import Erdos1135.Tao.Renewal.HoldExpectation
import Erdos1135.Tao.Renewal.HoldIID
import Mathlib.Tactic

/-!
# Section 7 Actual Q Finite Approximants

This module records terminal-one finite approximants for Tao's actual Section 7
`Q` function, plus the statement surface that a supplied `Q` is identified as
their finite limit.  It does not prove convergence of the approximants, construct
the infinite product, or prove Proposition 7.3.
-/

open scoped BigOperators Topology

namespace Erdos1135
namespace Tao

private theorem pmf_toReal_summable {α : Type*} (p : PMF α) :
    Summable fun a : α => (p a).toReal := by
  refine ENNReal.summable_toReal ?_
  rw [PMF.tsum_coe p]
  norm_num

private theorem pmf_toReal_tsum {α : Type*} (p : PMF α) :
    (∑' a : α, (p a).toReal) = 1 := by
  rw [← ENNReal.tsum_toReal_eq]
  · rw [PMF.tsum_coe p]
    norm_num
  · intro a
    exact PMF.apply_ne_top p a

theorem taoSection7HoldListPMF_summable_toReal
    (K : ℕ) :
    Summable fun hs : List TaoSection7RenewalPoint =>
      (taoSection7HoldListPMF K hs).toReal :=
  pmf_toReal_summable (taoSection7HoldListPMF K)

theorem taoSection7HoldListPMF_tsum_toReal
    (K : ℕ) :
    (∑' hs : List TaoSection7RenewalPoint,
      (taoSection7HoldListPMF K hs).toReal) = 1 :=
  pmf_toReal_tsum (taoSection7HoldListPMF K)

/--
Terminal-one finite approximation to Tao's actual `Q(j,l)`: sample `K` future
Hold increments, multiply the finite `QFinite` factors, and continue with
terminal value `1` after the last included increment.
-/
noncomputable def taoSection7QFiniteTerminalOneApprox
    (K : ℕ) (epsilon : ℝ) (W : TaoSection7RenewalPoint → Prop)
    (p : TaoSection7RenewalPoint) : ℝ :=
  ∑' hs : List TaoSection7RenewalPoint,
    (taoSection7HoldListPMF K hs).toReal *
      taoSection7QFinite epsilon W p hs

theorem taoSection7QFiniteTerminalOneApprox_summable
    {K : ℕ} {epsilon : ℝ} (hepsilon : 0 ≤ epsilon)
    (W : TaoSection7RenewalPoint → Prop) (p : TaoSection7RenewalPoint) :
    Summable fun hs : List TaoSection7RenewalPoint =>
      (taoSection7HoldListPMF K hs).toReal *
        taoSection7QFinite epsilon W p hs := by
  have hmass :
      Summable fun hs : List TaoSection7RenewalPoint =>
        (taoSection7HoldListPMF K hs).toReal :=
    taoSection7HoldListPMF_summable_toReal K
  refine Summable.of_norm_bounded hmass ?_
  intro hs
  have hm0 : 0 ≤ (taoSection7HoldListPMF K hs).toReal := ENNReal.toReal_nonneg
  have hq0 : 0 ≤ taoSection7QFinite epsilon W p hs :=
    taoSection7QFinite_nonneg epsilon W p hs
  have hq1 : taoSection7QFinite epsilon W p hs ≤ 1 :=
    taoSection7QFinite_le_one hepsilon W p hs
  have hqabs : |taoSection7QFinite epsilon W p hs| ≤ 1 :=
    abs_le.mpr ⟨by linarith [hq0], hq1⟩
  calc
    ‖(taoSection7HoldListPMF K hs).toReal *
        taoSection7QFinite epsilon W p hs‖
        = |(taoSection7HoldListPMF K hs).toReal *
            taoSection7QFinite epsilon W p hs| := Real.norm_eq_abs _
    _ = (taoSection7HoldListPMF K hs).toReal *
          |taoSection7QFinite epsilon W p hs| := by
        rw [abs_mul, abs_of_nonneg hm0]
    _ ≤ (taoSection7HoldListPMF K hs).toReal * 1 := by
        exact mul_le_mul_of_nonneg_left hqabs hm0
    _ = (taoSection7HoldListPMF K hs).toReal := by ring

