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import OAI.Computability.VertexCover.Analysis.BlockLawAc
import OAI.Computability.VertexCover.Analysis.LawNegInvariant
namespace OAI
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namespace VertexCover.Product
open MeasureTheory
theorem piMap_measurePreserving {ι X Y : Type*} [Fintype ι]
[MeasurableSpace X] [MeasurableSpace Y] (μ : Measure X) (ν : Measure Y)
[SigmaFinite μ] [SigmaFinite ν] (f : X → Y) (hf : MeasurePreserving f μ ν) :
MeasurePreserving (fun s : ι → X => fun i => f (s i))
(Measure.pi (fun _ : ι => μ)) (Measure.pi (fun _ : ι => ν)) := by
refine ⟨Measurable.of_eval (fun index => hf.measurable.comp (measurable_pi_apply index)), ?_⟩
let : SigmaFinite (Measure.map f μ) := by rw [hf.map_eq]; infer_instance
rw [Measure.pi_map_pi (fun _ => hf.aemeasurable)]
simp only [hf.map_eq]
end VertexCover.Product
namespace VertexCover.GridCoupling
open MeasureTheory ProbabilityTheory
open scoped ENNReal
theorem pi_uniformOn_univ {ι E : Type*} [Fintype ι] [Fintype E] [Nonempty E]
[MeasurableSpace E] [MeasurableSingletonClass E] :
Measure.pi (fun _ : ι => uniformOn (Set.univ : Set E)) =
uniformOn (Set.univ : Set (ι → E)) := by
classical
apply Measure.ext_of_singleton
intro s
rw [Measure.pi_singleton]
rw [uniformOn_univ]
simp [uniformOn_univ, ENNReal.inv_pow]
theorem vector_measurePreserving {ι : Type*} [Fintype ι] (p : ℕ) (hp : 0 < p) :
MeasurePreserving (fun s : ι → ℝ => fun i => index p hp (s i))
(VertexCover.Cube.law ι) (uniformOn (Set.univ : Set (ι → Fin p))) := by
let : NeZero p := ⟨hp.ne'⟩
have h := VertexCover.Product.piMap_measurePreserving (ι := ι)
VertexCover.Cube.intervalLaw (uniformOn (Set.univ : Set (Fin p)))
(index p hp) (index_measurePreserving p hp)
simpa only [pi_uniformOn_univ, VertexCover.Cube.law] using h
theorem integral_uniform_finite {E : Type*} [Fintype E] [Nonempty E]
[MeasurableSpace E] [MeasurableSingletonClass E] (f : E → ℝ) :
(∫ x, f x ∂uniformOn (Set.univ : Set E)) = (∑ x, f x) / Fintype.card E := by
rw [integral_fintype (Integrable.of_finite)]
simp only [measureReal_def, uniformOn_univ, Measure.count_singleton,
ENNReal.toReal_div, ENNReal.toReal_one, ENNReal.toReal_natCast, smul_eq_mul]
simp only [div_eq_mul_inv, ← Finset.sum_mul, mul_comm, mul_one]
end VertexCover.GridCoupling
namespace VertexCover.Cube
open MeasureTheory
instance blockLaw_negInvariant (ι κ : Type*) [Fintype ι] [Fintype κ] :
(blockLaw ι κ).IsNegInvariant := by
unfold blockLaw
exact Measure.pi.isNegInvariant (fun _ : ι => law κ)
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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