import Mathlib
import OAI.Computability.VertexCover.Reduction.SeparatorStarOscillation
import OAI.Computability.VertexCover.Analysis.SeparatorRoundingErrorAe

namespace OAI

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namespace VertexCover

theorem finiteMean_sub {α : Type*} [Fintype α] (f g : α → ℝ) :
    finiteMean (fun a => f a - g a) = finiteMean f - finiteMean g := by
  simp only [finiteMean, Finset.sum_sub_distrib, sub_div]

theorem finiteMean_abs_le {α : Type*} [Fintype α] (f : α → ℝ) :
    |finiteMean f| ≤ finiteMean (fun a => |f a|) := by
  unfold finiteMean
  rw [abs_div, abs_of_nonneg (show (0 : ℝ) ≤ Fintype.card α from Nat.cast_nonneg _)]
  exact div_le_div_of_nonneg_right (Finset.abs_sum_le_sum_abs _ _) (Nat.cast_nonneg _)

end VertexCover

namespace VertexCover.LabelCover
open MeasureTheory

def StarKey (Φ : LabelCover) (d : ℕ) :=
  PositionPair d → Option (Fin Φ.M) × Option (Fin Φ.u) × Option (Fin Φ.v)

noncomputable def starKey (Φ : LabelCover) {d : ℕ} (k : Fin d) (seed : Φ.Seeds d) :
    Φ.StarKey d := by
  classical
  exact fun e => (if e.1.1 = k ∨ e.1.2 = k then none else some (seed e),
    if e.1.1 = k then none else some (Φ.left (seed e)),
    if e.1.2 = k then none else some (Φ.right (seed e)))

noncomputable def starFiber (Φ : LabelCover) {d : ℕ} (k : Fin d) (seed : Φ.Seeds d) :
    Finset (Φ.Seeds d) := by
  classical
  exact Finset.univ.filter (fun seed' => Φ.starKey k seed' = Φ.starKey k seed)

theorem mem_starFiber (Φ : LabelCover) {d : ℕ} (k : Fin d) (seed seed' : Φ.Seeds d) :
    seed' ∈ Φ.starFiber k seed ↔ Φ.starKey k seed' = Φ.starKey k seed := by
  classical
  simp [starFiber]

theorem self_mem_starFiber (Φ : LabelCover) {d : ℕ} (k : Fin d) (seed : Φ.Seeds d) :
    seed ∈ Φ.starFiber k seed := (Φ.mem_starFiber k seed seed).mpr rfl

theorem starKey_eq_questions (Φ : LabelCover) {d : ℕ} (k : Fin d)
    (seed seed' : Φ.Seeds d) (h : Φ.starKey k seed = Φ.starKey k seed') :
    (∀ e : PositionPair d, e.1.1 ≠ k → Φ.left (seed e) = Φ.left (seed' e)) ∧
    (∀ e : PositionPair d, e.1.2 ≠ k → Φ.right (seed e) = Φ.right (seed' e)) := by
  constructor
  · intro e he
    have hh := congrArg (fun key : Φ.StarKey d => (key e).2.1) h
    simpa only [starKey, ite_eq_right he, Option.some.injEq] using hh
  · intro e he
    have hh := congrArg (fun key : Φ.StarKey d => (key e).2.2) h
    simpa only [starKey, ite_eq_right he, Option.some.injEq] using hh

noncomputable def starMean (Φ : LabelCover) {d : ℕ}
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (k : Fin d)
    (seed : Φ.Seeds d) (s : Fin d → Fin (Φ.WeightDimension d) → ℝ) : ℝ :=
  VertexCover.finiteMean (fun seed' : Φ.starFiber k seed =>
    Φ.separator A hA (Φ.continuousSum seed'.1 s))

theorem star_residual_abs_le (Φ : LabelCover) {d : ℕ}
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (k : Fin d)
    (seed : Φ.Seeds d) (s : Fin d → Fin (Φ.WeightDimension d) → ℝ)
    (hs : Φ.InWeightCube s) :
    |Φ.separator A hA (Φ.continuousSum seed s) - Φ.starMean A hA k seed s| ≤ 2 := by
  classical
  let : Nonempty (Φ.starFiber k seed) := ⟨⟨seed, Φ.self_mem_starFiber k seed⟩⟩
  have heq : Φ.separator A hA (Φ.continuousSum seed s) - Φ.starMean A hA k seed s =
      VertexCover.finiteMean (fun seed' : Φ.starFiber k seed =>
        Φ.separator A hA (Φ.continuousSum seed s) -
          Φ.separator A hA (Φ.continuousSum seed'.1 s)) := by
    rw [VertexCover.finiteMean_sub, VertexCover.finiteMean_const]
    rfl
  rw [heq]
  apply (VertexCover.finiteMean_abs_le _).trans
  calc
    VertexCover.finiteMean (fun seed' : Φ.starFiber k seed =>
        |Φ.separator A hA (Φ.continuousSum seed s) -
          Φ.separator A hA (Φ.continuousSum seed'.1 s)|) ≤
        VertexCover.finiteMean (fun _ : Φ.starFiber k seed => (2 : ℝ)) := by
      apply VertexCover.finiteMean_mono
      intro seed'
      have hkey := (Φ.mem_starFiber k seed seed'.1).mp seed'.2
      obtain ⟨hL, hR⟩ := Φ.starKey_eq_questions k seed seed'.1 hkey.symm
      exact Φ.separator_star_oscillation A hA seed seed'.1 s hs k hL hR
    _ = 2 := VertexCover.finiteMean_const _

theorem starMean_continuous (Φ : LabelCover) {d : ℕ}
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (k : Fin d)
    (seed : Φ.Seeds d) : Continuous (Φ.starMean A hA k seed) := by
  classical
  unfold starMean VertexCover.finiteMean
  exact (continuous_finsetSum _ (fun seed' _ =>
    Φ.separator_continuousSum_continuous A hA seed'.1)).div_const _

end VertexCover.LabelCover


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end OAI