theorem taoSection7QFiniteTerminalOneApprox_nonneg
    (K : ℕ) (epsilon : ℝ) (W : TaoSection7RenewalPoint → Prop)
    (p : TaoSection7RenewalPoint) :
    0 ≤ taoSection7QFiniteTerminalOneApprox K epsilon W p := by
  unfold taoSection7QFiniteTerminalOneApprox
  exact tsum_nonneg fun hs =>
    mul_nonneg ENNReal.toReal_nonneg
      (taoSection7QFinite_nonneg epsilon W p hs)

theorem taoSection7QFiniteTerminalOneApprox_le_one
    {K : ℕ} {epsilon : ℝ} (hepsilon : 0 ≤ epsilon)
    (W : TaoSection7RenewalPoint → Prop) (p : TaoSection7RenewalPoint) :
    taoSection7QFiniteTerminalOneApprox K epsilon W p ≤ 1 := by
  unfold taoSection7QFiniteTerminalOneApprox
  have hterm :
      Summable fun hs : List TaoSection7RenewalPoint =>
        (taoSection7HoldListPMF K hs).toReal *
          taoSection7QFinite epsilon W p hs :=
    taoSection7QFiniteTerminalOneApprox_summable hepsilon W p
  have hmass :
      Summable fun hs : List TaoSection7RenewalPoint =>
        (taoSection7HoldListPMF K hs).toReal :=
    taoSection7HoldListPMF_summable_toReal K
  calc
    (∑' hs : List TaoSection7RenewalPoint,
        (taoSection7HoldListPMF K hs).toReal *
          taoSection7QFinite epsilon W p hs)
        ≤ ∑' hs : List TaoSection7RenewalPoint,
            (taoSection7HoldListPMF K hs).toReal := by
          exact hterm.tsum_le_tsum
            (fun hs => by
              have hm0 : 0 ≤ (taoSection7HoldListPMF K hs).toReal :=
                ENNReal.toReal_nonneg
              have hq1 : taoSection7QFinite epsilon W p hs ≤ 1 :=
                taoSection7QFinite_le_one hepsilon W p hs
              calc
                (taoSection7HoldListPMF K hs).toReal *
                    taoSection7QFinite epsilon W p hs
                    ≤ (taoSection7HoldListPMF K hs).toReal * 1 := by
                    exact mul_le_mul_of_nonneg_left hq1 hm0
                _ = (taoSection7HoldListPMF K hs).toReal := by ring)
            hmass
    _ = 1 := taoSection7HoldListPMF_tsum_toReal K

theorem taoSection7QFiniteTerminalOneApprox_zero
    (epsilon : ℝ) (W : TaoSection7RenewalPoint → Prop)
    (p : TaoSection7RenewalPoint) :
    taoSection7QFiniteTerminalOneApprox 0 epsilon W p =
      taoSection7QWhiteFactor epsilon W p := by
  unfold taoSection7QFiniteTerminalOneApprox
  rw [tsum_eq_single ([] : List TaoSection7RenewalPoint)]
  · simp [taoSection7HoldListPMF_zero_apply_nil, taoSection7QFinite_nil]
  · intro hs hne
    cases hs with
    | nil => exact (hne rfl).elim
    | cons h hs =>
        rw [taoSection7HoldListPMF_zero_apply_cons]
        simp

/--
Deterministic prefix monotonicity: adding more sampled Hold increments to a
finite product can only decrease the value when `epsilon >= 0`.
-/
theorem taoSection7QFinite_append_le_left
    {epsilon : ℝ} (hepsilon : 0 ≤ epsilon)
    (W : TaoSection7RenewalPoint → Prop) :
    ∀ (p : TaoSection7RenewalPoint)
      (xs ys : List TaoSection7RenewalPoint),
      taoSection7QFinite epsilon W p (xs ++ ys) ≤
        taoSection7QFinite epsilon W p xs := by
  intro p xs
  induction xs generalizing p with
  | nil =>
      intro ys
      cases ys with
      | nil =>
          rfl
      | cons h hs =>
          rw [List.nil_append, taoSection7QFinite_cons, taoSection7QFinite_nil]
          calc
            taoSection7QWhiteFactor epsilon W p *
                taoSection7QFinite epsilon W (p + h) hs
                ≤ taoSection7QWhiteFactor epsilon W p * 1 := by
                exact mul_le_mul_of_nonneg_left
                  (taoSection7QFinite_le_one hepsilon W (p + h) hs)
                  (taoSection7QWhiteFactor_nonneg epsilon W p)
            _ = taoSection7QWhiteFactor epsilon W p := by ring
  | cons h xs ih =>
      intro ys
      rw [List.cons_append, taoSection7QFinite_cons, taoSection7QFinite_cons]
      exact mul_le_mul_of_nonneg_left (ih (p + h) ys)
        (taoSection7QWhiteFactor_nonneg epsilon W p)

