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import OAI.Computability.VertexCover.Reduction.DecodingUpperThreeHalves
import OAI.Computability.VertexCover.Analysis.OwnListEnergy
namespace OAI
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namespace VertexCover.LabelCover
open MeasureTheory
noncomputable def ownPrivateLists (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
(s : Φ.BatchWeights J) (β : ℝ) : Φ.PrivateLists d := by
classical
exact fun i => if hi : i.1 ∈ J then
Φ.ownList J frozen c0 A hA ⟨i.1, hi⟩ i (s ⟨i.1, hi⟩) β else []
theorem ownPrivateLists_length (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
(frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
(s : Φ.BatchWeights J) (m : ℕ) (hm : 0 < m) (i : Φ.Query d) :
(Φ.ownPrivateLists J frozen c0 A hA s (Parameters.β m) i).length ≤ Parameters.ell m := by
classical
unfold ownPrivateLists
split_ifs with hi
· exact Φ.ownList_length J frozen c0 A hA ⟨i.1, hi⟩ i _ m hm
· simp
theorem ownPrivateLists_membership_measurable (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (frozen : Φ.Seeds d) (c0 : Φ.Coordinate d → ℝ)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (β : ℝ)
(i : Φ.Query d) (a : i.LocalLabel) :
MeasurableSet {s | a ∈ Φ.ownPrivateLists J frozen c0 A hA s β i} := by
classical
unfold ownPrivateLists
by_cases hi : i.1 ∈ J
· simp only [dite_eq_left hi, mem_ownList]
exact measurableSet_le measurable_const
(((Φ.ownGradient_measurable J frozen c0 A hA ⟨i.1, hi⟩ i (i.slot a)).comp
(measurable_pi_apply (⟨i.1, hi⟩ : J))).abs)
· simp only [dite_eq_right hi, List.not_mem_nil, Set.ofPred_false]
exact MeasurableSet.empty
theorem ownPrivateLists_decoding_upper (Φ : LabelCover) (m : ℕ) (hm : 4 ≤ m)
(hval : Φ.value ≤ Parameters.σ m)
(J : Finset (Fin (Parameters.d m))) (hJ : J.card = Parameters.h m)
(frozen : Φ.Seeds (Parameters.d m)) (c0 : Φ.Coordinate (Parameters.d m) → ℝ)
(A : Finset (Φ.Coordinate (Parameters.d m) → ℝ)) (hA : A.Nonempty) :
(∫ s, finiteMean (fun hidden : Φ.HiddenSeeds J =>
(Φ.listHitMaximum (Φ.ownPrivateLists J frozen c0 A hA s (Parameters.β m))
(Φ.spliceSeeds J frozen hidden) J : ℝ)) ∂Φ.batchLaw J) < 3/2 := by
classical
let B : ℝ := 1 + ((Parameters.h m).choose 2 : ℝ) *
(Parameters.ell m : ℝ)^2 * Parameters.σ m
have hbound : ∀ s : Φ.BatchWeights J, finiteMean (fun hidden : Φ.HiddenSeeds J =>
(Φ.listHitMaximum (Φ.ownPrivateLists J frozen c0 A hA s (Parameters.β m))
(Φ.spliceSeeds J frozen hidden) J : ℝ)) ≤ B := by
intro s
have hh := Φ.decoding_upper_bound hval
(Φ.ownPrivateLists J frozen c0 A hA s (Parameters.β m))
(Φ.ownPrivateLists_length J frozen c0 A hA s m (by omega)) J frozen
simpa only [hJ] using hh
have hint := integral_mono_of_nonneg (μ := Φ.batchLaw J)
(f := fun s => finiteMean (fun hidden : Φ.HiddenSeeds J =>
(Φ.listHitMaximum (Φ.ownPrivateLists J frozen c0 A hA s (Parameters.β m))
(Φ.spliceSeeds J frozen hidden) J : ℝ)))
(g := fun _ => B) (Filter.Eventually.of_forall (fun s => by
unfold finiteMean
exact div_nonneg (Finset.sum_nonneg (fun _ _ => Nat.cast_nonneg _))
(Nat.cast_nonneg _))) (integrable_const B) (Filter.Eventually.of_forall hbound)
simp only [integral_const, probReal_univ, smul_eq_mul, one_mul] at hint
have hp := Parameters.pair_budget_lt m hm
dsimp [B] at hint
linarith
end VertexCover.LabelCover
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
end OAI
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