lean/OAI/Analysis/CharacterCriterion/EndpointMultiplicity.lean

import OAI.Analysis.CharacterCriterion.InfiniteTensor

namespace OAI

noncomputable section
open scoped TensorProduct
open scoped InnerProductSpace
open scoped BigOperators
open Set Module
open Topology
open NormedSpace
open scoped ComplexOrder
open scoped ComplexOrder
open scoped InnerProductSpace
open scoped InnerProductSpace
open NormedSpace
open scoped TensorProduct InnerProductSpace
open UniformSpace
variable [KirchbergRordam.GenericRW.Size]
local notation "rankNum" => KirchbergRordam.GenericRW.Size.n
variable [KirchbergRordam.GenericRW.PauliData]
omit [KirchbergRordam.GenericRW.Size] [KirchbergRordam.GenericRW.PauliData]
open Filter Topology BoundedContinuousFunction
open Filter Topology
open Topology
open scoped TensorProduct CStarAlgebra
open scoped CStarAlgebra ComplexOrder
open NormedSpace
open scoped ComplexOrder
open scoped ComplexOrder
open scoped InnerProductSpace
open scoped CStarAlgebra ComplexOrder
open scoped TensorProduct InnerProductSpace
open UniformSpace

namespace KirchbergRordam.DropMatrixRepresentations
open MatrixDiagonalPath MatrixBlockReindex
open scoped CStarAlgebra ComplexOrder
variable {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq ι] [DecidableEq κ]
lemma left_injective [Nonempty κ] : Function.Injective (MatrixTensor.leftMatrix (ι:=ι) (κ:=κ)) := by
  intro a b h
  ext i j
  let k : κ := Classical.choice inferInstance
  have he := congrArg (fun m : Mat (ι×κ) => m (i,k) (j,k)) h
  simpa [left_apply] using he
lemma right_injective [Nonempty ι] : Function.Injective (MatrixTensor.rightMatrix (ι:=ι) (κ:=κ)) := by
  intro a b h
  ext i j
  let k : ι := Classical.choice inferInstance
  have he := congrArg (fun m : Mat (ι×κ) => m (k,i) (k,j)) h
  simpa [right_apply] using he
variable (p q : ℕ) [instNeZerop : NeZero p] [instNeZeroq : NeZero q]
def leftValue (f : PrimeDimensionDrop.Algebra p q) : Mat (Fin p) := Classical.choose f.property.1
omit instNeZerop instNeZeroq in
lemma leftValue_spec
    (p : ℕ)
    (q : ℕ)
    [NeZero p]
    [NeZero q]
    (f : PrimeDimensionDrop.Algebra p q) :
    MatrixTensor.leftMatrix (κ:=Fin q) (leftValue p q f)=f.val 0 := Classical.choose_spec f.property.1
def rightValue (f : PrimeDimensionDrop.Algebra p q) : Mat (Fin q) := Classical.choose f.property.2
omit instNeZerop instNeZeroq in
lemma rightValue_spec
    (p : ℕ)
    (q : ℕ)
    [NeZero p]
    [NeZero q]
    (f : PrimeDimensionDrop.Algebra p q) :
    MatrixTensor.rightMatrix (ι:=Fin p) (rightValue p q f)=f.val 1 := Classical.choose_spec f.property.2
def leftEvaluation : PrimeDimensionDrop.Algebra p q →⋆ₐ[ℂ] Mat (Fin p) where
