Datasets:

roseking
/

openAI-math-dataset

import Mathlib
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

	import Mathlib
	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)νh :=
	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