import Mathlib
import OAI.Computability.VertexCover.Repetition.CompletedSampling
import OAI.Computability.VertexCover.Repetition.GameConditioning

namespace OAI

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namespace UniqueGames.Foundations.Repetition
open scoped BigOperators
open Games CorrelatedSampling
noncomputable section
variable {Q₁ Q₂ A₁ A₂ S : Type*}
  [Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂] [Fintype S]
  [Nonempty A₁] [Nonempty A₂] [Nonempty S]
  [DecidableEq Q₁] [DecidableEq Q₂] [DecidableEq S] {n : Nat}

theorem coordinate_probability_le_value_of_local_completion
    (G : Game Q₁ Q₂ A₁ A₂) (j : Fin n)
    (strategy : Strategy (Fin n → Q₁) (Fin n → Q₂) (Fin n → A₁) (Fin n → A₂))
    (σ : FiniteDistribution ((Q₁ × Q₂) × S))
    (profileL : Q₁ → FiniteDistribution S) (profileR : Q₂ → FiniteDistribution S)
    (completionL : Q₁ × S → FiniteDistribution (Fin n → Q₁))
    (completionR : Q₂ × S → FiniteDistribution (Fin n → Q₂))
    (supportL : ∀ q xs, (completionL q).weight xs ≠ 0 → xs j = q.1)
    (supportR : ∀ q ys, (completionR q).weight ys ≠ 0 → ys j = q.1)
    (fallback : S) :
    (σ.mixture (fun z => (completionL (z.1.1,z.2)).product
      (completionR (z.1.2,z.2)))).probability (G.coordinateWin strategy j) ≤
      G.value +
        (2 * Information.totalVariation σ.weight
          (fun z => G.questions.weight z.1 * (profileL z.1.1).weight z.2) +
         2 * Information.totalVariation σ.weight
          (fun z => G.questions.weight z.1 * (profileR z.1.2).weight z.2)) := by
  let targetLaw := σ.mixture (fun z => (completionL (z.1.1,z.2)).product (completionR (z.1.2,z.2)))
  let δ := 2 * Information.totalVariation σ.weight
      (fun z => G.questions.weight z.1 * (profileL z.1.1).weight z.2) +
    2 * Information.totalVariation σ.weight
      (fun z => G.questions.weight z.1 * (profileR z.1.2).weight z.2)
  change targetLaw.probability (G.coordinateWin strategy j) ≤ G.value + δ
  by_contra h
  have gap : 0 < targetLaw.probability (G.coordinateWin strategy j) - (G.value + δ) :=
    sub_pos.mpr (lt_of_not_ge h)
  let η := (targetLaw.probability (G.coordinateWin strategy j) - (G.value + δ)) / 2
  have hη : 0 < η := by dsimp [η]; linarith
  obtain ⟨N,hN⟩ := samplerOutputLaw_arbitrarily_close σ G.questions profileL profileR fallback η hη
  let thresholds := sharedProfileThresholds profileL profileR
  let γ := traceSeedLaw (rectangleDistribution (α := S) thresholds) N
  let sL := samplerLocal thresholds profileL fallback N
  let sR := samplerLocal thresholds profileR fallback N
  have hvariation := CompletedSampling.coordinateOutputLaw_totalVariation_le
    G.questions γ j sL sR completionL completionR supportL supportR σ
  have hclose : (CompletedSampling.coordinateOutputLaw G.questions γ j sL sR completionL completionR).totalVariation targetLaw ≤ δ + η :=
    hvariation.trans hN
  have hsuccess := G.coordinate_probability_le_value_add_embedding_distance j targetLaw
    (CompletedSampling.seedLaw γ completionL completionR)
    (CompletedSampling.coordinateLeft j sL) (CompletedSampling.coordinateRight j sR)
    (CompletedSampling.coordinateLeft_preserves j sL)
    (CompletedSampling.coordinateRight_preserves j sR) strategy
  have hsym : targetLaw.totalVariation
      (G.localEmbeddingLaw (CompletedSampling.seedLaw γ completionL completionR)
        (CompletedSampling.coordinateLeft j sL) (CompletedSampling.coordinateRight j sR)) =
      (CompletedSampling.coordinateOutputLaw G.questions γ j sL sR completionL completionR).totalVariation targetLaw := by
    unfold FiniteDistribution.totalVariation
    congr 1
    apply Finset.sum_congr rfl
    intro q _
    exact abs_sub_comm _ _
  rw [hsym] at hsuccess
  dsimp [η] at hclose
  linarith

end
end UniqueGames.Foundations.Repetition
end


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end OAI