theorem taoSection7_tsum_list_eq_tsum_cons_of_nil_zero
    {f : List TaoSection7RenewalPoint → ℝ} (h0 : f [] = 0) :
    (∑' hs : List TaoSection7RenewalPoint, f hs) =
      ∑' x : TaoSection7RenewalPoint × List TaoSection7RenewalPoint,
        f (x.1 :: x.2) := by
  classical
  let g : TaoSection7RenewalPoint × List TaoSection7RenewalPoint →
      List TaoSection7RenewalPoint := fun x => x.1 :: x.2
  have hg : Function.Injective g := by
    intro x y hxy
    cases x with
    | mk xh xt =>
        cases y with
        | mk yh yt =>
            simp [g] at hxy ⊢
            exact hxy
  have hsupp : Function.support f ⊆ Set.range g := by
    intro hs hhs
    cases hs with
    | nil =>
        exfalso
        exact hhs h0
    | cons h hs =>
        exact ⟨(h, hs), rfl⟩
  exact (hg.tsum_eq (f := f) hsupp).symm

theorem taoSection7HoldPMF_mul_holdListPMF_summable_toReal
    (K : ℕ) :
    Summable fun x : TaoSection7RenewalPoint × List TaoSection7RenewalPoint =>
      (taoSection7HoldPMF x.1).toReal *
        (taoSection7HoldListPMF K x.2).toReal := by
  have hf : Summable fun h : TaoSection7RenewalPoint =>
      (taoSection7HoldPMF h).toReal := taoSection7HoldPMF_summable_toReal
  have hg : Summable fun hs : List TaoSection7RenewalPoint =>
      (taoSection7HoldListPMF K hs).toReal :=
    taoSection7HoldListPMF_summable_toReal K
  exact Summable.mul_of_nonneg
    (f := fun h : TaoSection7RenewalPoint => (taoSection7HoldPMF h).toReal)
    (g := fun hs : List TaoSection7RenewalPoint =>
      (taoSection7HoldListPMF K hs).toReal)
    hf hg
    (fun _h => ENNReal.toReal_nonneg)
    (fun _hs => ENNReal.toReal_nonneg)

