Datasets:

roseking
/

openAI-math-dataset

import Mathlib
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


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end OAI

	import Mathlib
	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


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	end OAI