  toFun := leftValue p q
  map_zero' := left_injective (κ:=Fin q) (by rw [map_zero,leftValue_spec]; rfl)
  map_one' := left_injective (κ:=Fin q) (by rw [map_one,leftValue_spec]; rfl)
  map_add' a b := left_injective (κ:=Fin q) (by rw [leftValue_spec,map_add,leftValue_spec,leftValue_spec]; rfl)
  map_mul' a b := left_injective (κ:=Fin q) (by rw [leftValue_spec,map_mul,leftValue_spec,leftValue_spec]; rfl)
  commutes' z := left_injective (κ:=Fin q) (by rw [AlgHomClass.commutes,leftValue_spec]; rfl)
  map_star' a := left_injective (κ:=Fin q) (by rw [leftValue_spec,map_star,leftValue_spec]; rfl)
def rightEvaluation : PrimeDimensionDrop.Algebra p q →⋆ₐ[ℂ] Mat (Fin q) where
  toFun := rightValue p q
  map_zero' := right_injective (ι:=Fin p) (by rw [map_zero,rightValue_spec]; rfl)
  map_one' := right_injective (ι:=Fin p) (by rw [map_one,rightValue_spec]; rfl)
  map_add' a b := right_injective (ι:=Fin p) (by rw [rightValue_spec,map_add,rightValue_spec,rightValue_spec]; rfl)
  map_mul' a b := right_injective (ι:=Fin p) (by rw [rightValue_spec,map_mul,rightValue_spec,rightValue_spec]; rfl)
  commutes' z := right_injective (ι:=Fin p) (by rw [AlgHomClass.commutes,rightValue_spec]; rfl)
  map_star' a := right_injective (ι:=Fin p) (by rw [rightValue_spec,map_star,rightValue_spec]; rfl)
lemma two_three (r : ℕ) (hr : 2≤r) : ∃ a b : ℕ, 2*a+3*b=r := by
  by_cases h : r%2=0
  · exact ⟨r/2,0,by omega⟩
  · have hr3 : 3≤r := by omega
    exact ⟨(r-3)/2,1,by omega⟩
def blockRepresentation (a b : ℕ) : DimensionDrop23 →⋆ₐ[ℂ]
    Mat ((Fin 2×Fin a) ⊕ (Fin 3×Fin b)) :=
  sumBlock.comp (((MatrixTensor.leftMatrix (κ:=Fin a)).comp (leftEvaluation 2 3)).prod
    ((MatrixTensor.leftMatrix (κ:=Fin b)).comp (rightEvaluation 2 3)))
def representation (ι : Type*) [Fintype ι] [DecidableEq ι] (hι : 2≤Fintype.card ι) :
    DimensionDrop23 →⋆ₐ[ℂ] Mat ι := by
  let a := (two_three (Fintype.card ι) hι).choose
  let b := (two_three (Fintype.card ι) hι).choose_spec.choose
  have h : 2*a+3*b=Fintype.card ι := (two_three (Fintype.card ι) hι).choose_spec.choose_spec
  let e : ((Fin 2×Fin a) ⊕ (Fin 3×Fin b)) ≃ ι :=
    BalancedDropSystem.cardEquiv (by simpa using h)
  exact (CStarMatrix.reindexₐ ℂ ℂ e).toStarAlgHom.comp (blockRepresentation a b)
end KirchbergRordam.DropMatrixRepresentations
namespace KirchbergRordam.EndpointMultiplicity
open MatrixDiagonalPath MatrixBlockReindex DiagonalIteration
open scoped ComplexOrder CStarAlgebra
variable {P Q R : Type*} [Fintype P] [Fintype Q] [Fintype R]
  [DecidableEq P] [DecidableEq Q] [DecidableEq R]
def leftExpand : P ⊕ ((P×Q)×R) ≃ P × (Unit ⊕ (Q×R)) where
  toFun x := match x with
    | .inl a => (a,.inl ())
