lean/OAI/Analysis/InterpolatedFactors/RelativeClosure.lean

import OAI.Analysis.InterpolatedFactors.CornerGeneration

namespace OAI

noncomputable section

universe u_1 u_2 u_3 u_4

namespace FreeGroupFactorMain.Amplification
variable {A : Type u_1} [CStarAlgebra A]

 theorem compress_mul_corner (p : A) (hp : IsStarProjection p) (B : A) (hB : Commute p B)
    (a : Corner p hp) : (p*B*p)*(a:A)=B*(a:A) := by
  rw [mul_assoc _ p,a.property.1,hB.eq,mul_assoc,a.property.1]

 theorem corner_mul_compress (p : A) (hp : IsStarProjection p) (B : A) (hB : Commute p B)
    (a : Corner p hp) : (a:A)*(p*B*p)=(a:A)*B := by
  calc
    _ = ((a:A)*p)*(B*p) := by simp only [mul_assoc]
    _ = (a:A)*(p*B) := by rw [a.property.2,← hB.eq]
    _ = _ := by rw [← mul_assoc,a.property.2]

 theorem relative_corner_bicommutant (s : Set A) (p : A) (hp : IsStarProjection p)
    (hpM : p∈StarSubalgebra.centralizer ℂ (StarSubalgebra.centralizer ℂ s : Set A)) :
    StarSubalgebra.centralizer ℂ
      (StarSubalgebra.centralizer ℂ
        (subalgebraCorner (StarSubalgebra.centralizer ℂ
          (StarSubalgebra.centralizer ℂ s : Set A)) p hp hpM : Set (Corner p hp)) : Set (Corner p hp)) ≤
      subalgebraCorner (StarSubalgebra.centralizer ℂ (StarSubalgebra.centralizer ℂ s : Set A)) p hp hpM := by
  intro a ha
  change (a:A)∈StarSubalgebra.centralizer ℂ (StarSubalgebra.centralizer ℂ s : Set A)
  rw [StarSubalgebra.mem_centralizer_iff]
  have htest (B : A) (hB : B∈StarSubalgebra.centralizer ℂ s) : B*(a:A)=(a:A)*B := by
    have hBp : Commute p B := ((StarSubalgebra.mem_centralizer_iff ℂ).mp hpM B hB).1.symm
    let b : Corner p hp := ⟨p*B*p,by
      constructor
      · rw [← mul_assoc,← mul_assoc,hp.isIdempotentElem.eq]
      · rw [mul_assoc _ p,hp.isIdempotentElem.eq]⟩
    have hb : b∈StarSubalgebra.centralizer ℂ
        (subalgebraCorner (StarSubalgebra.centralizer ℂ
          (StarSubalgebra.centralizer ℂ s : Set A)) p hp hpM : Set (Corner p hp)) := by
      rw [StarSubalgebra.mem_centralizer_iff]
      intro x hx
      have hh := (StarSubalgebra.mem_centralizer_iff ℂ).mp hx B hB
      constructor
      · apply Subtype.ext
        change (x:A)*(p*B*p)=(p*B*p)*(x:A)
        rw [corner_mul_compress p hp B hBp,compress_mul_corner p hp B hBp]
        exact hh.1.symm
      · apply Subtype.ext
        change ((star x : Corner p hp):A)*(p*B*p)=(p*B*p)*((star x : Corner p hp):A)
        rw [corner_mul_compress p hp B hBp,compress_mul_corner p hp B hBp]
        exact ((StarSubalgebra.mem_centralizer_iff ℂ).mp (star_mem hx) B hB).1.symm
    have hh := congrArg (fun z : Corner p hp => (z:A))
      (((StarSubalgebra.mem_centralizer_iff ℂ).mp ha b hb).1)
    change (p*B*p)*(a:A)=(a:A)*(p*B*p) at hh
    rwa [compress_mul_corner p hp B hBp,corner_mul_compress p hp B hBp] at hh
  intro B hB
  exact ⟨htest B hB,htest (star B) (star_mem hB)⟩
end FreeGroupFactorMain.Amplification

namespace FreeGroupFactorMain.FreeFock

section
open scoped ENNReal
variable {α : Type u_2} [DecidableEq α]

 def creationWord : List α → Operator α
  | [] => 1
  | a::w => creation a*creationWord w

