lean/OAI/Analysis/InterpolatedFactors/AmbientSystem.lean

import OAI.Analysis.InterpolatedFactors.Ambient

namespace OAI

noncomputable section

universe u_1 u_2

namespace FreeGroupFactorMain.WeightedFock

section
open FreeFock MeasureTheory CyclicSpectral
variable {ι : Type u_1} [DecidableEq ι]

theorem vacuum_norm_one (u v : ℝ) (hunit : u^2+v^2=1) : ‖(vacuum u v:H ι)‖=1 := by
  have hh := vacuumTrace_one (ι := ι) u v
  rw [hunit,Complex.ofReal_one] at hh
  change inner ℂ (vacuum u v:H ι) (vacuum u v)=1 at hh
  have hr := congrArg Complex.re hh
  rw [inner_self_eq_norm_sq_to_K] at hr
  have hr : ‖(vacuum u v:H ι)‖^2=1 := by exact_mod_cast hr
  nlinarith [norm_nonneg (vacuum u v:H ι)]

theorem diagonalGenerator_noAtoms (u v : ℝ) (hu : u≠0) (hv : v≠0) (i : ι) :
    NullSingletonClass (measure (diagonalGenerator u v i) (diagonalGenerator_selfAdjoint u v i) (vacuum u v)) := by
  let : InnerProductSpace ℝ (H ι) := InnerProductSpace.rclikeToReal ℂ (H ι)
  apply noAtoms
  intro r η hη
  have he : diagonalGenerator u v i η=(r:ℂ) • η := by
    exact hη.trans (by
      apply Subtype.ext
      funext w
      change r • η w=(r:ℂ)*(η w)
      exact Complex.real_smul)
  have hh := diagonalGenerator_eigen_root_zero u v hu hv i (r:ℂ) η he
  change (inner ℂ (vacuum u v) η).re=0
  rw [hh,Complex.zero_re]

variable (u v : ℝ) (hu : u≠0) (hv : v≠0) (hunit : u^2+v^2=1)

def ambientUniform : UniformAlgebra →⋆ₐ[ℂ] Operator ℕ := by
  letI := probability _ (diagonalGenerator_selfAdjoint u v (0:ℕ)) (vacuum u v) (vacuum_norm_one u v hunit)
  letI := diagonalGenerator_noAtoms u v hu hv (0:ℕ)
  exact uniformHom (diagonalGenerator u v (0:ℕ)) (diagonalGenerator_selfAdjoint u v 0) (vacuum u v)

theorem ambientUniform_mem (a : UniformAlgebra) : ambientUniform u v hu hv hunit a∈diagonalBase u v (0:ℕ) := by
  let := probability _ (diagonalGenerator_selfAdjoint u v (0:ℕ)) (vacuum u v) (vacuum_norm_one u v hunit)
  let := diagonalGenerator_noAtoms u v hu hv (0:ℕ)
  exact uniformHom_mem _ _ _ _ (Set.isClosed_centralizer _) (diagonalGenerator_mem u v 0) a

def ambientUniformBase : UniformAlgebra →⋆ₐ[ℂ] diagonalBase u v (0:ℕ) :=
  (ambientUniform u v hu hv hunit).codRestrict _ (ambientUniform_mem u v hu hv hunit)

theorem ambientUniform_state (a : UniformAlgebra) :
    vacuumTrace u v (ambientUniform u v hu hv hunit a)=uniformState a := by
  let := probability _ (diagonalGenerator_selfAdjoint u v (0:ℕ)) (vacuum u v) (vacuum_norm_one u v hunit)
  let := diagonalGenerator_noAtoms u v hu hv (0:ℕ)
  exact uniformHom_state _ _ _ a
end

