lean/OAI/Computability/VertexCover/Reduction/AverageCommute.lean

import Mathlib
import OAI.Computability.VertexCover.Analysis.IntegralUpdate
import OAI.Computability.VertexCover.Analysis.Restriction

namespace OAI

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namespace VertexCover.Restriction
open MeasureTheory
open scoped BigOperators

namespace EnergyForm

section Integral
variable {X : Type*} [TopologicalSpace X] [CompactSpace X]
  [MeasurableSpace X] [BorelSpace X] (μ : Measure X) [IsFiniteMeasure μ]

theorem continuous_integrable (f : C(X, ℝ)) : Integrable f μ :=
  f.continuous.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)

theorem continuous_mul_integrable (f g : C(X, ℝ)) :
    Integrable (fun x => f x * g x) μ :=
  (f.continuous.mul g.continuous).integrable_of_hasCompactSupport
    (HasCompactSupport.of_compactSpace _)

noncomputable def integral : EnergyForm C(X, ℝ) where
  form :=
    { toFun := fun f =>
        { toFun := fun g => ∫ x, f x * g x ∂μ
          map_add' := fun g h => by
            simp only [ContinuousMap.add_apply, mul_add]
            exact integral_add (continuous_mul_integrable μ f g)
              (continuous_mul_integrable μ f h)
          map_smul' := fun a g => by
            simp only [ContinuousMap.smul_apply, smul_eq_mul]
            simp_rw [mul_left_comm (f _) a]
            exact integral_const_mul _ _ }
      map_add' := fun f g => by
        ext h
        simp only [LinearMap.add_apply, ContinuousMap.add_apply, add_mul]
        exact integral_add (continuous_mul_integrable μ f h)
          (continuous_mul_integrable μ g h)
      map_smul' := fun a f => by
        ext g
        simp only [LinearMap.smul_apply, ContinuousMap.smul_apply, smul_eq_mul]
        simp_rw [mul_assoc]
        exact integral_const_mul _ _ }
  symm f g := by
    change (∫ x, f x * g x ∂μ) = ∫ x, g x * f x ∂μ
    simp only [mul_comm]
  nonneg f := integral_nonneg (fun x => mul_self_nonneg (f x))

@[simp] theorem integral_form (f g : C(X, ℝ)) :
    (integral μ).form f g = ∫ x, f x * g x ∂μ := rfl

@[simp] theorem integral_energy (f : C(X, ℝ)) :
    (integral μ).energy f = ∫ x, (f x)^2 ∂μ := by
  simp only [energy, integral_form, pow_two]

end Integral

namespace Projection
section Averaging
variable {ι X : Type*} [Fintype ι] [DecidableEq ι]
  [TopologicalSpace X] [CompactSpace X] [SecondCountableTopology X]
  [MeasurableSpace X] [BorelSpace X]
  (μ : Measure X) [IsProbabilityMeasure μ]

noncomputable def updateFunction (i : ι) (f : C(ι → X, ℝ)) : C(X, C(ι → X, ℝ)) :=
  ContinuousMap.curry ⟨fun p : X × (ι → X) => f (Function.update p.2 i p.1),
    f.continuous.comp (continuous_snd.update i continuous_fst)⟩

omit [Fintype ι] [CompactSpace X] [SecondCountableTopology X] [MeasurableSpace X]
  [BorelSpace X] in
@[simp] theorem updateFunction_apply (i : ι) (f : C(ι → X, ℝ)) (x : X) (s : ι → X) :
    updateFunction i f x s = f (Function.update s i x) := rfl

omit [Fintype ι] [SecondCountableTopology X] in
theorem updateFunction_integrable (i : ι) (f : C(ι → X, ℝ)) :
    Integrable (updateFunction i f) μ :=
  (updateFunction i f).continuous.integrable_of_hasCompactSupport
    (HasCompactSupport.of_compactSpace _)

noncomputable def averageMap (i : ι) : C(ι → X, ℝ) →ₗ[ℝ] C(ι → X, ℝ) where
  toFun f := ∫ x, updateFunction i f x ∂μ
  map_add' f g := by
    have he : (fun x => updateFunction i (f + g) x) =
        (fun x => updateFunction i f x + updateFunction i g x) := by
      funext x
      ext s
      rfl
    rw [he, integral_add (updateFunction_integrable μ i f)
      (updateFunction_integrable μ i g)]
  map_smul' a f := by
    have he : (fun x => updateFunction i (a • f) x) =
        (fun x => a • updateFunction i f x) := by
      funext x
      ext s
      rfl
    rw [he, integral_smul]
    rfl