theorem taoSection7HoldPMF_mul_holdListPMF_mul_QFinite_summable
    {epsilon : ℝ} (hepsilon : 0 ≤ epsilon)
    (K : ℕ) (W : TaoSection7RenewalPoint → Prop)
    (p : TaoSection7RenewalPoint) :
    Summable fun x : TaoSection7RenewalPoint × List TaoSection7RenewalPoint =>
      (taoSection7HoldPMF x.1).toReal *
        (taoSection7HoldListPMF K x.2).toReal *
        taoSection7QFinite epsilon W (p + x.1) x.2 := by
  have hmass :
      Summable fun x : TaoSection7RenewalPoint × List TaoSection7RenewalPoint =>
        (taoSection7HoldPMF x.1).toReal *
          (taoSection7HoldListPMF K x.2).toReal :=
    taoSection7HoldPMF_mul_holdListPMF_summable_toReal K
  refine Summable.of_norm_bounded hmass ?_
  intro x
  have hm0 : 0 ≤ (taoSection7HoldPMF x.1).toReal *
      (taoSection7HoldListPMF K x.2).toReal :=
    mul_nonneg ENNReal.toReal_nonneg ENNReal.toReal_nonneg
  have hq0 : 0 ≤ taoSection7QFinite epsilon W (p + x.1) x.2 :=
    taoSection7QFinite_nonneg epsilon W (p + x.1) x.2
  have hq1 : taoSection7QFinite epsilon W (p + x.1) x.2 ≤ 1 :=
    taoSection7QFinite_le_one hepsilon W (p + x.1) x.2
  have hqabs : |taoSection7QFinite epsilon W (p + x.1) x.2| ≤ 1 := by
    rw [abs_of_nonneg hq0]
    exact hq1
  calc
    ‖(taoSection7HoldPMF x.1).toReal *
        (taoSection7HoldListPMF K x.2).toReal *
        taoSection7QFinite epsilon W (p + x.1) x.2‖
        = |((taoSection7HoldPMF x.1).toReal *
          (taoSection7HoldListPMF K x.2).toReal) *
          taoSection7QFinite epsilon W (p + x.1) x.2| := by
            rw [Real.norm_eq_abs]
    _ = ((taoSection7HoldPMF x.1).toReal *
          (taoSection7HoldListPMF K x.2).toReal) *
          |taoSection7QFinite epsilon W (p + x.1) x.2| := by
            rw [abs_mul, abs_of_nonneg hm0]
    _ ≤ ((taoSection7HoldPMF x.1).toReal *
          (taoSection7HoldListPMF K x.2).toReal) * 1 := by
            exact mul_le_mul_of_nonneg_left hqabs hm0
    _ = (taoSection7HoldPMF x.1).toReal *
          (taoSection7HoldListPMF K x.2).toReal := by ring

theorem taoSection7QFiniteTerminalOneApprox_succ
    {epsilon : ℝ} (hepsilon : 0 ≤ epsilon)
    (K : ℕ) (W : TaoSection7RenewalPoint → Prop)
    (p : TaoSection7RenewalPoint) :
    taoSection7QFiniteTerminalOneApprox (K + 1) epsilon W p =
      taoSection7QWhiteFactor epsilon W p *
        taoSection7HoldExpectationFull
          (fun h => taoSection7QFiniteTerminalOneApprox K epsilon W (p + h)) := by
  unfold taoSection7QFiniteTerminalOneApprox
  unfold taoSection7HoldExpectationFull
  have hnil :
      (taoSection7HoldListPMF (K + 1) ([] : List TaoSection7RenewalPoint)).toReal *
          taoSection7QFinite epsilon W p [] = 0 := by
    rw [taoSection7HoldListPMF_succ_apply_nil]
    simp
  calc
    (∑' hs : List TaoSection7RenewalPoint,
        (taoSection7HoldListPMF (K + 1) hs).toReal *
          taoSection7QFinite epsilon W p hs)
        = ∑' x : TaoSection7RenewalPoint × List TaoSection7RenewalPoint,
            (taoSection7HoldListPMF (K + 1) (x.1 :: x.2)).toReal *
              taoSection7QFinite epsilon W p (x.1 :: x.2) := by
          exact taoSection7_tsum_list_eq_tsum_cons_of_nil_zero
            (f := fun hs : List TaoSection7RenewalPoint =>
              (taoSection7HoldListPMF (K + 1) hs).toReal *
                taoSection7QFinite epsilon W p hs)
            hnil
    _ = ∑' x : TaoSection7RenewalPoint × List TaoSection7RenewalPoint,
          taoSection7QWhiteFactor epsilon W p *
            ((taoSection7HoldPMF x.1).toReal *
              (taoSection7HoldListPMF K x.2).toReal *
              taoSection7QFinite epsilon W (p + x.1) x.2) := by
          apply tsum_congr
          intro x
          rw [taoSection7HoldListPMF_succ_apply_cons]
          rw [taoSection7QFinite_cons]
          rw [ENNReal.toReal_mul]
          ring
    _ = taoSection7QWhiteFactor epsilon W p *
          (∑' x : TaoSection7RenewalPoint × List TaoSection7RenewalPoint,
            (taoSection7HoldPMF x.1).toReal *
              (taoSection7HoldListPMF K x.2).toReal *
              taoSection7QFinite epsilon W (p + x.1) x.2) := by
          rw [tsum_mul_left]
    _ = taoSection7QWhiteFactor epsilon W p *
          (∑' h : TaoSection7RenewalPoint,
            (taoSection7HoldPMF h).toReal *
              (∑' hs : List TaoSection7RenewalPoint,
                (taoSection7HoldListPMF K hs).toReal *
                  taoSection7QFinite epsilon W (p + h) hs)) := by
          congr 1
          have hsum :=
            taoSection7HoldPMF_mul_holdListPMF_mul_QFinite_summable
              hepsilon K W p
          have hfiber :
              ∀ h : TaoSection7RenewalPoint,
              Summable fun hs : List TaoSection7RenewalPoint =>
                (taoSection7HoldPMF h).toReal *
                    (taoSection7HoldListPMF K hs).toReal *
                    taoSection7QFinite epsilon W (p + h) hs := by
            intro h
            have hterm :
                Summable fun hs : List TaoSection7RenewalPoint =>
                  (taoSection7HoldListPMF K hs).toReal *
                    taoSection7QFinite epsilon W (p + h) hs :=
              taoSection7QFiniteTerminalOneApprox_summable hepsilon W (p + h)
            simpa [mul_assoc] using hterm.mul_left (taoSection7HoldPMF h).toReal
          rw [Summable.tsum_prod' hsum hfiber]
          apply tsum_congr
          intro h
          simpa [mul_assoc] using
            (tsum_mul_left
              (a := (taoSection7HoldPMF h).toReal)
              (f := fun hs : List TaoSection7RenewalPoint =>
                (taoSection7HoldListPMF K hs).toReal *
                  taoSection7QFinite epsilon W (p + h) hs))
    _ = taoSection7QWhiteFactor epsilon W p *
          (∑' h : TaoSection7RenewalPoint,
            (taoSection7HoldPMF h).toReal *
              (∑' hs : List TaoSection7RenewalPoint,
                (taoSection7HoldListPMF K hs).toReal *
                  taoSection7QFinite epsilon W (p + h) hs)) := by
          rfl

