import OAI.Analysis.FiniteFactor.Matrix.Entries

namespace OAI

namespace FiniteFactor
open MeasureTheory Filter
open scoped ENNReal ComplexConjugate
noncomputable section

theorem source_fiber_restrict (A : Set (SourceCoordinate → ℝ)) (n : ℤ) :
    (sourceMeasure SourceCoordinate).restrict (A ×ˢ {n}) =
      (sourceBaseMeasure.restrict A).map (fun x => (x,n)) := by
  rw [← Measure.prod_restrict]
  simp only [Measure.restrict_singleton, Measure.count_singleton, one_smul]
  exact Measure.prod_dirac _

theorem source_fiber_integral (f : SourceHilbert SourceCoordinate)
    (A : Set (SourceCoordinate → ℝ)) (n : ℤ) :
    ∫ z in A ×ˢ {n}, f z ∂sourceMeasure SourceCoordinate =
      ∫ x in A, f (x,n) ∂sourceBaseMeasure := by
  rw [source_fiber_restrict]
  exact integral_map (measurable_id.prodMk measurable_const).aemeasurable
    (Lp.stronglyMeasurable f).aestronglyMeasurable

theorem source_fiber_integrable (f : SourceHilbert SourceCoordinate) (n : ℤ) :
    Integrable (fun x => f (x,n)) sourceBaseMeasure := by
  have hi := integrableOn_Lp_of_measure_ne_top f (by norm_num : (1 : ℝ≥0∞) ≤ 2)
    (sourceFiberMeasure_ne_top Set.univ n)
  rw [IntegrableOn, source_fiber_restrict, Measure.restrict_univ] at hi
  exact (integrable_map_measure (Lp.stronglyMeasurable f).aestronglyMeasurable
    (measurable_id.prodMk measurable_const).aemeasurable).mp hi

theorem sourceMatrixEntry_measurable
    (H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) (k n : ℤ) :
    Measurable (sourceMatrixEntry H k n) :=
  (Lp.stronglyMeasurable (H (sourceBasis n))).measurable.comp
    (measurable_id.prodMk measurable_const)

theorem sourceMatrixEntry_integrable
    (H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) (k n : ℤ) :
    Integrable (sourceMatrixEntry H k n) sourceBaseMeasure :=
  source_fiber_integrable _ _

theorem sourceMatrixEntry_setIntegral
    (H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate)
    (A : Set (SourceCoordinate → ℝ)) (hA : MeasurableSet A) (k n : ℤ) :
    ∫ x in A, sourceMatrixEntry H k n x ∂sourceBaseMeasure =
      inner ℂ (sourceFiberIndicator A hA k) (H (sourceBasis n)) := by
  rw [sourceFiberIndicator, L2.inner_indicatorConstLp_one, source_fiber_integral]
  rfl

/-- The displayed trace integral is the actual previously proved normal state. -/
theorem sourceVacuumFunctional_matrixIntegral
    (H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate) :
    sourceVacuumFunctional H = ∫ x, sourceMatrixEntry H 0 0 x ∂sourceBaseMeasure := by
  rw [← setIntegral_univ, sourceMatrixEntry_setIntegral H Set.univ MeasurableSet.univ]
  rfl

/-- Adjoint matrix coefficients agree almost everywhere, on the actual source base. -/
theorem sourceMatrixEntry_star (θ : SourceCoordinate → ℝ)
    {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
    (hH : H ∈ sourceAlgebra θ) (k n : ℤ) :
    sourceMatrixEntry (star H) k n =ᵐ[sourceBaseMeasure]
      fun x => star (sourceMatrixEntry H n k x) := by
  apply Integrable.ae_eq_of_forall_setIntegral_eq _ _
    (sourceMatrixEntry_integrable _ _ _)
    (Complex.conjCLE.integrable_comp_iff.mpr (sourceMatrixEntry_integrable H n k))
  intro A hA _
  change (∫ x in A, sourceMatrixEntry (star H) k n x ∂sourceBaseMeasure) =
    ∫ x in A, conj (sourceMatrixEntry H n k x) ∂sourceBaseMeasure
  rw [sourceMatrixEntry_setIntegral _ A hA, integral_conj, sourceMatrixEntry_setIntegral _ A hA,
    inner_conj_symm]
  rw [ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_right]
  have he : H (sourceFiberIndicator A hA k) =
      sourceScalar (sourceBaseIndicator A hA) (H (sourceBasis k)) := by
    rw [← sourceScalar_indicator_basis]
    exact (congrArg (fun T : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate =>
      T (sourceBasis k)) (sourceAlgebra_commutes_scalar θ hH (sourceBaseIndicator A hA))).symm
  rw [he]
  have hs : star (sourceScalar (sourceBaseIndicator A hA)) =
      sourceScalar (sourceBaseIndicator A hA) := by
    rw [sourceScalar_star]
    congr 1
    unfold sourceBaseStar sourceBaseIndicator
    congr 1
    funext x
    by_cases hx : x ∈ A <;> simp [hx]
  rw [← ContinuousLinearMap.adjoint_inner_right, ← ContinuousLinearMap.star_eq_adjoint,
    hs, sourceScalar_indicator_basis]

end
end FiniteFactor

end OAI