omit [Fintype ι] [SecondCountableTopology X] in
@[simp] theorem averageMap_apply (i : ι) (f : C(ι → X, ℝ)) (s : ι → X) :
    averageMap μ i f s = ∫ x, f (Function.update s i x) ∂μ :=
  ContinuousMap.integral_apply (updateFunction_integrable μ i f) s

omit [Fintype ι] [SecondCountableTopology X] in
theorem average_update (i : ι) (f : C(ι → X, ℝ)) (s : ι → X) (x : X) :
    averageMap μ i f (Function.update s i x) = averageMap μ i f s := by
  simp only [averageMap_apply, Function.update_idem]

omit [Fintype ι] [SecondCountableTopology X] in
theorem average_idem (i : ι) (f : C(ι → X, ℝ)) :
    averageMap μ i (averageMap μ i f) = averageMap μ i f := by
  ext s
  simp only [averageMap_apply, Function.update_idem]
  simp

theorem integral_average_mul_fixed (i : ι) (f g : C(ι → X, ℝ))
    (hg : ∀ s x, g (Function.update s i x) = g s) :
    (∫ s, averageMap μ i f s * g s ∂Measure.pi (fun _ : ι => μ)) =
      ∫ s, f s * g s ∂Measure.pi (fun _ : ι => μ) := by
  have hi : Integrable (fun p : (ι → X) × X =>
      f (Function.update p.1 i p.2) * g (Function.update p.1 i p.2))
      ((Measure.pi (fun _ : ι => μ)).prod μ) :=
    ((f.continuous.comp (continuous_fst.update i continuous_snd)).mul
      (g.continuous.comp (continuous_fst.update i continuous_snd))).integrable_of_hasCompactSupport
        (HasCompactSupport.of_compactSpace _)
  calc
    _ = ∫ s, ∫ x, f (Function.update s i x) * g (Function.update s i x)
        ∂μ ∂Measure.pi (fun _ : ι => μ) := by
      simp only [hg, integral_mul_const, averageMap_apply]
    _ = ∫ x, ∫ s, f (Function.update s i x) * g (Function.update s i x)
        ∂Measure.pi (fun _ : ι => μ) ∂μ := integral_integral_swap hi
    _ = _ := VertexCover.Product.integral_update μ i (continuous_mul_integrable _ f g)

noncomputable def average (i : ι) :
    (integral (Measure.pi (fun _ : ι => μ))).Projection where
  toLinearMap := averageMap μ i
  idem := average_idem μ i
  selfAdjoint f g := by
    change (∫ s, averageMap μ i f s * g s ∂Measure.pi (fun _ : ι => μ)) =
      ∫ s, f s * averageMap μ i g s ∂Measure.pi (fun _ : ι => μ)
    have h₁ := integral_average_mul_fixed μ i g (averageMap μ i f)
      (average_update μ i f)
    have h₂ := integral_average_mul_fixed μ i f (averageMap μ i g)
      (average_update μ i g)
    calc
      _ = ∫ s, g s * averageMap μ i f s ∂Measure.pi (fun _ : ι => μ) := by
        apply integral_congr_ae
        exact Filter.Eventually.of_forall (fun s => mul_comm _ _)
      _ = ∫ s, averageMap μ i g s * averageMap μ i f s
          ∂Measure.pi (fun _ : ι => μ) := h₁.symm
      _ = ∫ s, averageMap μ i f s * averageMap μ i g s
          ∂Measure.pi (fun _ : ι => μ) := by
        apply integral_congr_ae
        exact Filter.Eventually.of_forall (fun s => mul_comm _ _)
      _ = _ := h₂

@[simp] theorem average_apply (i : ι) (f : C(ι → X, ℝ)) (s : ι → X) :
    average μ i f s = ∫ x, f (Function.update s i x) ∂μ := averageMap_apply μ i f s

theorem average_invariant (i : ι) (f : C(ι → X, ℝ)) (s : ι → X) (x : X) :
    average μ i f (Function.update s i x) = average μ i f s := average_update μ i f s x

theorem average_commute (i j : ι) (f : C(ι → X, ℝ)) :
    average μ i (average μ j f) = average μ j (average μ i f) := by
  by_cases hij : i = j
  · subst j
    rfl
  ext s
  simp only [average_apply]
  have hi : Integrable (fun p : X × X =>
      f (Function.update (Function.update s i p.1) j p.2)) (μ.prod μ) :=
    (f.continuous.comp ((continuous_const.update i continuous_fst).update j
      continuous_snd)).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)
  rw [integral_integral_swap hi]
  simp only [Function.update_comm hij]

end Averaging
end Projection
end EnergyForm
end VertexCover.Restriction


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end

end OAI