@[simp] theorem creationWord_nil : creationWord ([] : List α)=1 := rfl
@[simp] theorem creationWord_cons (a : α) (w : List α) :
    creationWord (a::w)=creation a*creationWord w := rfl

 theorem creationWord_append (v w : List α) :
    creationWord (v++w)=creationWord v*creationWord w := by
  induction v with
  | nil => simp
  | cons a v ih => simp only [List.cons_append,creationWord_cons,ih,mul_assoc]

 theorem creationWord_vacuum (w : List α) : creationWord w vacuum=lp.single 2 w 1 := by
  induction w with
  | nil => rfl
  | cons a w ih =>
    change creation a (creationWord w vacuum)=_
    rw [ih,creation_single]

 theorem creationWord_star_vacuum (w : List α) :
    star (creationWord w) vacuum=if w=[] then vacuum else 0 := by
  cases w with
  | nil => simp
  | cons a w =>
    simp only [creationWord_cons,star_mul,List.cons_ne_nil,ite_false]
    change star (creationWord w) (star (creation a) (lp.single 2 [] 1))=0
    rw [annihilation_single_nil,map_zero]

 def wick (w : List α) : Operator α :=
  ∑ k ∈ Finset.range (w.length+1), creationWord (w.take k)*star (creationWord (w.drop k).reverse)

 @[simp] theorem wick_nil : wick ([] : List α)=1 := by simp [wick]

 theorem wick_star (w : List α) : star (wick w)=wick w.reverse := by
  rw [wick,star_sum (R := Operator α)]
  simp only [star_mul,star_star]
  have hs := Finset.sum_range_reflect
    (fun k => creationWord (w.reverse.take k)*star (creationWord (w.reverse.drop k).reverse)) (w.length+1)
  calc
    _ = ∑ k ∈ Finset.range (w.length+1),
        creationWord (w.reverse.take (w.length+1-1-k))*
          star (creationWord (w.reverse.drop (w.length+1-1-k)).reverse) := by
      apply Finset.sum_congr rfl
      intro k hk
      have hk' : k ≤ w.length := Nat.le_of_lt_succ (Finset.mem_range.mp hk)
      simp only [Nat.add_sub_cancel,List.take_reverse,List.drop_reverse,List.reverse_reverse]
      rw [Nat.sub_sub_self hk']
    _ = wick w.reverse := by simpa only [wick,List.length_reverse] using hs

 theorem wick_vacuum (w : List α) : wick w vacuum=lp.single 2 w 1 := by
  classical
  rw [wick,sum_apply]
  calc
    _ = (creationWord (w.take w.length)*star (creationWord (w.drop w.length).reverse)) vacuum := by
      apply Finset.sum_eq_single w.length
      · intro k hk hkn
        have hk' : k < w.length := lt_of_le_of_ne (Nat.le_of_lt_succ (Finset.mem_range.mp hk)) hkn
        change creationWord (w.take k) (star (creationWord (w.drop k).reverse) vacuum)=0
        rw [creationWord_star_vacuum]
        have hn : (w.drop k).reverse ≠ [] := by
          intro he
          have hh := congrArg List.length he
          simp only [List.length_reverse,List.length_drop,List.length_nil] at hh
          omega
        rw [ite_eq_right hn,map_zero]
      · intro h
        exact False.elim (h (Finset.mem_range.mpr (Nat.lt_succ_self _)))
    _ = _ := by
      simp only [List.take_length,List.drop_length,List.reverse_nil,creationWord_nil,star_one,mul_one]
      exact creationWord_vacuum w