section
open FreeFock WeightedCornerIndex GaussianRecognition CyclicSpectral

def diagonalAnnihilators (u v : ℝ) : Submodule ℂ (Operator ℕ) where
  carrier := {X | ∀ a : diagonalBase u v (0:ℕ),star X ((a:Operator ℕ) (vacuum u v))=0}
  zero_mem' := by intro a;simp [star_zero (R := Operator ℕ)]
  add_mem' := by intro X Y hX hY a;rw [star_add (R := Operator ℕ),add_apply,hX a,hY a,add_zero]
  smul_mem' := by intro c X hX a;rw [star_smul,smul_apply,hX a,smul_zero]

theorem diagonalAnnihilators_sandwich (u v : ℝ) (p q : diagonalBase u v (0:ℕ)) (X : Operator ℕ)
    (hX : X∈diagonalAnnihilators u v) : (p:Operator ℕ)*X*(q:Operator ℕ)∈diagonalAnnihilators u v := by
  intro a
  rw [star_mul,star_mul,mul_apply_eq_comp,mul_apply_eq_comp]
  have hh := hX (star p*a)
  change star X (star (p:Operator ℕ) ((a:Operator ℕ) (vacuum u v)))=0 at hh
  rw [hh,map_zero]

theorem ambientExtra_annihilator (u v : ℝ) (i : WeightedCornerIndex.Index) : ambientExtra u v i∈diagonalAnnihilators u v := by
  intro a
  unfold ambientExtra
  cases residualSlot i with
  | none => simp [star_zero (R := Operator ℕ)]
  | some b => exact diagonal_base_annihilate u v 0 b a

theorem ambientRaw_annihilator (u v : ℝ) (i : WeightedCornerIndex.Index) (n : ℕ) : ambientRaw u v i n∈diagonalAnnihilators u v := by
  intro a
  exact fresh_C_annihilates_old_vacuum u v {0} (reservoirCode (i,n)+1) (by simp)
    ⟨a,diagonalBase_le_singleton u v 0 a.property⟩

variable (u v : ℝ) (hu : u≠0) (hv : v≠0) (hunit : u^2+v^2=1)

def ambientCreation (i : WeightedCornerIndex.Index × ℕ) : Operator ℕ := (ambientAbsorption u v hu hv).output i.1 i.2

theorem ambientCreation_annihilator (i : WeightedCornerIndex.Index × ℕ) : ambientCreation u v hu hv i∈diagonalAnnihilators u v :=
  (ambientAbsorption u v hu hv).output_mem (diagonalAnnihilators u v) (diagonalAnnihilators_sandwich u v)
    (ambientExtra_annihilator u v) (ambientRaw_annihilator u v) i.1 i.2

theorem ambientCreation_covariance (i j : WeightedCornerIndex.Index × ℕ) (a : UniformAlgebra) :
    star (ambientCreation u v hu hv i)*ambientUniform u v hu hv hunit a*ambientCreation u v hu hv j=
      if i=j then uniformState a • (1:Operator ℕ) else 0 := by
  have hh := (ambientAbsorption u v hu hv).output_cov i.1 j.1 i.2 j.2 (ambientUniformBase u v hu hv hunit a)
  have hs : (ambientAbsorption u v hu hv).τ (ambientUniformBase u v hu hv hunit a)=uniformState a :=
    ambientUniform_state u v hu hv hunit a
  rw [hs] at hh
  change star (ambientCreation u v hu hv i)*ambientUniform u v hu hv hunit a*ambientCreation u v hu hv j=
    (if i.1=j.1 ∧ i.2=j.2 then uniformState a • (1:Operator ℕ) else 0) at hh
  simpa only [← Prod.ext_iff] using hh

def ambientSystem : GaussianRecognition.System UniformAlgebra (H ℕ)
    (WeightedCornerIndex.Index × ℕ) uniformCovariance where
  coeff := ambientUniform u v hu hv hunit
  root := vacuum u v
  create := ambientCreation u v hu hv
  contract i j a := by
    rw [ambientCreation_covariance]
    by_cases h : i=j
    · simp only [uniformCovariance,h,ite_true,map_smul,map_one]
    · simp only [uniformCovariance,ite_eq_right h,map_zero]
  annihilate i a := ambientCreation_annihilator u v hu hv i (ambientUniformBase u v hu hv hunit a)
end

