import Mathlib
import OAI.Computability.VertexCover.Reduction.PrivateProjectionSurvival
import OAI.Computability.VertexCover.Reduction.BatchesTripleSum

namespace OAI

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namespace VertexCover.LabelCover
open VertexCover.Restriction
open EnergyForm.Projection
open scoped BigOperators

theorem weightGroup_residual_budget (Φ : LabelCover) {d : ℕ}
    (Q : (Φ.restrictionForm d).Projection)
    (hQ : ∀ k f, Φ.weightProjection k (Q f) = Q (Φ.weightProjection k f))
    (f : Φ.RestrictionSpace d) :
    (∑ j : Fin d, (Φ.restrictionForm d).energy (Q.residual (Φ.weightGroup j f))) ≤
      (Φ.restrictionForm d).energy (Q.residual f) :=
  finDifference_residual_budget _ Φ.weightProjection_commute Q hQ f

theorem weightGroup_pair_budget (Φ : LabelCover) {d : ℕ}
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) :
    (∑ j : Fin d, ∑ k : Fin d,
      ((Φ.restrictionForm d).energy ((Φ.weightProjection k).residual
        (Φ.weightGroup j (Φ.restrictionFunction A hA))) +
       (Φ.restrictionForm d).energy ((Φ.starProjection k).residual
        (Φ.weightGroup j (Φ.restrictionFunction A hA))))) ≤ 8*d := by
  rw [Finset.sum_comm]
  calc
    _ ≤ ∑ k : Fin d,
        ((Φ.restrictionForm d).energy ((Φ.weightProjection k).residual
          (Φ.restrictionFunction A hA)) +
         (Φ.restrictionForm d).energy ((Φ.starProjection k).residual
          (Φ.restrictionFunction A hA))) := by
      apply Finset.sum_le_sum
      intro k _
      rw [Finset.sum_add_distrib]
      apply add_le_add
      · exact Φ.weightGroup_residual_budget (Φ.weightProjection k)
          (fun i f => Φ.weightProjection_commute i k f) _
      · exact Φ.weightGroup_residual_budget (Φ.starProjection k)
          (fun i f => (Φ.condition_weightProjection_commute (Φ.starKey k) i f).symm) _
    _ ≤ _ := by
      rw [Finset.sum_add_distrib]
      have hw := Φ.restriction_weight_budget A hA
      have hs := Φ.restriction_star_budget A hA
      linarith

theorem weightGroup_seed_budget (Φ : LabelCover) {d : ℕ}
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) :
    (∑ j : Fin d, ∑ e : PositionPair d,
      (Φ.restrictionForm d).energy ((Φ.seedProjection e).residual
        (Φ.weightGroup j (Φ.restrictionFunction A hA)))) ≤ 8*(d:ℝ)^2 := by
  rw [Finset.sum_comm]
  calc
    _ ≤ ∑ e : PositionPair d, (Φ.restrictionForm d).energy
        ((Φ.seedProjection e).residual (Φ.restrictionFunction A hA)) := by
      apply Finset.sum_le_sum
      intro e _
      exact Φ.weightGroup_residual_budget (Φ.seedProjection e)
        (fun i f => (Φ.condition_weightProjection_commute
          (fun seed : Φ.Seeds d => Function.update seed e ⟨0, Φ.M_pos⟩) i f).symm) _
    _ ≤ _ := Φ.restriction_seed_budget A hA

theorem privateProjection_batch_bound (Φ : LabelCover) {d h : ℕ} (hh : 3 ≤ h)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) :
    ((d-1).choose (h-1) : ℝ) * (Φ.restrictionForm d).energy (Φ.restrictionFunction A hA) ≤
      (∑ J ∈ batches d h, ∑ j ∈ J, (Φ.restrictionForm d).energy
        (Φ.privateProjection J j (Φ.restrictionFunction A hA))) +
      ((d-2).choose (h-2) : ℝ)*(8*d) + ((d-3).choose (h-3) : ℝ)*(8*(d:ℝ)^2) := by
  classical
  let B := Φ.restrictionForm d
  let f := Φ.restrictionFunction A hA
  let pairLoss := fun j k : Fin d => B.energy ((Φ.weightProjection k).residual (Φ.weightGroup j f)) +
    B.energy ((Φ.starProjection k).residual (Φ.weightGroup j f))
  let seedLoss := fun (j : Fin d) (e : PositionPair d) =>
    B.energy ((Φ.seedProjection e).residual (Φ.weightGroup j f))
  have hsurv : (∑ J ∈ batches d h, ∑ j ∈ J, B.energy (Φ.weightGroup j f)) ≤
      (∑ J ∈ batches d h, ∑ j ∈ J, B.energy (Φ.privateProjection J j f)) +
      (∑ J ∈ batches d h, ∑ j ∈ J, ∑ k ∈ J.erase j, pairLoss j k) +
      (∑ J ∈ batches d h, ∑ j ∈ J,
        ∑ e ∈ (internalPairs J).filter (fun e => ¬ (e.1.1 = j ∨ e.1.2 = j)), seedLoss j e) := by
    simp only [← Finset.sum_add_distrib]
    apply Finset.sum_le_sum
    intro J _
    apply Finset.sum_le_sum
    intro j _
    exact Φ.privateProjection_survival J j f
  have hpair := batches_pair_sum_le (by omega : 2 ≤ h) pairLoss
    (fun j k => add_nonneg (B.energy_nonneg _) (B.energy_nonneg _))
  have htriple := batches_triple_sum_le hh
    (fun e : PositionPair d => e.1.1) (fun e : PositionPair d => e.1.2)
    (fun e => ne_of_lt e.2) seedLoss (fun _ _ => B.energy_nonneg _)
  have hsets (J : Finset (Fin d)) (j : Fin d) :
      (internalPairs J).filter (fun e => ¬ (e.1.1 = j ∨ e.1.2 = j)) =
        Finset.univ.filter (fun e : PositionPair d =>
          e.1.1 ∈ J ∧ e.1.2 ∈ J ∧ ¬ (e.1.1 = j ∨ e.1.2 = j)) := by
    ext e
    simp only [internalPairs, Finset.mem_filter, Finset.mem_univ, true_and]
    tauto
  simp_rw [hsets] at hsurv
  rw [batches_singleton_sum (by omega : 1 ≤ h), Φ.weightGroup_energy_sum A hA] at hsurv
  have hpb : (∑ j, ∑ k, pairLoss j k) ≤ 8*d := Φ.weightGroup_pair_budget A hA
  have hsb : (∑ j, ∑ e, seedLoss j e) ≤ 8*(d:ℝ)^2 := Φ.weightGroup_seed_budget A hA
  have hpb' := mul_le_mul_of_nonneg_left hpb (Nat.cast_nonneg ((d-2).choose (h-2)) : (0:ℝ) ≤ _)
  have hsb' := mul_le_mul_of_nonneg_left hsb (Nat.cast_nonneg ((d-3).choose (h-3)) : (0:ℝ) ≤ _)
  dsimp only [B, f] at hsurv
  linarith

end VertexCover.LabelCover


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

end OAI