 theorem wick_creation_formula (a : α) (w : List α) :
    wick (a::w)=creation a*wick w+star (creation a)*star (creationWord w.reverse) := by
  rw [wick,List.length_cons,Finset.sum_range_succ']
  have he (k : ℕ) :
      creationWord ((a::w).take (k+1))*star (creationWord ((a::w).drop (k+1)).reverse)=
      creation a*(creationWord (w.take k)*star (creationWord (w.drop k).reverse)) := by
    simp only [List.take_succ_cons,List.drop_succ_cons,creationWord_cons,mul_assoc]
  simp_rw [he]
  rw [← Finset.mul_sum]
  simp only [List.take_zero,List.drop_zero,creationWord_nil,one_mul,List.reverse_cons,    creationWord_append,creationWord_cons,creationWord_nil,mul_one,star_mul]
  rfl

 theorem wick_semicircular_cons (a : α) (w : List α) :
    wick (a::w)=semicircular a*wick w-
      (match w with | [] => 0 | b::v => if a=b then wick v else 0) := by
  cases w with
  | nil => simp [wick_creation_formula,semicircular]
  | cons b v =>
    rw [wick_creation_formula a,wick_creation_formula b]
    change creation a*(creation b*wick v+star (creation b)*star (creationWord v.reverse))+
       star (creation a)*star (creationWord (b::v).reverse)=
      (creation a+star (creation a))*(creation b*wick v+star (creation b)*star (creationWord v.reverse))-
        (if a=b then wick v else 0)
    simp only [List.reverse_cons,creationWord_append,creationWord_cons,creationWord_nil,mul_one,star_mul]
    have he : (star (creation a)*creation b)*wick v=if a=b then wick v else 0 := by
      rw [annihilation_creation]
      by_cases h : a=b <;> simp [h]
    rw [← he]
    noncomm_ring

end

section
variable {α : Type u_3} [DecidableEq α]

def runSpan (i : α) : Submodule ℂ (Operator α) :=
  Submodule.span ℂ (Set.range (fun n : ℕ => wick (List.replicate n i)))

def positiveRunSpan (i : α) : Submodule ℂ (Operator α) :=
  Submodule.span ℂ (Set.range (fun n : ℕ => wick (List.replicate (n+1) i)))

 theorem run_mem (i : α) (n : ℕ) : wick (List.replicate n i) ∈ runSpan i :=
  Submodule.subset_span (Set.mem_range_self n)

 theorem semicircular_mul_runSpan (i : α) (x : Operator α) (hx : x∈runSpan i) :
    semicircular i*x∈runSpan i := by
  induction hx using Submodule.span_induction with
  | mem x hx =>
    obtain ⟨n,rfl⟩ := hx
    have h := wick_semicircular_cons i (List.replicate n i)
    cases n with
    | zero => simpa [wick_semicircular_cons] using run_mem i 1
    | succ n =>
      have he : semicircular i*wick (List.replicate (n+1) i)=
          wick (List.replicate (n+2) i)+wick (List.replicate n i) := by
        simpa only [List.replicate_succ,ite_true] using (eq_add_of_sub_eq h.symm)
      rw [he]
      exact (runSpan i).add_mem (run_mem i (n+2)) (run_mem i n)
  | zero => simp
  | add x y _ _ hx hy => simpa only [mul_add] using (runSpan i).add_mem hx hy
  | smul c x _ hx => simpa only [mul_smul_comm] using (runSpan i).smul_mem c hx

