import Mathlib
import OAI.Computability.VertexCover.Sampling.LabelBound

namespace OAI

section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
                                                                                                      
section

namespace UniqueGames.Foundations.CorrelatedSampling

open scoped BigOperators

noncomputable section

def profileThresholds {κ S : Type*} [Fintype κ] [Fintype S]
    (P : κ → S → ℝ) : List ℝ := by
  classical
  exact (Finset.univ.toList : List (κ × S)).map (fun ks => P ks.1 ks.2)

theorem mem_profileThresholds {κ S : Type*} [Fintype κ] [Fintype S]
    (P : κ → S → ℝ) (k : κ) (s : S) : P k s ∈ profileThresholds P := by
  classical
  exact List.mem_map.mpr ⟨(k, s), by simp, rfl⟩

theorem distribution_weight_le_one {S : Type*} [Fintype S]
    (μ : Games.FiniteDistribution S) (s : S) : μ.weight s ≤ 1 := by
  calc
    μ.weight s ≤ ∑ t, μ.weight t :=
      Finset.single_le_sum (fun t _ => μ.nonnegative t) (Finset.mem_univ s)
    _ = 1 := μ.normalized

variable {Q₁ Q₂ S : Type*} [Fintype Q₁] [Fintype Q₂] [Fintype S]

def sharedProfileThresholds (L : Q₁ → Games.FiniteDistribution S)
    (R : Q₂ → Games.FiniteDistribution S) : List ℝ :=
  profileThresholds (fun (k : Q₁ ⊕ Q₂) s => (Sum.elim L R k).weight s)

theorem left_mem_sharedProfileThresholds (L : Q₁ → Games.FiniteDistribution S)
    (R : Q₂ → Games.FiniteDistribution S) (x : Q₁) (s : S) :
    (L x).weight s ∈ sharedProfileThresholds L R := by
  exact mem_profileThresholds _ (Sum.inl x) s

theorem right_mem_sharedProfileThresholds (L : Q₁ → Games.FiniteDistribution S)
    (R : Q₂ → Games.FiniteDistribution S) (y : Q₂) (s : S) :
    (R y).weight s ∈ sharedProfileThresholds L R := by
  exact mem_profileThresholds _ (Sum.inr y) s

abbrev SharedProfileSeed (L : Q₁ → Games.FiniteDistribution S)
    (R : Q₂ → Games.FiniteDistribution S) :=
  RectangleSeed (sharedProfileThresholds L R) S

def sharedProfileLeft (L : Q₁ → Games.FiniteDistribution S)
    (R : Q₂ → Games.FiniteDistribution S) (x : Q₁) : SharedProfileSeed L R → Bool :=
  rectangleAccept (sharedProfileThresholds L R) (L x).weight

def sharedProfileRight (L : Q₁ → Games.FiniteDistribution S)
    (R : Q₂ → Games.FiniteDistribution S) (y : Q₂) : SharedProfileSeed L R → Bool :=
  rectangleAccept (sharedProfileThresholds L R) (R y).weight

def sharedProfileReject (L : Q₁ → Games.FiniteDistribution S)
    (R : Q₂ → Games.FiniteDistribution S) (x : Q₁) (y : Q₂) : ℝ :=
  eventMass (rectangleWeight (sharedProfileThresholds L R))
    (fun s => !(sharedProfileLeft L R x s || sharedProfileRight L R y s))

def uniformRejectionRate (S : Type*) [Fintype S] : ℝ :=
  1 - 1 / (Fintype.card S : ℝ)

theorem uniformRejectionRate_nonnegative (S : Type*) [Fintype S] [Nonempty S] :
    0 ≤ uniformRejectionRate S := by
  have hc : (0 : ℝ) < Fintype.card S := by
    exact_mod_cast (Fintype.card_pos : 0 < Fintype.card S)
  have hc₁ : (1 : ℝ) ≤ Fintype.card S := by
    exact_mod_cast (Nat.succ_le_of_lt (Fintype.card_pos : 0 < Fintype.card S))
  apply sub_nonneg.mpr
  exact (div_le_iff₀ hc).2 (by simpa using hc₁)

theorem uniformRejectionRate_lt_one (S : Type*) [Fintype S] [Nonempty S] :
    uniformRejectionRate S < 1 := by
  have hc : (0 : ℝ) < Fintype.card S := by
    exact_mod_cast (Fintype.card_pos : 0 < Fintype.card S)
  have hp : (0 : ℝ) < 1 / (Fintype.card S : ℝ) := div_pos (by norm_num) hc
  dsimp [uniformRejectionRate]
  linarith

theorem sharedProfileReject_nonnegative [Nonempty S]
    (L : Q₁ → Games.FiniteDistribution S) (R : Q₂ → Games.FiniteDistribution S)
    (x : Q₁) (y : Q₂) : 0 ≤ sharedProfileReject L R x y := by
  exact eventMass_nonneg _ _
    (rectangleDistribution (α := S) (sharedProfileThresholds L R)).nonnegative