    | .inr ((a,b),r) => (a,.inr (b,r))
  invFun x := match x.2 with
    | .inl _ => .inl x.1
    | .inr (b,r) => .inr ((x.1,b),r)
  left_inv x := by rcases x with a|⟨⟨a,b⟩,r⟩ <;> rfl
  right_inv x := by
    rcases x with ⟨a,u|⟨b,r⟩⟩
    · cases u; rfl
    · rfl
def rightExpand : Q ⊕ ((P×Q)×R) ≃ Q × (Unit ⊕ (P×R)) where
  toFun x := match x with
    | .inl b => (b,.inl ())
    | .inr ((a,b),r) => (b,.inr (a,r))
  invFun x := match x.2 with
    | .inl _ => .inl x.1
    | .inr (a,r) => .inr ((a,x.1),r)
  left_inv x := by rcases x with b|⟨⟨a,b⟩,r⟩ <;> rfl
  right_inv x := by
    rcases x with ⟨b,u|⟨a,r⟩⟩
    · cases u; rfl
    · rfl
lemma leftExpand_matrix (a : Mat P) :
    CStarMatrix.reindexₐ ℂ ℂ (leftExpand (Q:=Q) (R:=R))
      (sumBlock (a,block (fun _ : R => MatrixTensor.leftMatrix (κ:=Q) a))) =
      MatrixTensor.leftMatrix (κ:=Unit ⊕ (Q×R)) a := by
  ext ⟨i,k⟩ ⟨j,l⟩
  rcases k with u|⟨b,r⟩ <;> rcases l with v|⟨c,s⟩
  · cases u; cases v; change a i j = a i j * 1; simp
  · cases u; change 0 = a i j * 0; simp
  · cases v; change 0 = a i j * 0; simp
  · change (if r=s then a i j * (if b=c then 1 else 0) else 0) =
      a i j * (if (Sum.inr (b,r) : Unit ⊕ Q×R)=.inr (c,s) then 1 else 0)
    by_cases hb : b=c <;> by_cases hr : r=s <;> simp [hb,hr,Prod.ext_iff]
lemma rightExpand_matrix (a : Mat Q) :
    CStarMatrix.reindexₐ ℂ ℂ (rightExpand (P:=P) (R:=R))
      (sumBlock (a,block (fun _ : R => MatrixTensor.rightMatrix (ι:=P) a))) =
      MatrixTensor.leftMatrix (κ:=Unit ⊕ (P×R)) a := by
  ext ⟨i,k⟩ ⟨j,l⟩
  rcases k with u|⟨b,r⟩ <;> rcases l with v|⟨c,s⟩
  · cases u; cases v; change a i j = a i j * 1; simp
  · cases u; change 0 = a i j * 0; simp
  · cases v; change 0 = a i j * 0; simp
  · change (if r=s then (if b=c then 1 else 0) * a i j else 0) =
      a i j * (if (Sum.inr (b,r) : Unit ⊕ P×R)=.inr (c,s) then 1 else 0)
    by_cases hb : b=c <;> by_cases hr : r=s <;> simp [hb,hr,Prod.ext_iff]
variable {A B C : Type*} [CStarAlgebra A] [CStarAlgebra B] [CStarAlgebra C]
def constHom (ι : Type*) : A →⋆ₐ[ℂ] (ι→A) where
  toFun a _ := a
  map_zero' := rfl
  map_one' := rfl
  map_add' _ _ := rfl
  map_mul' _ _ := rfl
  commutes' _ := rfl
  map_star' _ := rfl
def prodUnitary (u : unitary A) (v : unitary B) : unitary (A×B) :=
  ⟨(u,v),⟨Prod.ext u.property.1 v.property.1,Prod.ext u.property.2 v.property.2⟩⟩
lemma map_conj (φ : A →⋆ₐ[ℂ] B) (u : unitary A) (a : A) :
    φ (conj u a) = conj (Unitary.map (StarMonoidHom.ofClass φ) u) (φ a) := by
  change φ ((u:A)*a*star (u:A)) = φ (u:A)*φ a*star (φ (u:A))
  rw [map_mul,map_mul,map_star]
def duplicate : Mat P × Mat Q →⋆ₐ[ℂ] Mat (P ⊕ (Q×R)) :=
  sumBlock.comp ((StarAlgHom.fst ℂ (Mat P) (Mat Q)).prod
