import OAI.Analysis.FiniteFactor.Matrix.Integrals
import Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

namespace OAI

namespace FiniteFactor
open MeasureTheory Filter
open scoped ENNReal ComplexConjugate
noncomputable section

def sourceFiberInner (f g : SourceHilbert SourceCoordinate) (x : SourceCoordinate → ℝ) : ℂ :=
  ∫ n : ℤ, inner ℂ (f (x,n)) (g (x,n)) ∂Measure.count

theorem sourceFiberInner_integrable (f g : SourceHilbert SourceCoordinate) :
    Integrable (sourceFiberInner f g) sourceBaseMeasure :=
  (L2.integrable_inner (𝕜 := ℂ) f g).integral_prod_left

theorem source_inner_fiber (f g : SourceHilbert SourceCoordinate) :
    inner ℂ f g = ∫ x, sourceFiberInner f g x ∂sourceBaseMeasure :=
  integral_prod _ (L2.integrable_inner (𝕜 := ℂ) f g)

theorem sourceScalar_indicator_inner (A : Set (SourceCoordinate → ℝ)) (hA : MeasurableSet A)
    (f g : SourceHilbert SourceCoordinate) :
    inner ℂ (sourceScalar (sourceBaseIndicator A hA) f) g =
      ∫ x in A, sourceFiberInner f g x ∂sourceBaseMeasure := by
  rw [source_inner_fiber, ← integral_indicator hA]
  apply integral_congr_ae
  filter_upwards [source_ae_all_fibers (sourceScalar_apply_ae (sourceBaseIndicator A hA) f)] with x hx
  unfold sourceFiberInner
  by_cases ha : x ∈ A
  · simp only [Set.indicator_of_mem ha]
    apply integral_congr_ae
    exact Eventually.of_forall fun n => by dsimp only; rw [hx]; simp [sourceBaseIndicator, ha]
  · simp only [Set.indicator_of_notMem ha]
    have he : (fun n : ℤ => inner ℂ ((sourceScalar (sourceBaseIndicator A hA) f) (x,n)) (g (x,n))) =
        fun _ => (0 : ℂ) := by
      funext n
      rw [hx]
      simp [sourceBaseIndicator, ha]
    rw [he, integral_zero]

/-- The scalar distribution identity determining every output coordinate of H. -/
theorem source_coordinate_action_inner (θ : SourceCoordinate → ℝ)
    {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
    (hH : H ∈ sourceAlgebra θ) (f : SourceHilbert SourceCoordinate) (k : ℤ) :
    (fun x => (H f) (x,k)) =ᵐ[sourceBaseMeasure]
      sourceFiberInner ((star H) (sourceBasis k)) f := by
  apply Integrable.ae_eq_of_forall_setIntegral_eq _ _ (source_fiber_integrable _ _)
    (sourceFiberInner_integrable _ _)
  intro A hA _
  rw [← source_fiber_integral, ← L2.inner_indicatorConstLp_one]
  change inner ℂ (sourceFiberIndicator A hA k) (H f) = _
  rw [← ContinuousLinearMap.adjoint_inner_left, ← ContinuousLinearMap.star_eq_adjoint]
  have he : (star H) (sourceFiberIndicator A hA k) =
      sourceScalar (sourceBaseIndicator A hA) ((star 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 (star_mem hH)
      (sourceBaseIndicator A hA))).symm
  rw [he, sourceScalar_indicator_inner]

/-- Actual action on arbitrary L2 input, expressed through the constructed
matrix entries. The counting integral is an absolutely convergent series a.e. -/
theorem sourceMatrix_action (θ : SourceCoordinate → ℝ)
    {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
    (hH : H ∈ sourceAlgebra θ) (f : SourceHilbert SourceCoordinate) (k : ℤ) :
    (fun x => (H f) (x,k)) =ᵐ[sourceBaseMeasure]
      fun x => ∫ n : ℤ, sourceMatrixEntry H k n x * f (x,n) ∂Measure.count := by
  have hs : ∀ᵐ x ∂sourceBaseMeasure, ∀ n : ℤ,
      sourceMatrixEntry (star H) n k x = star (sourceMatrixEntry H k n x) := by
    rw [ae_all_iff]
    intro n
    exact sourceMatrixEntry_star θ hH n k
  filter_upwards [source_coordinate_action_inner θ hH f k, hs] with x hx hs
  rw [hx]
  unfold sourceFiberInner
  apply integral_congr_ae
  exact Eventually.of_forall fun n => by
    change inner ℂ (sourceMatrixEntry (star H) n k x) (f (x,n)) = _
    rw [hs]
    simp [RCLike.inner_apply, mul_comm]

/-- The matrix action is absolutely integrable over the counting measure. -/
theorem sourceMatrix_action_integrable (θ : SourceCoordinate → ℝ)
    {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
    (hH : H ∈ sourceAlgebra θ) (f : SourceHilbert SourceCoordinate) (k : ℤ) :
    ∀ᵐ x ∂sourceBaseMeasure,
      Integrable (fun n : ℤ => sourceMatrixEntry H k n x * f (x,n)) Measure.count := by
  have hs : ∀ᵐ x ∂sourceBaseMeasure, ∀ n : ℤ,
      sourceMatrixEntry (star H) n k x = star (sourceMatrixEntry H k n x) := by
    rw [ae_all_iff]
    intro n
    exact sourceMatrixEntry_star θ hH n k
  filter_upwards [(L2.integrable_inner (𝕜 := ℂ) ((star H) (sourceBasis k)) f).prod_right_ae,
    hs] with x hi hs
  apply hi.congr
  exact Eventually.of_forall fun n => by
    change inner ℂ (sourceMatrixEntry (star H) n k x) (f (x,n)) = _
    rw [hs]
    simp [RCLike.inner_apply, mul_comm]

theorem sourceMatrix_action_tsum (θ : SourceCoordinate → ℝ)
    {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
    (hH : H ∈ sourceAlgebra θ) (f : SourceHilbert SourceCoordinate) (k : ℤ) :
    ∀ᵐ x ∂sourceBaseMeasure,
      (H f) (x,k) = ∑' n : ℤ, sourceMatrixEntry H k n x * f (x,n) := by
  filter_upwards [sourceMatrix_action θ hH f k, sourceMatrix_action_integrable θ hH f k]
    with x hx hi
  rw [hx, integral_countable hi]
  simp [measureReal_def]

/-- Products are given by the matrix product on a single conull base set. -/
theorem sourceMatrix_mul (θ : SourceCoordinate → ℝ)
    {H B : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
    (hH : H ∈ sourceAlgebra θ) :
    ∀ᵐ x ∂sourceBaseMeasure, ∀ k n : ℤ,
      sourceMatrixEntry (H * B) k n x =
        ∫ r : ℤ, sourceMatrixEntry H k r x * sourceMatrixEntry B r n x ∂Measure.count := by
  rw [ae_all_iff]
  intro k
  rw [ae_all_iff]
  intro n
  exact sourceMatrix_action θ hH (B (sourceBasis n)) k

end
end FiniteFactor

end OAI