theorem sharedProfile_reject_le_rate [Nonempty S]
    (L : Q₁ → Games.FiniteDistribution S) (R : Q₂ → Games.FiniteDistribution S)
    (x : Q₁) (y : Q₂) : sharedProfileReject L R x y ≤ uniformRejectionRate S := by
  let thresholds := sharedProfileThresholds L R
  have hsum : (∑ s : RectangleSeed thresholds S, rectangleWeight thresholds s) = 1 :=
    (rectangleDistribution (α := S) thresholds).normalized
  have hcomp := eventMass_complement (rectangleWeight thresholds)
    (fun s => sharedProfileLeft L R x s || sharedProfileRight L R y s) hsum
  have hu : eventMass (rectangleWeight thresholds)
      (fun s => sharedProfileLeft L R x s || sharedProfileRight L R y s) =
      (1 + totalVariation (L x).weight (R y).weight) / (Fintype.card S : ℝ) := by
    change eventMass (rectangleWeight thresholds)
      (fun s => rectangleAccept thresholds (L x).weight s ||
        rectangleAccept thresholds (R y).weight s) = _
    rw [rectangle_union_mass thresholds (L x).weight (R y).weight
      (left_mem_sharedProfileThresholds L R x)
      (right_mem_sharedProfileThresholds L R y)
      (L x).nonnegative (R y).nonnegative
      (distribution_weight_le_one (L x)) (distribution_weight_le_one (R y))]
    rw [unionMass_eq (L x).weight (R y).weight (L x).normalized (R y).normalized]
  have hc : (0 : ℝ) < Fintype.card S := by
    exact_mod_cast (Fintype.card_pos : 0 < Fintype.card S)
  have htv := totalVariation_nonneg (L x).weight (R y).weight
  have hquot : 1 / (Fintype.card S : ℝ) ≤
      (1 + totalVariation (L x).weight (R y).weight) / (Fintype.card S : ℝ) :=
    (div_le_div_iff_of_pos_right hc).2 (by linarith)
  rw [hu] at hcomp
  change eventMass (rectangleWeight thresholds)
    (fun s => !(sharedProfileLeft L R x s || sharedProfileRight L R y s)) ≤
      1 - 1 / (Fintype.card S : ℝ)
  linarith

private theorem nonnegative_powers_mono (a b : ℝ) (ha : 0 ≤ a) (hab : a ≤ b)
    (n : ℕ) : a ^ n ≤ b ^ n := by
  have hb : 0 ≤ b := le_trans ha hab
  induction n with
  | zero => simp
  | succ n ih =>
      rw [pow_succ, pow_succ]
      exact mul_le_mul ih hab ha (pow_nonneg hb n)

theorem sharedProfile_label_diagonal_bound [Nonempty S] [DecidableEq S]
    (L : Q₁ → Games.FiniteDistribution S) (R : Q₂ → Games.FiniteDistribution S)
    (x : Q₁) (y : Q₂) (fallback a : S) (n : ℕ) :
    min ((L x).weight a) ((R y).weight a) /
        (1 + totalVariation (L x).weight (R y).weight) *
        (1 - uniformRejectionRate S ^ n) ≤
      traceAverage (rectangleWeight (sharedProfileThresholds L R)) n
        (localDiagonal (sharedProfileLeft L R x) (sharedProfileRight L R y)
          Prod.fst fallback a) := by
  have hp := nonnegative_powers_mono (sharedProfileReject L R x y)
    (uniformRejectionRate S) (sharedProfileReject_nonnegative L R x y)
    (sharedProfile_reject_le_rate L R x y) n
  have htv := totalVariation_nonneg (L x).weight (R y).weight
  have hm : 0 ≤ min ((L x).weight a) ((R y).weight a) /
      (1 + totalVariation (L x).weight (R y).weight) :=
    div_nonneg (le_min ((L x).nonnegative a) ((R y).nonnegative a)) (by linarith)
  have hsurv :
      min ((L x).weight a) ((R y).weight a) /
          (1 + totalVariation (L x).weight (R y).weight) *
          (1 - uniformRejectionRate S ^ n) ≤
        min ((L x).weight a) ((R y).weight a) /
          (1 + totalVariation (L x).weight (R y).weight) *
          (1 - sharedProfileReject L R x y ^ n) :=
    mul_le_mul_of_nonneg_left (by linarith) hm
  refine hsurv.trans ?_
  simpa only [sharedProfileReject, sharedProfileLeft, sharedProfileRight] using
    rectangle_label_diagonal_bound (sharedProfileThresholds L R)
      (L x).weight (R y).weight fallback a
      (left_mem_sharedProfileThresholds L R x)
      (right_mem_sharedProfileThresholds L R y)
      (L x).nonnegative (R y).nonnegative
      (distribution_weight_le_one (L x)) (distribution_weight_le_one (R y))
      (L x).normalized (R y).normalized n

theorem exists_uniformRejectionRate_pow_lt (S : Type*) [Fintype S] [Nonempty S]
    (η : ℝ) (hη : 0 < η) : ∃ n : ℕ, uniformRejectionRate S ^ n < η := by
  have ht := tendsto_pow_atTop_nhds_zero_of_lt_one
    (uniformRejectionRate_nonnegative S) (uniformRejectionRate_lt_one S)
  exact (ht.eventually (gt_mem_nhds hη)).exists

end

end UniqueGames.Foundations.CorrelatedSampling
end


end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI