import OAI.Probability.InvariantIsing.Spectral.CompactSpectralPartition
import OAI.Probability.InvariantIsing.Spectral.SpectralPartitionPressure
import OAI.Probability.InvariantIsing.Cavity.CavitySpectralSqueeze

namespace OAI

/-! The finite positive spectral partitions bracket physical mean pressure. -/

noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter Set
open scoped Topology Classical Function

namespace InvariantIsing.CompactSpectralPartition

variable {ν : ProbabilityMeasure ℝ} {a b δ : ℝ} (D : CompactSpectralPartition ν a b δ)

def lowerValue (x : ℝ) : ℝ :=
  D.value (positiveSpectralLabel D.weight D.lower
    (spectralPartitionIndex D.cells (spectralBallCell_cover _) x))

def upperValue (x : ℝ) : ℝ :=
  D.value (positiveSpectralLabel D.weight D.upper
    (spectralPartitionIndex D.cells (spectralBallCell_cover _) x))

lemma rounding_bounds (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
    (x : ℝ) (hx : x∈Icc a b) : D.lowerValue x-δ ≤ x ∧ x ≤ D.upperValue x+δ := by
  have he := D.endpoint_bounds ha hb
  exact positive_spectral_rounding_bounds (ν : Measure ℝ) D.cells (spectralBallCell_cover _)
    D.value a b δ D.lower D.upper he.1 he.2
    (positive_spectral_ball_cell_close (ν : Measure ℝ) D.center D.radius a δ
      D.radius_le D.residual_zero) x hx

lemma rounded_mem (hab : a ≤ b) (x : ℝ) :
    D.lowerValue x∈Icc a b ∧ D.upperValue x∈Icc a b :=
  ⟨D.value_mem hab _,D.value_mem hab _⟩

lemma pressure_tendsto
    (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput)
    (hpub : PanchenkoTalagrandFieldPairInput)
    (μ : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (μ N)]
    [∀ N, (μ N).IsMulRightInvariant] (eig : (N : ℕ) → Fin N → ℝ)
    (hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1)))
      atTop (𝓝 ν)) :
    Tendsto (fun N => ∫ V, rotatedPressure (fun i => D.lowerValue (eig N i))
      (matrixRotation V⁻¹) (fun _ => 0) ∂μ N) atTop
      (𝓝 (variationalFunctional (measureR (D.law : Measure ℝ) D.edge)).toReal) ∧
    Tendsto (fun N => ∫ V, rotatedPressure (fun i => D.upperValue (eig N i))
      (matrixRotation V⁻¹) (fun _ => 0) ∂μ N) atTop
      (𝓝 (variationalFunctional (measureR (D.law : Measure ℝ) D.edge)).toReal) := by
  have hh (fallback : D.Positive) := spectral_partition_pressure_tendsto hhaar hgauss hpub μ eig ν hweak
    D.cells (spectralBallCell_cover _) (spectralBallCell_disjoint _)
    (spectralBallCell_measurable _ (fun _ => measurableSet_ball))
    (spectralBallCell_null_boundary (ν : Measure ℝ) _ D.null_boundary)
    D.value fallback D.upper D.le_upper
  exact ⟨hh D.lower,hh D.upper⟩

lemma mean_pressure_bounds {N : ℕ} (hN : 0 < N)
    (μ : Measure (Orthogonal N)) [IsProbabilityMeasure μ] (eig : Fin N → ℝ)
    (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support)
    (hab : a ≤ b) (heig : ∀ i, eig i∈Icc a b) :
    (∫ V, rotatedPressure (fun i => D.lowerValue (eig i))
      (matrixRotation V⁻¹) (fun _ => 0) ∂μ)-δ/2 ≤
        ∫ V, rotatedPressure eig (matrixRotation V⁻¹) (fun _ => 0) ∂μ ∧
    (∫ V, rotatedPressure eig (matrixRotation V⁻¹) (fun _ => 0) ∂μ) ≤
      (∫ V, rotatedPressure (fun i => D.upperValue (eig i))
        (matrixRotation V⁻¹) (fun _ => 0) ∂μ)+δ/2 := by
  let K := max |a| |b|
  have hbound (x : ℝ) (hx : x∈Icc a b) : |x| ≤ K := by
    apply abs_le.mpr
    constructor
    · have hh := neg_abs_le a
      have hk : |a| ≤ K := le_max_left _ _
      linarith [hx.1]
    · exact hx.2.trans ((le_abs_self b).trans (le_max_right _ _))
  exact cavity_meanPressure_squeeze hN μ eig (fun i => D.lowerValue (eig i))
    (fun i => D.upperValue (eig i)) (fun _ => 0) δ K
    (fun i => hbound _ (heig i))
    (fun i => hbound _ (D.rounded_mem hab (eig i)).1)
    (fun i => hbound _ (D.rounded_mem hab (eig i)).2)
    (fun i => (D.rounding_bounds ha hb _ (heig i)).1)
    (fun i => (D.rounding_bounds ha hb _ (heig i)).2)

end InvariantIsing.CompactSpectralPartition

end

end OAI