 theorem run_mul_runSpan (i : α) (n : ℕ) (x : Operator α) (hx : x∈runSpan i) :
    wick (List.replicate n i)*x∈runSpan i := by
  suffices ∀ n, (wick (List.replicate n i)*x∈runSpan i) ∧
      (wick (List.replicate (n+1) i)*x∈runSpan i) from (this n).1
  intro n
  induction n with
  | zero =>
    constructor
    · simpa using hx
    · simpa [wick_semicircular_cons] using semicircular_mul_runSpan i x hx
  | succ n ih =>
    refine ⟨ih.2,?_⟩
    have he : wick (List.replicate (n+1+1) i)=
        semicircular i*wick (List.replicate (n+1) i)-wick (List.replicate n i) := by
      conv_lhs => rw [List.replicate_succ,wick_semicircular_cons]
      simp only [List.replicate_succ,ite_true]
    rw [he,sub_mul,mul_assoc]
    exact (runSpan i).sub_mem (semicircular_mul_runSpan i _ ih.2) ih.1

def runAlgebra (i : α) : StarSubalgebra ℂ (Operator α) where
  carrier := (runSpan i : Set (Operator α))
  zero_mem' := (runSpan i).zero_mem
  add_mem' := (runSpan i).add_mem
  algebraMap_mem' c := by
    change algebraMap ℂ (Operator α) c∈runSpan i
    simpa only [Algebra.algebraMap_eq_smul_one] using
      (runSpan i).smul_mem c (show (1 : Operator α)∈runSpan i by simpa using run_mem i 0)
  one_mem' := by simpa using run_mem i 0
  mul_mem' {a b} hx hy := by
    change a∈runSpan i at hx
    change b∈runSpan i at hy
    change a*b∈runSpan i
    induction hx using Submodule.span_induction with
    | mem x hx => obtain ⟨n,rfl⟩ := hx; exact run_mul_runSpan i n _ hy
    | zero => simp
    | add x y _ _ hx hy => simpa only [add_mul] using (runSpan i).add_mem hx hy
    | smul c x _ hx => simpa only [smul_mul_assoc] using (runSpan i).smul_mem c hx
  star_mem' {a} hx := by
    change a∈runSpan i at hx
    change star a∈runSpan i
    induction hx using Submodule.span_induction with
    | mem x hx => obtain ⟨n,rfl⟩ := hx; simpa only [wick_star,List.reverse_replicate] using run_mem i n
    | zero => simpa only [star_zero (R := Operator α)] using (runSpan i).zero_mem
    | add x y _ _ hx hy => simpa only [star_add (R := Operator α)] using (runSpan i).add_mem hx hy
    | smul c x _ hx => simpa only [star_smul] using (runSpan i).smul_mem (star c) hx

 theorem vacuumTrace_linear : IsLinearMap ℂ (@vacuumTrace α _) where
  map_add := by intro x y; simp [vacuumTrace]
  map_smul := by intro c x; simp [vacuumTrace]

 theorem vacuumTrace_continuous : Continuous (@vacuumTrace α _) := by
  exact continuous_const.inner (continuous_id.clm_apply continuous_const)

 theorem vacuumTrace_run (i : α) (n : ℕ) :
    vacuumTrace (wick (List.replicate n i))=if n=0 then 1 else 0 := by
  cases n with
  | zero => simpa using (vacuumTrace_one (α := α))
  | succ n =>
    unfold vacuumTrace
    rw [wick_vacuum]
    simp [vacuum,lp.inner_single_left,lp.single_apply]

end
variable {α : Type u_4} [DecidableEq α]

theorem run_star_creation (a j : α) (h : a≠j) (n : ℕ) :
    star (creationWord (List.replicate (n+1) a))*creation j=0 := by
  rw [List.replicate_succ,creationWord_cons,star_mul,mul_assoc,annihilation_creation,ite_eq_right h,mul_zero]

