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import OAI.Computability.VertexCover.Analysis.PoincareOne
namespace OAI
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namespace VertexCover.Cube
open MeasureTheory ProbabilityTheory
open scoped ENNReal
theorem law_eq_restrict_volume (ι : Type*) [Fintype ι] :
law ι = (1/2 : ℝ≥0∞) ^ Fintype.card ι •
volume.restrict (Set.univ.pi (fun _ : ι => Set.Icc (-1 : ℝ) 1)) := by
apply Measure.pi_eq
intro s hs
rw [Measure.smul_apply, Measure.restrict_apply (MeasurableSet.univ_pi hs),
← Set.pi_inter_distrib, volume_pi_pi]
simp only [intervalLaw, Measure.smul_apply, Measure.restrict_apply (hs _),
smul_eq_mul, Finset.prod_mul_distrib, Finset.prod_const, Finset.card_univ]
theorem law_ac (ι : Type*) [Fintype ι] : law ι ≪ volume := by
rw [law_eq_restrict_volume]
intro s hs
have hz : volume.restrict (Set.univ.pi (fun _ : ι => Set.Icc (-1 : ℝ) 1)) s = 0 :=
le_antisymm ((Measure.restrict_le_self s).trans_eq hs) zero_le
simp only [Measure.smul_apply, hz, smul_zero]
end VertexCover.Cube
namespace VertexCover.Cube
open MeasureTheory ProbabilityTheory
open scoped ENNReal
noncomputable def blockLaw (ι κ : Type*) [Fintype ι] [Fintype κ] :
Measure (ι → κ → ℝ) := Measure.pi (fun _ : ι => law κ)
instance blockLaw_probability (ι κ : Type*) [Fintype ι] [Fintype κ] :
IsProbabilityMeasure (blockLaw ι κ) := by
unfold blockLaw
infer_instance
theorem blockLaw_eq_restrict_volume (ι κ : Type*) [Fintype ι] [Fintype κ] :
blockLaw ι κ = ((1/2 : ℝ≥0∞) ^ Fintype.card κ) ^ Fintype.card ι •
volume.restrict (Set.univ.pi (fun _ : ι =>
Set.univ.pi (fun _ : κ => Set.Icc (-1 : ℝ) 1))) := by
apply Measure.pi_eq
intro s hs
rw [Measure.smul_apply, Measure.restrict_apply (MeasurableSet.univ_pi hs),
← Set.pi_inter_distrib, volume_pi_pi]
simp only [law_eq_restrict_volume, Measure.smul_apply,
Measure.restrict_apply (hs _), smul_eq_mul, Finset.prod_mul_distrib,
Finset.prod_const, Finset.card_univ]
theorem blockLaw_ac (ι κ : Type*) [Fintype ι] [Fintype κ] :
blockLaw ι κ ≪ volume := by
rw [blockLaw_eq_restrict_volume]
intro s hs
have hz : volume.restrict (Set.univ.pi (fun _ : ι =>
Set.univ.pi (fun _ : κ => Set.Icc (-1 : ℝ) 1))) s = 0 :=
le_antisymm ((Measure.restrict_le_self s).trans_eq hs) zero_le
simp only [Measure.smul_apply, hz, smul_zero]
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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