section
open FreeFock CyclicSpectral GaussianRecognition Filter Topology
open scoped ENNReal
variable {ι : Type u_2} [DecidableEq ι]

theorem path_closed_total (K : Submodule ℂ (H ι)) (hK : IsClosed (K:Set (H ι)))
    (hδ : ∀ w : Path ι,lp.single 2 w (1:ℂ)∈K) : K=⊤ := by
  apply top_unique
  intro ξ _
  apply hK.mem_of_tendsto (lp.hasSum_single (by norm_num : (2:ℝ≥0∞)≠⊤) ξ)
  filter_upwards [] with s
  apply K.sum_mem
  intro w _
  have he : lp.single (E := fun _ : Path ι => ℂ) 2 w (ξ w)=(ξ w) • lp.single 2 w 1 := by
    ext v
    simp [lp.single_apply,Pi.single_apply]
  rw [he]
  exact K.smul_mem _ (hδ w)

theorem ambient_commutant_total (u v : ℝ) (hu : u≠0) (hv : v≠0) :
    (Submodule.span ℂ (Set.range (fun B : StarSubalgebra.centralizer ℂ
      (ambientAlgebra (ι := ι) u v:Set (Operator ι)) => (B:Operator ι) (vacuum u v)))).topologicalClosure=⊤ := by
  apply path_closed_total _ (Submodule.isClosed_topologicalClosure _)
  intro w
  obtain ⟨B,hB,heB,_⟩ := good_right_basis u v hu hv w
  have hh : B∈StarSubalgebra.centralizer ℂ (ambientAlgebra (ι := ι) u v:Set (Operator ι)) :=
    centralizer_bicommutant _ ⟨_,hB⟩
  apply Submodule.le_topologicalClosure _
  rw [← heB]
  exact Submodule.subset_span ⟨⟨B,hh⟩,rfl⟩

variable (u v : ℝ) (hu : u≠0) (hv : v≠0) (hunit : u^2+v^2=1)

theorem diagonalGenerator_mem_uniform : diagonalGenerator u v (0:ℕ)∈
    Bicommutant.algebra (Set.range (ambientUniform u v hu hv hunit)) := by
  let : Nontrivial (H ℕ) := nontrivial_of_ne (vacuum u v:H ℕ) 0 (by
    intro h
    have hh := vacuum_norm_one (ι := ℕ) u v hunit
    rw [h,norm_zero] at hh
    exact zero_ne_one hh)
  let := probability _ (diagonalGenerator_selfAdjoint u v (0:ℕ)) (vacuum u v) (vacuum_norm_one u v hunit)
  let := diagonalGenerator_noAtoms u v hu hv (0:ℕ)
  have h := operator_mem_haar_bicommutant (diagonalGenerator u v (0:ℕ)) (diagonalGenerator_selfAdjoint u v 0)
    (vacuum u v) (ambientAlgebra u v) (Set.isClosed_centralizer _)
    (diagonalBase_le_ambient u v 0 (diagonalGenerator_mem u v 0)) (ambient_commutant_total u v hu hv)
  apply Bicommutant.mono _ (Set.range (ambientUniform u v hu hv hunit)) ?_ h
  rintro T rfl
  exact ⟨uniformPhase,(uniformHom_phase (diagonalGenerator u v (0:ℕ)) (diagonalGenerator_selfAdjoint u v 0) (vacuum u v)).symm⟩

theorem diagonalBase_le_uniform : diagonalBase u v (0:ℕ)≤
    Bicommutant.algebra (Set.range (ambientUniform u v hu hv hunit)) := by
  rw [diagonalBase_eq_generated]
  apply Bicommutant.generated_le
  rintro T rfl
  exact diagonalGenerator_mem_uniform u v hu hv hunit
end