theorem taoSection7QFiniteTerminalOneApprox_succ_le
    {epsilon : ℝ} (hepsilon : 0 ≤ epsilon)
    (K : ℕ) (W : TaoSection7RenewalPoint → Prop)
    (p : TaoSection7RenewalPoint) :
    taoSection7QFiniteTerminalOneApprox (K + 1) epsilon W p ≤
      taoSection7QFiniteTerminalOneApprox K epsilon W p := by
  induction K generalizing p with
  | zero =>
      rw [taoSection7QFiniteTerminalOneApprox_succ hepsilon 0 W p]
      rw [taoSection7QFiniteTerminalOneApprox_zero]
      have hExpLe :
          taoSection7HoldExpectationFull
              (fun h : TaoSection7RenewalPoint =>
                taoSection7QFiniteTerminalOneApprox 0 epsilon W (p + h)) ≤ 1 :=
        taoSection7HoldExpectationFull_le_one_of_bounded01
          (fun h =>
            taoSection7QFiniteTerminalOneApprox_nonneg 0 epsilon W (p + h))
          (fun h =>
            taoSection7QFiniteTerminalOneApprox_le_one (K := 0) hepsilon W (p + h))
      calc
        taoSection7QWhiteFactor epsilon W p *
            taoSection7HoldExpectationFull
              (fun h : TaoSection7RenewalPoint =>
                taoSection7QFiniteTerminalOneApprox 0 epsilon W (p + h))
            ≤ taoSection7QWhiteFactor epsilon W p * 1 := by
              exact mul_le_mul_of_nonneg_left hExpLe
                (taoSection7QWhiteFactor_nonneg epsilon W p)
        _ = taoSection7QWhiteFactor epsilon W p := by ring
  | succ K ih =>
      rw [taoSection7QFiniteTerminalOneApprox_succ hepsilon (K + 1) W p]
      rw [taoSection7QFiniteTerminalOneApprox_succ hepsilon K W p]
      exact mul_le_mul_of_nonneg_left
        (taoSection7HoldExpectationFull_mono_of_bounded01
          (fun h =>
            taoSection7QFiniteTerminalOneApprox_nonneg (K + 1) epsilon W (p + h))
          (fun h =>
            taoSection7QFiniteTerminalOneApprox_le_one (K := K + 1)
              hepsilon W (p + h))
          (fun h =>
            taoSection7QFiniteTerminalOneApprox_nonneg K epsilon W (p + h))
          (fun h =>
            taoSection7QFiniteTerminalOneApprox_le_one (K := K)
              hepsilon W (p + h))
          (fun h => ih (p + h)))
        (taoSection7QWhiteFactor_nonneg epsilon W p)