    ((block.comp (constHom R)).comp (StarAlgHom.snd ℂ (Mat P) (Mat Q))))
def duplicateUnitary (v : unitary (Mat Q)) : unitary (Mat (P⊕(Q×R))) :=
  Unitary.map (StarMonoidHom.ofClass (duplicate (P:=P) (Q:=Q) (R:=R)))
    (prodUnitary (1:unitary (Mat P)) v)
lemma conj_duplicate (v : unitary (Mat Q)) (a : Mat P) (b : Mat Q) :
    conj (duplicateUnitary (P:=P) (Q:=Q) (R:=R) v) (sumBlock (a,block (fun _ : R => b))) =
      sumBlock (a,block (fun _ : R => conj v b)) := by
  have hm := map_conj (duplicate (P:=P) (Q:=Q) (R:=R)) (prodUnitary 1 v) (a,b)
  change duplicate (conj (prodUnitary 1 v) (a,b)) = _ at hm
  change conj (Unitary.map (StarMonoidHom.ofClass (duplicate (P:=P) (Q:=Q) (R:=R))) (prodUnitary 1 v))
    (duplicate (a,b)) = _
  rw [← hm]
  change duplicate ((1:Mat P)*a*star (1:Mat P),conj v b)=_
  simp only [star_one,one_mul,mul_one]
  rfl
end KirchbergRordam.EndpointMultiplicity
namespace KirchbergRordam.EndpointMultiplicity
open MatrixDiagonalPath MatrixBlockReindex DiagonalIteration
open scoped ComplexOrder CStarAlgebra
variable {P Q R : Type*} [Fintype P] [Fintype Q] [Fintype R]
  [DecidableEq P] [DecidableEq Q] [DecidableEq R]

def leftBase (h : 2≤Fintype.card (Unit ⊕ (Q×R))) :
    DimensionDrop23 →⋆ₐ[ℂ] Mat (P⊕((P×Q)×R)) :=
  (CStarMatrix.reindexₐ ℂ ℂ (leftExpand (P:=P) (Q:=Q) (R:=R)).symm).toStarAlgHom.comp
    (MatrixTensor.rightMatrix.comp (DropMatrixRepresentations.representation _ h))
def rightBase (h : 2≤Fintype.card (Unit ⊕ (P×R))) :
    DimensionDrop23 →⋆ₐ[ℂ] Mat (Q⊕((P×Q)×R)) :=
  (CStarMatrix.reindexₐ ℂ ℂ (rightExpand (P:=P) (Q:=Q) (R:=R)).symm).toStarAlgHom.comp
    (MatrixTensor.rightMatrix.comp (DropMatrixRepresentations.representation _ h))
lemma leftBase_commute (h : 2≤Fintype.card (Unit ⊕ (Q×R))) (a : Mat P)
    (d : DimensionDrop23) :
    Commute (leftBase (P:=P) h d)
      (sumBlock (a,block (fun _ : R => MatrixTensor.leftMatrix (κ:=Q) a))) := by
  let e := CStarMatrix.reindexₐ ℂ ℂ (leftExpand (P:=P) (Q:=Q) (R:=R))
  apply e.injective
  change e (leftBase h d * _)=e (_ * leftBase h d)
  rw [map_mul,map_mul,leftExpand_matrix]
  have he : e (leftBase (P:=P) h d)=MatrixTensor.rightMatrix
      (DropMatrixRepresentations.representation _ h d) := by
    simp only [leftBase,StarAlgHom.comp_apply,
      CStarMatrix.reindexₐ_symm]
    change e (e.symm (MatrixTensor.rightMatrix (DropMatrixRepresentations.representation _ h d))) = _
    exact e.apply_symm_apply _
  rw [he]
  exact (MatrixTensor.legs_commute a _).symm.eq
lemma rightBase_commute (h : 2≤Fintype.card (Unit ⊕ (P×R))) (a : Mat Q)
    (d : DimensionDrop23) :
    Commute (rightBase (Q:=Q) h d)