section
open FreeFock WeightedCornerIndex GaussianRecognition
variable (u v : ℝ) (hu : u≠0) (hv : v≠0) (hunit : u^2+v^2=1)

theorem ambientExtra_pair_mem (i : WeightedCornerIndex.Index) :
    ambientExtra u v i+star (ambientExtra u v (CircularRotation.flip i))∈ambientAlgebra u v := by
  simp only [ambientExtra,residualSlot_flip]
  cases residualSlot i with
  | none => simp only [Option.map_none,Option.elim_none,star_zero (Operator ℕ),add_zero];exact zero_mem _
  | some b =>
    simp only [Option.map_some,Option.elim_some,Bool.not_not]
    rw [rect_paired_eq]
    have hm (r : Bool) : (vertex r : Operator ℕ)∈ambientAlgebra u v := by
      cases r
      · exact sub_mem (one_mem _) (P_mem_ambient u v)
      · exact P_mem_ambient u v
    exact mul_mem (mul_mem (hm _) (S_mem_ambient u v 0)) (hm _)

theorem ambientOutput_pair_mem (i : WeightedCornerIndex.Index) (n : ℕ) :
    (ambientAbsorption u v hu hv).output i n+
      star ((ambientAbsorption u v hu hv).output (CircularRotation.flip i) n)∈ambientAlgebra u v := by
  apply ((ambientAbsorption u v hu hv).output_generation (ambientAlgebra u v)
    (fun i => diagonalBase_le_ambient u v 0 (ambientLeft u v i).property)
    (fun i => diagonalBase_le_ambient u v 0 (ambientRight u v i).property)).mp ?_ i n
  exact ⟨ambientExtra_pair_mem u v,fun i n => S_mem_ambient u v (reservoirCode (i,n)+1)⟩

theorem ambientSystem_le_ambient : (ambientSystem u v hu hv hunit).generated scalarFlip≤ambientAlgebra u v := by
  apply Bicommutant.generated_le
  rintro T (⟨a,rfl⟩|⟨i,rfl⟩)
  · exact diagonalBase_le_ambient u v 0 (ambientUniformBase u v hu hv hunit a).property
  · exact ambientOutput_pair_mem u v hu hv i.1 i.2

theorem diagonalBase_le_system : diagonalBase u v (0:ℕ)≤(ambientSystem u v hu hv hunit).generated scalarFlip :=
  (diagonalBase_le_uniform u v hu hv hunit).trans
    (Bicommutant.mono _ _ (fun _ h => Or.inl h))

theorem ambient_generator_decomposition :
    S u v (0:ℕ)=diagonalSelf u v 0+
      (ambientExtra u v (residualIndex true)+star (ambientExtra u v (residualIndex false)))+
      (ambientExtra u v (residualIndex false)+star (ambientExtra u v (residualIndex true))) := by
  change S u v 0=diagonalSelf u v 0+
    (ambientRect u v 0 true false+star (ambientRect u v 0 false true))+
    (ambientRect u v 0 false true+star (ambientRect u v 0 true false))
  rw [diagonalSelf,diagonalLeg,diagonalLeg,rect_paired_eq,rect_paired_eq,rect_paired_eq,rect_paired_eq]
  change S u v 0=P*S u v 0*P+(1-P)*S u v 0*(1-P)+P*S u v 0*(1-P)+(1-P)*S u v 0*P
  simp only [sub_mul,mul_sub,one_mul,mul_one]
  abel

