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import OAI.Computability.VertexCover.Reduction.PrivateTotalEnergyLaw
import OAI.Computability.VertexCover.Reduction.PrivateProjectionBatchLower
namespace OAI
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namespace VertexCover.LabelCover
open MeasureTheory ProbabilityTheory
open VertexCover.Restriction
open EnergyForm.Projection
open scoped BigOperators
theorem restrictionFunction_energy_variance (Φ : LabelCover) {d : ℕ}
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) :
(Φ.restrictionForm d).energy (Φ.restrictionFunction A hA) =
VertexCover.finiteMean (fun seed : Φ.Seeds d =>
variance (fun s => Φ.separator A hA (Φ.continuousSum seed s))
(VertexCover.Cube.blockLaw (Fin d) (Fin (Φ.WeightDimension d)))) := by
rw [restrictionForm_energy]
apply congrArg VertexCover.finiteMean
funext seed
rw [variance_eq_integral
(Φ.separator_continuousSum_continuous A hA seed).measurable.aemeasurable,
Φ.separator_continuousSum_mean_zero]
simp only [sub_zero, restrictionFunction_apply]
exact CompactCube.integral_realWeights
((Φ.separator_continuousSum_continuous A hA seed).pow 2).measurable.aestronglyMeasurable
theorem restriction_frozen_exists (Φ : LabelCover) {d h : ℕ}
(hd : 4 ≤ d) (hh : 3 ≤ h) (hhd : h ≤ d) (ν : ℝ) (hν : 0 < ν)
(hscale : 16*(h:ℝ)+32*(h:ℝ)^2 ≤ ν*d/2)
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
(henergy : ν*d ≤ (Φ.restrictionForm d).energy (Φ.restrictionFunction A hA)) :
∃ J : Finset (Fin d), J.card = h ∧ ∃ frozen : Φ.Seeds d,
∃ s : Φ.RestrictionWeights d,
ν*h/4 ≤ Φ.ownTotalVariance J frozen
(Φ.outsideSum J frozen (CompactCube.realWeights s)) A hA := by
classical
have : Nonempty (Fin Φ.M) := ⟨⟨0, Φ.M_pos⟩⟩
by_contra hno
push Not at hno
have hbound (J : Finset (Fin d)) (hJ : J ∈ batches d h) :
(∑ j ∈ J, (Φ.restrictionForm d).energy
(Φ.privateProjection J j (Φ.restrictionFunction A hA))) ≤ ν*h/4 := by
rw [← Finset.sum_coe_sort J, Φ.privateTotal_energy_law J A hA]
calc
_ ≤ VertexCover.finiteMean (fun _ : Φ.Seeds d => ν*h/4) := by
apply VertexCover.finiteMean_mono
intro frozen
have hb := integral_mono (Φ.ownTotalVariance_frozen_integrable J frozen A hA)
(integrable_const (ν*h/4))
(fun s => (hno J (mem_batches.mp hJ) frozen s).le)
simpa using hb
_ = _ := VertexCover.finiteMean_const _
have hsum := Finset.sum_le_sum hbound
have hl := Φ.privateProjection_batch_lower hd hh ν hscale A hA henergy
simp only [Finset.sum_const, nsmul_eq_mul, card_batches] at hsum
have hpos : 0 < (d.choose h:ℝ)*ν*h :=
mul_pos (mul_pos (Nat.cast_pos.mpr (Nat.choose_pos hhd)) hν)
(Nat.cast_pos.mpr (by omega))
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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