import OAI.Analysis.FiniteFactor.CrossedProduct.FiniteVectors
import Mathlib.Topology.Instances.Complex

namespace OAI

namespace FiniteFactor
open MeasureTheory Filter
open scoped ENNReal
noncomputable section

/-- Continuity in a separable parameter lets a countable dense family of
almost-everywhere inequalities determine every parameter on one conull set. -/
theorem ae_forall_le_of_continuous {X E : Type*} [MeasurableSpace X]
    (μ : Measure X) [TopologicalSpace E] [TopologicalSpace.SeparableSpace E]
    (f g : E → X → ℝ) (hf : ∀ x, Continuous (fun e => f e x))
    (hg : ∀ x, Continuous (fun e => g e x))
    (h : ∀ e, ∀ᵐ x ∂μ, f e x ≤ g e x) :
    ∀ᵐ x ∂μ, ∀ e, f e x ≤ g e x := by
  obtain ⟨s, hsc, hsd⟩ := TopologicalSpace.exists_countable_dense E
  have hs : ∀ᵐ x ∂μ, ∀ e ∈ s, f e x ≤ g e x :=
    (ae_ball_iff hsc).mpr fun e _ => h e
  filter_upwards [hs] with x hx e
  exact le_on_closure hx (hf x).continuousOn (hg x).continuousOn (hsd e)

theorem sourceFiberEnergy_finite_le (f : SourceHilbert SourceCoordinate) (s : Finset ℤ) :
    ∀ᵐ x ∂sourceBaseMeasure, ∑ n ∈ s, ‖f (x,n)‖ ^ 2 ≤ sourceFiberEnergy f x := by
  filter_upwards [(source_norm_sq_integrable f).prod_right_ae] with x hx
  have h := setIntegral_le_integral (μ := Measure.count) (s := (s : Set ℤ))
    hx (Eventually.of_forall fun n => sq_nonneg ‖f (x,n)‖)
  rw [setIntegral_finset s hx.integrableOn] at h
  simpa [measureReal_def, sourceFiberEnergy] using h

/-- Each finite scalar matrix compression satisfies the actual operator norm bound. -/
theorem sourceMatrix_finite_bound (θ : SourceCoordinate → ℝ)
    {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
    (hH : H ∈ sourceAlgebra θ) (s t : Finset ℤ) (a : ℤ → ℂ) :
    ∀ᵐ x ∂sourceBaseMeasure,
      (∑ k ∈ t, ‖∑ n ∈ s, a n * sourceMatrixEntry H k n x‖ ^ 2) ≤
        ‖H‖ ^ 2 * ∑ n ∈ s, ‖a n‖ ^ 2 := by
  filter_upwards [sourceFiniteVector_image_coe H s a,
    sourceFiniteVector_energy s a,
    sourceFiberEnergy_finite_le (H (sourceFiniteVector s a)) t,
    sourceAlgebra_fiberEnergy_le θ hH (sourceFiniteVector s a)] with x hc he hf hh
  calc
    _ = ∑ k ∈ t, ‖(H (sourceFiniteVector s a)) (x,k)‖ ^ 2 := by simp_rw [hc]
    _ ≤ sourceFiberEnergy (H (sourceFiniteVector s a)) x := hf
    _ ≤ ‖H‖ ^ 2 * sourceFiberEnergy (sourceFiniteVector s a) x := hh
    _ = _ := by rw [he]

/-- One conull set works for every complex finite input vector and every finite
output truncation. No field or boundedness hypothesis has been substituted. -/
theorem sourceMatrix_finite_bound_all (θ : SourceCoordinate → ℝ)
    {H : SourceHilbert SourceCoordinate →L[ℂ] SourceHilbert SourceCoordinate}
    (hH : H ∈ sourceAlgebra θ) :
    ∀ᵐ x ∂sourceBaseMeasure, ∀ (s t : Finset ℤ) (a : ℤ → ℂ),
      (∑ k ∈ t, ‖∑ n ∈ s, a n * sourceMatrixEntry H k n x‖ ^ 2) ≤
        ‖H‖ ^ 2 * ∑ n ∈ s, ‖a n‖ ^ 2 := by
  rw [ae_all_iff]
  intro s
  rw [ae_all_iff]
  intro t
  apply ae_forall_le_of_continuous (E := ℤ → ℂ) sourceBaseMeasure
    (fun a x => ∑ k ∈ t, ‖∑ n ∈ s, a n * sourceMatrixEntry H k n x‖ ^ 2)
    (fun a _ => ‖H‖ ^ 2 * ∑ n ∈ s, ‖a n‖ ^ 2)
  · intro x; fun_prop
  · intro x; fun_prop
  · exact sourceMatrix_finite_bound θ hH s t

end
end FiniteFactor

end OAI