import OAI.Probability.InvariantIsing.Cavity.CavityPhysicalLogLimit
import OAI.Probability.InvariantIsing.Cavity.CavityCanonicalHaarLog

namespace OAI

/-! The spectral/GG logarithmic limit in the canonical coordinates of the
physical pressure increment. -/

noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter
open scoped Topology BigOperators

namespace InvariantIsing

theorem cavity_canonical_capped_log_limit {m d n : ℕ}
    (N depth : ℕ → ℕ) (hN : ∀ r, 0<N r) (k : ℕ → Fin m → ℕ)
    (e : (r : ℕ) → (((a : Fin m) × Fin (k r a)) ⊕ Fin d) ≃ Fin (N r))
    (a₀ : Fin d → Fin m) (hk : ∀ r a, d ≤ k r a)
    (hgroups : ∀ r a, 0<cavityBaseGroupDimension (k r) a₀ a)
    (hdims : ∀ a, Tendsto (fun r => cavityBaseGroupDimension (k r) a₀ a) atTop atTop)
    (μ : (r : ℕ) → Measure (Orthogonal (N r))) [∀ r, IsProbabilityMeasure (μ r)]
    (θ : (r : ℕ) → Measure (LabeledTree (depth r))) [∀ r, IsProbabilityMeasure (θ r)]
    (μG : (r : ℕ) → (a : Fin m) → Measure (Orthogonal (cavityBaseGroupDimension (k r) a₀ a)))
    [∀ r a, IsProbabilityMeasure (μG r a)] [∀ r a, (μG r a).IsMulRightInvariant]
    (lam : Fin m → ℝ) (v : ℕ → Fin m → ℝ) (u : ℕ → ℕ → ℝ)
    {c : ℝ} (hc : 0<c)
    (hfrac : ∀ r a, c ≤ (cavityBaseGroupDimension (k r) a₀ a : ℝ)/N r)
    (ρ : Fin m → ℝ) (hρ : ∀ a, 0<ρ a) (hρsum : ∑ a, ρ a=1)
    (hρlim : Tendsto (fun r a => (cavityBaseGroupDimension (k r) a₀ a : ℝ)/N r) atTop (𝓝 ρ))
    (Q₀ : ProbabilityMeasure (SpectralArray (m+1)))
    (hlim : Tendsto (fun r => cavityRotationArrayLaw (μ r) (θ r)
      (diagonalPerturbedEigenvalues
        (fun i => lam ((cavityBaseGroupEquiv (k r) (e r) a₀).symm i).1)
        (cavitySpectralGroup (fun i => ((cavityBaseGroupEquiv (k r) (e r) a₀).symm i).1)) (v r) 1)
      (cavitySpectralGroup (fun i => ((cavityBaseGroupEquiv (k r) (e r) a₀).symm i).1))
      (u r)) atTop (𝓝 Q₀))
    (hgg : HasEntryGhirlandaGuerra (fun x i j => x (i,j)) (Q₀ : Measure (SpectralArray (m+1))))
    (hG : ∀ᵐ x ∂(Q₀ : Measure (SpectralArray (m+1))), SpectralGram x)
    (d₀ : Fin (m+1) → ℝ) (hd0 : ∀ j, 0≤d₀ j)
    (hd : ∀ᵐ x ∂(Q₀ : Measure (SpectralArray (m+1))), ∀ i j, (x (i,i) j : ℝ)=d₀ j)
    (hE : ∀ e : ℕ → ℕ, Function.Injective e →
      (Q₀ : Measure (SpectralArray (m+1))).map (permuteSpectralArray e)=Q₀)
    (hP : ∀ᵐ x ∂(Q₀ : Measure (SpectralArray (m+1))), SpectralPartitionGeometry m x)
    (hn : ∀ᵐ x ∂(Q₀ : Measure (SpectralArray (m+1))), ∀ j, 0≤(x (0,1) j : ℝ))
    (hoff : ∀ j l, ∀ Φ : ℝ → ℝ, Continuous Φ → ∀ B : ℝ, 0≤B → (∀ t, |Φ t|≤B) →
      spectralOffWardResidual Q₀ ρ lam j l Φ=0)
    (hdiag : ∀ j l, spectralDiagonalWardResidual Q₀ ρ lam j l=0)
    (A : CavityFactorBlocks d n) (cap : ℝ) (hcap : 0≤cap) :
    let p := spectralSpinQuantilePath Q₀ hP hn
    let Qf := fun r => cavityLabeledDisorderLaw r (chainExponent (uniformCut r))
      (cavityFiniteRootCovariance ρ lam hρ hρsum a₀ (cavityStrictUniformPath p r)
        (cavityStrictUniformLevels p r))
      (cavityFiniteNoiseCovariance ρ lam hρ hρsum a₀ (cavityStrictUniformPath p r)
        (uniformCut r) (cavityStrictUniformLevels p r))
    let η := fun r => cavityLabeledPriorKernel r
      (cavityFiniteCovariancePath ρ lam hρ hρsum a₀ (cavityStrictUniformPath p r)
        (cavityStrictUniformLevels p r) r) (uniformSpinPrior n)
    Tendsto (fun r =>
      (∫ ω, cavityCanonicalHaarLog (k r) (e r) a₀ (hk r) lam (v r) (u r) 1 cap 0 A ω
        ∂(((μ r).prod (θ r)).prod gaussianCoordinates).prod (Measure.pi (μG r))) -
      ∫ ω, Real.log (∫ x, Real.exp (min (cavityLabeledPotential r A.1 A.2.1 A.2.2 (ω,x)) cap)
        ∂η r ω) ∂Qf r) atTop (𝓝 0) := by
  intro p Qf η
  let dims := fun r => cavityBaseGroupDimension (k r) a₀
  let E := fun r => cavityBaseGroupEquiv (k r) (e r) a₀
  let frames := fun r => cavityCanonicalGroupFrame (k r) (e r) a₀ (hk r)
  let eig := fun r => diagonalPerturbedEigenvalues (fun i => lam ((E r).symm i).1)
    (cavitySpectralGroup (fun i => ((E r).symm i).1)) (v r) 1
  have hinj : Function.Injective (fun j : Fin d => (a₀ j,j)) := by
    intro i j hij
    exact congrArg Prod.snd hij
  have h := cavity_physical_capped_log_limit N depth hN dims hgroups hdims E μ θ μG frames
    (fun r a => cavityCanonicalGroupFrame_gram (k r) (e r) a₀ (hk r) a) eig u hc hfrac
    ρ lam hρ hρsum hρlim Q₀ hlim hgg hG d₀ hd0 hd hE hP hn hoff hdiag
    a₀ (fun j => (a₀ j,j)) hinj (fun _ => rfl) A.1 A.2.1 A.2.2 cap hcap
  simpa only [cavityCanonicalHaarLog, cavityHaarLogObservable, cavityHaarSpinPotential,
    one_mul, zero_mul, sub_zero, dims, E, frames, eig, p, Qf, η] using h

end InvariantIsing

end

end OAI