theorem ambient_le_ambientSystem : ambientAlgebra u v≤(ambientSystem u v hu hv hunit).generated scalarFlip := by
  let G := (ambientSystem u v hu hv hunit).generated scalarFlip
  have hbase := diagonalBase_le_system u v hu hv hunit
  have hout (i : WeightedCornerIndex.Index) (n : ℕ) :
      (ambientAbsorption u v hu hv).output i n+
        star ((ambientAbsorption u v hu hv).output (CircularRotation.flip i) n)∈G :=
    Bicommutant.subset _ (Or.inr (Set.mem_range_self (i,n)))
  have hgen := ((ambientAbsorption u v hu hv).output_generation G
    (fun i => hbase (ambientLeft u v i).property)
    (fun i => hbase (ambientRight u v i).property)).mpr hout
  apply Bicommutant.generated_le
  rintro T (rfl|⟨n,rfl⟩)
  · exact hbase (P_mem_diagonalBase u v 0)
  · cases n with
    | zero =>
      rw [ambient_generator_decomposition u v]
      exact add_mem (add_mem (hbase (diagonalSelf_mem u v 0)) (hgen.1 (residualIndex true)))
        (hgen.1 (residualIndex false))
    | succ n =>
      obtain ⟨⟨i,k⟩,he⟩ := reservoirCode.surjective n
      subst n
      exact hgen.2 i k

theorem ambient_generated_eq : (ambientSystem u v hu hv hunit).generated scalarFlip=ambientAlgebra u v :=
  le_antisymm (ambientSystem_le_ambient u v hu hv hunit) (ambient_le_ambientSystem u v hu hv hunit)

theorem ambientSystem_total :
    (Submodule.span ℂ (Set.range (ambientSystem u v hu hv hunit).family)).topologicalClosure=⊤ := by
  apply path_closed_total _ (Submodule.isClosed_topologicalClosure _)
  intro w
  obtain ⟨B,hB,hh⟩ := good_basis u v hu hv w
  rw [← hh]
  exact (ambientSystem u v hu hv hunit).normalHull_bicommutant scalarFlip B
    (ambient_le_ambientSystem u v hu hv hunit hB.1)
end

open FreeFock GaussianRecognition CyclicSpectral
variable (u v : ℝ) (hu : u≠0) (hv : v≠0) (hunit : u^2+v^2=1)

theorem scalar_ambient_state (a : UniformAlgebra) :
    inner ℂ scalarSystem.root (scalarSystem.coeff a scalarSystem.root)=
      inner ℂ (ambientSystem u v hu hv hunit).root
        ((ambientSystem u v hu hv hunit).coeff a (ambientSystem u v hu hv hunit).root) := by
  exact (scalarUniform_state (none:ScalarIndex) a).trans (ambientUniform_state u v hu hv hunit a).symm

def ambientGaussianIso : (ambientSystem u v hu hv hunit).generated scalarFlip ≃⋆ₐ[ℂ]
    scalarSystem.generated scalarFlip :=
  (ambientSystem u v hu hv hunit).generatedIso scalarSystem
    (scalar_ambient_state u v hu hv hunit)
    (ambientSystem_total u v hu hv hunit) scalarSystem_total scalarFlip

theorem ambientGaussianIso_normal :
    @Continuous _ _ ((ambientSystem u v hu hv hunit).generatedTopology scalarFlip)
      (scalarSystem.generatedTopology scalarFlip) (ambientGaussianIso u v hu hv hunit) :=
  (ambientSystem u v hu hv hunit).generatedIso_normal scalarSystem
    (scalar_ambient_state u v hu hv hunit)
    (ambientSystem_total u v hu hv hunit) scalarSystem_total scalarFlip

theorem ambientGaussianIso_trace (T : (ambientSystem u v hu hv hunit).generated scalarFlip) :
    inner ℂ scalarSystem.root ((ambientGaussianIso u v hu hv hunit T : FreeFock.Operator ScalarIndex) scalarSystem.root)=
      vacuumTrace u v (T:Operator ℕ) :=
  (ambientSystem u v hu hv hunit).generatedIso_trace scalarSystem
    (scalar_ambient_state u v hu hv hunit)
    (ambientSystem_total u v hu hv hunit) scalarSystem_total scalarFlip T
end FreeGroupFactorMain.WeightedFock

end

end OAI