have hp0 : 0 ≤ (p x).toReal := ENNReal.toReal_nonneg
    by_cases hx : x ∈ E
    · have heg : 1 ≤ Real.exp (g x) := Real.one_le_exp (hg0 x)
      have hscalar :
          Real.exp (-d) * Real.exp (g x) + c ≤ Real.exp (g x) := by
        calc
          Real.exp (-d) * Real.exp (g x) + c ≤
              (1 - c) * Real.exp (g x) + c := by
            gcongr
          _ ≤ Real.exp (g x) := by
            nlinarith [mul_nonneg hc0 (sub_nonneg.mpr heg)]
      dsimp [discounted, eventTerm, base]
      simp only [Set.indicator_of_mem hx, mul_one]
      calc
        (p x).toReal * (Real.exp (-d) * Real.exp (g x)) +
            c * (p x).toReal =
            (p x).toReal *
              (Real.exp (-d) * Real.exp (g x) + c) := by ring
        _ ≤ (p x).toReal * Real.exp (g x) :=
          mul_le_mul_of_nonneg_left hscalar hp0
    · dsimp [discounted, eventTerm, base]
      simp [Set.indicator, hx]
  have htotal :
      (∑' x : alpha, discounted x) + ∑' x : alpha, eventTerm x ≤
        ∑' x : alpha, base x := by
    rw [← hdiscountedSum.tsum_add heventSum]
    exact (hdiscountedSum.add heventSum).tsum_le_tsum hpoint (by
      simpa [base] using hsum)
  have heventMass :
      (∑' x : alpha, eventTerm x) = c * (p.toOuterMeasure E).toReal := by
    calc
      (∑' x : alpha, eventTerm x) =
          c * ∑' x : alpha, E.indicator (fun x => (p x).toReal) x := by
        rw [← tsum_mul_left]
      _ = c * (p.toOuterMeasure E).toReal := by
        rw [pmf_eventMass_toReal_eq_tsum_indicator]
  dsimp [discounted, base] at htotal ⊢
  rw [heventMass] at htotal
  linarith

/-- Canonical first-passage specialization of Tao's `(7.47)` bound. -/
theorem lemma77CanonicalFirstPassageEndpointPMF_discount_le_747
    (start : TaoSection7RenewalPoint) (s : ℕ)
    (whiteExit : Set (ℕ × ℤ)) (g : (ℕ × ℤ) → ℝ)
    {epsilon : ℝ} (hepsilon0 : 0 ≤ epsilon) (hepsilon1 : epsilon ≤ 1)
    (hg0 : ∀ x, 0 ≤ g x)
    (hsum : Summable fun x : ℕ × ℤ =>
      (TaoSection7Lemma77.lemma77CanonicalFirstPassageEndpointPMF start s x).toReal *
        Real.exp (g x)) :
    (∑' x : ℕ × ℤ,
      (TaoSection7Lemma77.lemma77CanonicalFirstPassageEndpointPMF start s x).toReal *
        (Real.exp
            (-(epsilon ^ 3 / 2 *
              whiteExit.indicator (fun _ => (1 : ℝ)) x)) *
          Real.exp (g x))) ≤
      (∑' x : ℕ × ℤ,
        (TaoSection7Lemma77.lemma77CanonicalFirstPassageEndpointPMF start s x).toReal *
          Real.exp (g x)) -
        epsilon ^ 3 / 4 *
          ((TaoSection7Lemma77.lemma77CanonicalFirstPassageEndpointPMF start s).toOuterMeasure
            whiteExit).toReal := by
  apply pmf_discounted_exp_expectation_le_sub_eventMass
    (p := TaoSection7Lemma77.lemma77CanonicalFirstPassageEndpointPMF start s)
    (E := whiteExit) (g := g)
    (d := epsilon ^ 3 / 2) (c := epsilon ^ 3 / 4)
  · positivity
  · exact taoSection7Prop78_exp_neg_epsilon_cube_half_le
      hepsilon0 hepsilon1
  · exact hg0
  · exact hsum

end

end Tao
end Erdos1135