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import OAI.Computability.VertexCover.PCP.SpectralCut
import OAI.Computability.VertexCover.Analysis.IntegralUpdate
import OAI.Computability.VertexCover.Analysis.VariancePi
namespace OAI
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namespace VertexCover.Product
open MeasureTheory
theorem update_right_measurePreserving {ι E : Type*} [Fintype ι] [DecidableEq ι]
[MeasurableSpace E] (μ : Measure E) [IsProbabilityMeasure μ] (j : ι) :
MeasurePreserving (fun p : (ι → E) × E => Function.update p.1 j p.2)
((Measure.pi (fun _ : ι => μ)).prod μ) (Measure.pi (fun _ : ι => μ)) := by
exact (update_measurePreserving μ j).comp Measure.measurePreserving_swap
theorem update_right_ae {ι E : Type*} [Fintype ι] [DecidableEq ι]
[MeasurableSpace E] (μ : Measure E) [IsProbabilityMeasure μ] (j : ι)
{P : (ι → E) → Prop} (hP : ∀ᵐ s ∂Measure.pi (fun _ : ι => μ), P s) :
∀ᵐ s ∂Measure.pi (fun _ : ι => μ), ∀ᵐ x ∂μ, P (Function.update s j x) := by
exact Measure.ae_ae_of_ae_prod ((update_right_measurePreserving μ j).quasiMeasurePreserving.ae hP)
theorem integral_update_right {ι E : Type*} [Fintype ι] [DecidableEq ι]
[MeasurableSpace E] (μ : Measure E) [IsProbabilityMeasure μ] (j : ι)
{f : (ι → E) → ℝ} (hf : Integrable f (Measure.pi (fun _ : ι => μ))) :
(∫ s, ∫ x, f (Function.update s j x) ∂μ ∂Measure.pi (fun _ : ι => μ)) =
∫ s, f s ∂Measure.pi (fun _ : ι => μ) := by
have hp := update_right_measurePreserving μ j
calc
_ = ∫ p, f (Function.update p.1 j p.2)
∂((Measure.pi (fun _ : ι => μ)).prod μ) := by
simpa only [Function.comp_apply] using
(integral_prod _ (hp.integrable_comp_of_integrable hf)).symm
_ = ∫ s, f s ∂Measure.map (fun p : (ι → E) × E => Function.update p.1 j p.2)
((Measure.pi (fun _ : ι => μ)).prod μ) := by
exact (integral_map hp.measurable.aemeasurable
(by simpa only [hp.map_eq] using hf.aestronglyMeasurable)).symm
_ = _ := by rw [hp.map_eq]
end VertexCover.Product
namespace VertexCover.Cube
open MeasureTheory ProbabilityTheory
open scoped ENNReal
abbrev carrier (n : ℕ) : Set (Fin n → ℝ) := Set.univ.pi (fun _ => Set.Icc (-1 : ℝ) 1)
theorem measurableSet_carrier (n : ℕ) : MeasurableSet (carrier n) :=
MeasurableSet.univ_pi (fun _ => measurableSet_Icc)
theorem ae_carrier (n : ℕ) : ∀ᵐ s ∂law (Fin n), s ∈ carrier n := by
rw [law_eq_restrict_volume]
exact (Measure.ae_ennreal_smul_measure_iff (pow_ne_zero _ (by norm_num) :
(1/2 : ℝ≥0∞) ^ Fintype.card (Fin n) ≠ 0)).mpr
(ae_restrict_mem (measurableSet_carrier n))
theorem norm_le_one_of_mem_carrier {n : ℕ} {s : Fin n → ℝ} (hs : s ∈ carrier n) :
‖s‖ ≤ 1 := by
apply (pi_norm_le_iff_of_nonneg (by norm_num : (0:ℝ) ≤ 1)).mpr
intro k
exact (abs_le.mpr (Set.mem_univ_pi.mp hs k))
noncomputable def boundedVersion {n : ℕ} (f : (Fin n → ℝ) → ℝ) : (Fin n → ℝ) → ℝ :=
(carrier n).indicator f
theorem boundedVersion_measurable {n : ℕ} {f : (Fin n → ℝ) → ℝ}
(hf : Measurable f) : Measurable (boundedVersion f) :=
hf.indicator (measurableSet_carrier n)
theorem boundedVersion_ae {n : ℕ} (f : (Fin n → ℝ) → ℝ) :
boundedVersion f =ᵐ[law (Fin n)] f := by
filter_upwards [ae_carrier n] with s hs
exact Set.indicator_of_mem hs f
theorem boundedVersion_bound {n : ℕ} {f : (Fin n → ℝ) → ℝ}
(hf : LipschitzWith 1 f) (s : Fin n → ℝ) :
‖boundedVersion f s‖ ≤ ‖f 0‖ + 1 := by
by_cases hs : s ∈ carrier n
· rw [boundedVersion, Set.indicator_of_mem hs]
have h := hf.dist_le_mul s 0
simp only [NNReal.coe_one, one_mul, dist_zero_right, Real.dist_eq] at h
calc
‖f s‖ ≤ ‖f s - f 0‖ + ‖f 0‖ := by simpa using norm_add_le (f s - f 0) (f 0)
_ ≤ 1 + ‖f 0‖ := by simpa only [Real.norm_eq_abs, add_comm] using add_le_add_right (h.trans (norm_le_one_of_mem_carrier hs)) ‖f 0‖
_ = _ := add_comm _ _
· rw [boundedVersion, Set.indicator_of_notMem hs]
simp only [norm_zero]
positivity
theorem variance_le_coordinate_sum {n : ℕ} {f : (Fin n → ℝ) → ℝ}
(hf : LipschitzWith 1 f) :
variance f (law (Fin n)) ≤
∑ j, ∫ s, VertexCover.Product.coordinateVariance intervalLaw f j s ∂law (Fin n) := by
have he := boundedVersion_ae f
have hb := VertexCover.Product.variance_pi_le intervalLaw n (boundedVersion f)
(boundedVersion_measurable hf.continuous.measurable) (boundedVersion_bound hf)
have hv : variance (boundedVersion f) (law (Fin n)) = variance f (law (Fin n)) :=
variance_congr he
change variance (boundedVersion f) (law (Fin n)) ≤ _ at hb
rw [hv] at hb
convert hb using 1
apply Finset.sum_congr rfl
intro j hj
apply integral_congr_ae
filter_upwards [VertexCover.Product.update_right_ae intervalLaw j he] with s hs
exact (variance_congr hs).symm
end VertexCover.Cube
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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