theorem run_creation (a j : α) (h : a≠j) (n : ℕ) :
    wick (List.replicate n a)*creation j=creationWord (List.replicate n a)*creation j := by
  induction n with
  | zero => simp
  | succ n ih =>
    rw [List.replicate_succ,wick_creation_formula,add_mul,mul_assoc,ih]
    have hz : star (creation a)*star (creationWord (List.replicate n a).reverse)*creation j=0 := by
      rw [List.reverse_replicate]
      have he : star (creation a)*star (creationWord (List.replicate n a))=
          star (creationWord (List.replicate (n+1) a)) := by
        rw [show List.replicate (n+1) a=List.replicate n a++[a] from (List.replicate_add _ _ _).trans (by simp),
          creationWord_append,star_mul,creationWord_cons,creationWord_nil,mul_one]
      rw [he,run_star_creation a j h]
    rw [hz,add_zero,creationWord_cons,mul_assoc]

theorem run_coarse_covariance (a i j : α) (hi : i≠a) (hj : j≠a) (n : ℕ) :
    star (creation i)*wick (List.replicate n a)*creation j=
      vacuumTrace (wick (List.replicate n a)) • (if i=j then (1:Operator α) else 0) := by
  rw [mul_assoc,run_creation a j hj.symm,vacuumTrace_run]
  cases n with
  | zero => simp [annihilation_creation]
  | succ n =>
    rw [List.replicate_succ,creationWord_cons,← mul_assoc,← mul_assoc,annihilation_creation,      ite_eq_right hi,zero_mul,zero_mul]
    simp [zero_smul ℂ (A := Operator α)]

theorem polynomial_coarse_covariance (a i j : α) (hi : i≠a) (hj : j≠a)
    (T : Operator α) (hT : T∈runSpan a) :
    star (creation i)*T*creation j=vacuumTrace T • (if i=j then (1:Operator α) else 0) := by
  induction hT using Submodule.span_induction with
  | mem T hT => obtain ⟨n,rfl⟩ := hT; exact run_coarse_covariance a i j hi hj n
  | zero => simp [vacuumTrace,zero_smul ℂ (A := Operator α)]
  | add T U _ _ hT hU =>
    simp only [mul_add,add_mul,hT,hU,(vacuumTrace_linear (α := α)).map_add,      add_smul (R := ℂ) (M := Operator α)]
  | smul c T _ hT =>
    simp only [mul_smul_comm,smul_mul_assoc,hT,(vacuumTrace_linear (α := α)).map_smul,smul_smul,smul_eq_mul]

theorem closed_coarse_covariance (a i j : α) (hi : i≠a) (hj : j≠a)
    (T : (runAlgebra a).topologicalClosure) :
    star (creation i)*(T:Operator α)*creation j=vacuumTrace (T:Operator α) • (if i=j then (1:Operator α) else 0) := by
  exact closure_minimal (fun x hx => polynomial_coarse_covariance a i j hi hj x hx)
    (isClosed_eq ((continuous_const.mul continuous_id).mul continuous_const)
      (vacuumTrace_continuous.smul continuous_const)) T.property

theorem closed_coefficient_annihilation (a i : α) (hi : i≠a)
    (T : (runAlgebra a).topologicalClosure) : star (creation i) ((T:Operator α) vacuum)=0 := by
  apply closure_minimal (t := {T:Operator α | star (creation i) (T vacuum)=0}) ?_
    (isClosed_eq ((star (creation i)).continuous.comp (continuous_id.clm_apply continuous_const)) continuous_const) T.property
  intro T hT
  change T∈runSpan a at hT
  change star (creation i) (T vacuum)=0
  induction hT using Submodule.span_induction with
  | mem T hT =>
    obtain ⟨n,rfl⟩ := hT
    rw [wick_vacuum]
    cases n with
    | zero => exact annihilation_single_nil i 1
    | succ n => rw [List.replicate_succ,annihilation_single_cons,ite_eq_right hi]
  | zero => simp
  | add T U _ _ hT hU => simp only [add_apply,map_add,hT,hU,add_zero]
  | smul c T _ hT => simp only [smul_apply,map_smul,hT,smul_zero]
end FreeGroupFactorMain.FreeFock

end

end OAI