theorem taoSection7QFiniteTerminalOneApprox_add_le_left
    {epsilon : ℝ} (hepsilon : 0 ≤ epsilon)
    (K L : ℕ) (W : TaoSection7RenewalPoint → Prop)
    (p : TaoSection7RenewalPoint) :
    taoSection7QFiniteTerminalOneApprox (K + L) epsilon W p ≤
      taoSection7QFiniteTerminalOneApprox K epsilon W p := by
  induction L with
  | zero =>
      rw [Nat.add_zero]
  | succ L ih =>
      rw [Nat.add_succ]
      exact
        (taoSection7QFiniteTerminalOneApprox_succ_le
          hepsilon (K + L) W p).trans ih

theorem taoSection7QFiniteTerminalOneApprox_antitone_add
    {epsilon : ℝ} (hepsilon : 0 ≤ epsilon)
    (W : TaoSection7RenewalPoint → Prop)
    (p : TaoSection7RenewalPoint) (K L : ℕ) :
    taoSection7QFiniteTerminalOneApprox (K + L) epsilon W p ≤
      taoSection7QFiniteTerminalOneApprox K epsilon W p :=
  taoSection7QFiniteTerminalOneApprox_add_le_left hepsilon K L W p

/--
Statement surface: a supplied `Q` is Tao's actual Section 7 `Q` when it is the
pointwise limit of the terminal-one finite approximants.
-/
def TaoSection7ActualQFiniteLimitStatement
    (epsilon : ℝ) (W : TaoSection7RenewalPoint → Prop)
    (Q : TaoSection7RenewalPoint → ℝ) : Prop :=
  ∀ p : TaoSection7RenewalPoint,
    Filter.Tendsto
      (fun K : ℕ => taoSection7QFiniteTerminalOneApprox K epsilon W p)
      Filter.atTop
      (𝓝 (Q p))

/-- Compatibility alias for route checkers that classify finite-limit `Q` surfaces. -/
def TaoSection7QApproximationLimitStatement
    (epsilon : ℝ) (W : TaoSection7RenewalPoint → Prop)
    (Q : TaoSection7RenewalPoint → ℝ) : Prop :=
  TaoSection7ActualQFiniteLimitStatement epsilon W Q

theorem taoSection7ActualQFiniteLimit_unique
    {epsilon : ℝ} {W : TaoSection7RenewalPoint → Prop}
    {Q R : TaoSection7RenewalPoint → ℝ}
    (hQ : TaoSection7ActualQFiniteLimitStatement epsilon W Q)
    (hR : TaoSection7ActualQFiniteLimitStatement epsilon W R) :
    Q = R := by
  funext p
  exact tendsto_nhds_unique (hQ p) (hR p)

theorem taoSection7ActualQ_nonneg_of_finiteLimit
    {epsilon : ℝ} {W : TaoSection7RenewalPoint → Prop}
    {Q : TaoSection7RenewalPoint → ℝ}
    (hQ : TaoSection7ActualQFiniteLimitStatement epsilon W Q)
    (p : TaoSection7RenewalPoint) :
    0 ≤ Q p := by
  exact ge_of_tendsto (hQ p)
    (Filter.Eventually.of_forall fun K =>
      taoSection7QFiniteTerminalOneApprox_nonneg K epsilon W p)

theorem taoSection7ActualQ_le_one_of_finiteLimit
    {epsilon : ℝ} (hepsilon : 0 ≤ epsilon)
    {W : TaoSection7RenewalPoint → Prop}
    {Q : TaoSection7RenewalPoint → ℝ}
    (hQ : TaoSection7ActualQFiniteLimitStatement epsilon W Q)
    (p : TaoSection7RenewalPoint) :
    Q p ≤ 1 := by
  exact le_of_tendsto (hQ p)
    (Filter.Eventually.of_forall fun K =>
      taoSection7QFiniteTerminalOneApprox_le_one hepsilon W p)

end Tao
end Erdos1135