import Erdos1135.Tao.Renewal.GeometryBridge
import Erdos1135.Tao.Renewal.QmStatement

/-!
# Section 7 Proposition 7.8 Boundary Partition Surface

This module records the source-domain and real-threshold partition interface
for the `Q_m` boundary slice.  It does not prove the boundary estimate `(7.41)`,
any of the three case estimates, Proposition 7.8, or outer decay.
-/

namespace Erdos1135
namespace Tao

theorem taoSection7QmBoundary_toPoint_sourceDomain
    {J m : ℕ} {p : TaoSection7RenewalPoint}
    (hboundary : taoSection7QmBoundary J m p) :
    taoSection7SourcePointInDomain J p.toPoint := by
  unfold taoSection7SourcePointInDomain taoSection7QmBoundary at *
  simp [TaoSection7RenewalPoint.toPoint]
  omega

/-- Real vertical-depth threshold used to split the Prop. 7.8 boundary cases. -/
noncomputable def taoSection7Prop78BoundaryThreshold (m : ℕ) : ℝ :=
  (m : ℝ) / (Real.log (m : ℝ)) ^ 2

/-- Boundary point lies near the top of a triangle, relative to Tao's real threshold. -/
noncomputable def taoSection7Prop78NearTop
    (m : ℕ) (Δ : TaoSection7Triangle) (p : TaoSection7Point) : Prop :=
  ((Δ.verticalDepth p : ℤ) : ℝ) ≤ taoSection7Prop78BoundaryThreshold m

/-- Boundary point lies far below the top of a triangle, relative to Tao's real threshold. -/
noncomputable def taoSection7Prop78FarBelowTop
    (m : ℕ) (Δ : TaoSection7Triangle) (p : TaoSection7Point) : Prop :=
  taoSection7Prop78BoundaryThreshold m < ((Δ.verticalDepth p : ℤ) : ℝ)

theorem taoSection7Prop78NearTop_or_FarBelowTop
    (m : ℕ) (Δ : TaoSection7Triangle) (p : TaoSection7Point) :
    taoSection7Prop78NearTop m Δ p ∨
      taoSection7Prop78FarBelowTop m Δ p := by
  unfold taoSection7Prop78NearTop taoSection7Prop78FarBelowTop
  exact le_or_gt _ _

/-- Near-top branch for a renewal boundary point already covered by a triangle. -/
noncomputable def TaoSection7QmBoundaryNearTop
    (threshold : ℝ) (family : Set TaoSection7Triangle)
    (p : TaoSection7RenewalPoint) : Prop :=
  ∃ Δ ∈ family, Δ.Mem p.toPoint ∧
    ((Δ.verticalDepth p.toPoint : ℤ) : ℝ) ≤ threshold

/-- Far-below branch for a renewal boundary point already covered by a triangle. -/
noncomputable def TaoSection7QmBoundaryFarBelow
    (threshold : ℝ) (family : Set TaoSection7Triangle)
    (p : TaoSection7RenewalPoint) : Prop :=
  ∃ Δ ∈ family, Δ.Mem p.toPoint ∧
    threshold < ((Δ.verticalDepth p.toPoint : ℤ) : ℝ)

theorem taoSection7SourceTriangleCover_renewalPoint_of_qmBoundary_not_cutoffWhite
    {n J m : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ}
    {family : Set TaoSection7Triangle}
    (hcover : TaoSection7TriangleFamilyCoverBlack
      (taoSection7SourceBlackInDomain n xi epsilon J) family)
    {p : TaoSection7RenewalPoint}
    (hboundary : taoSection7QmBoundary J m p)
    (hnotWhite : ¬ taoSection7SourceWhiteRenewal
      (taoSection7SourceWhiteWCutoff n xi epsilon J) p) :
    ∃ Δ ∈ family, Δ.Mem p.toPoint :=
  taoSection7SourceTriangleCover_renewalPoint_of_not_cutoffWhite hcover
    (taoSection7QmBoundary_toPoint_sourceDomain hboundary) hnotWhite

/--
Concrete source partition for a boundary renewal point: after a source triangle
cover and failure of cutoff-white, the point is covered by a triangle and falls
on one side of the real vertical-depth threshold.  This is a partition surface
only, not the estimate `(7.41)`.
-/
theorem taoSection7QmBoundary_partition_of_source_triangle_cover_concrete
    {n J m : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ}
    {family : Set TaoSection7Triangle}
    (hcover : TaoSection7TriangleFamilyCoverBlack
      (taoSection7SourceBlackInDomain n xi epsilon J) family)
    {p : TaoSection7RenewalPoint}
    (hboundary : taoSection7QmBoundary J m p) :
    taoSection7SourceWhiteRenewal
        (taoSection7SourceWhiteWCutoff n xi epsilon J) p ∨
      TaoSection7QmBoundaryNearTop
        (taoSection7Prop78BoundaryThreshold m) family p ∨
        TaoSection7QmBoundaryFarBelow
          (taoSection7Prop78BoundaryThreshold m) family p := by
  classical
  by_cases hwhite :
      taoSection7SourceWhiteRenewal
        (taoSection7SourceWhiteWCutoff n xi epsilon J) p
  · exact Or.inl hwhite
  · right
    rcases taoSection7SourceTriangleCover_renewalPoint_of_qmBoundary_not_cutoffWhite
        hcover hboundary hwhite with ⟨Δ, hΔ, hpΔ⟩
    rcases taoSection7Prop78NearTop_or_FarBelowTop m Δ p.toPoint with hnear | hfar
    · left
      exact ⟨Δ, hΔ, hpΔ, hnear⟩
    · right
      exact ⟨Δ, hΔ, hpΔ, hfar⟩

theorem taoSection7QmBoundaryAtCutoff_partition_of_source_triangle_cover_concrete
    {n m : ℕ} {xi : ZMod (3 ^ n)} {epsilon : ℝ}
    {family : Set TaoSection7Triangle}
    (hcover : TaoSection7TriangleFamilyCoverBlack
      (taoSection7SourceBlackInDomain n xi epsilon (n / 2)) family)
    {p : TaoSection7RenewalPoint}
    (hboundary : taoSection7QmBoundary (n / 2) m p) :
    taoSection7SourceActualW n xi epsilon p ∨
      TaoSection7QmBoundaryNearTop
        (taoSection7Prop78BoundaryThreshold m) family p ∨
        TaoSection7QmBoundaryFarBelow
          (taoSection7Prop78BoundaryThreshold m) family p := by
  simpa [taoSection7SourceActualW] using
    (taoSection7QmBoundary_partition_of_source_triangle_cover_concrete
    (n := n) (J := n / 2) (m := m) (xi := xi) (epsilon := epsilon)
    (family := family) hcover hboundary)

end Tao
end Erdos1135