      (sumBlock (a,block (fun _ : R => MatrixTensor.rightMatrix (ι:=P) a))) := by
  let e := CStarMatrix.reindexₐ ℂ ℂ (rightExpand (P:=P) (Q:=Q) (R:=R))
  apply e.injective
  change e (rightBase h d * _)=e (_ * rightBase h d)
  rw [map_mul,map_mul,rightExpand_matrix]
  have he : e (rightBase (Q:=Q) h d)=MatrixTensor.rightMatrix
      (DropMatrixRepresentations.representation _ h d) := by
    simp only [rightBase,StarAlgHom.comp_apply,
      CStarMatrix.reindexₐ_symm]
    change e (e.symm (MatrixTensor.rightMatrix (DropMatrixRepresentations.representation _ h d))) = _
    exact e.apply_symm_apply _
  rw [he]
  exact (MatrixTensor.legs_commute a _).symm.eq

def leftRep (h : 2≤Fintype.card (Unit ⊕ (Q×R))) (v : unitary (Mat (P×Q))) :
    DimensionDrop23 →⋆ₐ[ℂ] Mat (P⊕((P×Q)×R)) :=
  (conj (duplicateUnitary (P:=P) (Q:=P×Q) (R:=R) v)).toStarAlgHom.comp (leftBase (P:=P) (Q:=Q) (R:=R) h)
def rightRep (h : 2≤Fintype.card (Unit ⊕ (P×R))) (v : unitary (Mat (P×Q))) :
    DimensionDrop23 →⋆ₐ[ℂ] Mat (Q⊕((P×Q)×R)) :=
  (conj (duplicateUnitary (P:=Q) (Q:=P×Q) (R:=R) v)).toStarAlgHom.comp (rightBase (P:=P) (Q:=Q) (R:=R) h)
lemma leftRep_commute (h : 2≤Fintype.card (Unit ⊕ (Q×R))) (v : unitary (Mat (P×Q)))
    (a : Mat P) (d : DimensionDrop23) :
    Commute (leftRep h v d)
      (sumBlock (a,block (fun _ : R => conj v (MatrixTensor.leftMatrix (κ:=Q) a)))) := by
  have hc := (leftBase_commute h a d).map
    (conj (duplicateUnitary (P:=P) (Q:=P×Q) (R:=R) v)).toRingHom
  change Commute (conj (duplicateUnitary (P:=P) (Q:=P×Q) (R:=R) v) (leftBase h d))
    (conj (duplicateUnitary (P:=P) (Q:=P×Q) (R:=R) v)
      (sumBlock (a,block (fun _ : R => MatrixTensor.leftMatrix (κ:=Q) a)))) at hc
  rw [conj_duplicate] at hc
  exact hc
lemma rightRep_commute (h : 2≤Fintype.card (Unit ⊕ (P×R))) (v : unitary (Mat (P×Q)))
    (a : Mat Q) (d : DimensionDrop23) :
    Commute (rightRep h v d)
      (sumBlock (a,block (fun _ : R => conj v (MatrixTensor.rightMatrix (ι:=P) a)))) := by
  have hc := (rightBase_commute h a d).map
    (conj (duplicateUnitary (P:=Q) (Q:=P×Q) (R:=R) v)).toRingHom
  change Commute (conj (duplicateUnitary (P:=Q) (Q:=P×Q) (R:=R) v) (rightBase h d))
    (conj (duplicateUnitary (P:=Q) (Q:=P×Q) (R:=R) v)
      (sumBlock (a,block (fun _ : R => MatrixTensor.rightMatrix (ι:=P) a)))) at hc
  rw [conj_duplicate] at hc
  exact hc

variable {D B : Type*} [CStarAlgebra D] [CStarAlgebra B]
lemma commutator_approx (ρ : D →⋆ₐ[ℂ] B) (a b : B) (d : D) {ε : ℝ}
    (hc : Commute (ρ d) b) (h : ‖a-b‖≤ε) :
    ‖ρ d*a-a*ρ d‖≤2*ε*‖d‖ := by
  have he : ρ d*a-a*ρ d=ρ d*(a-b)-(a-b)*ρ d := by
    rw [mul_sub,sub_mul,hc.eq]
    abel
  rw [he]
  calc
    _ ≤ ‖ρ d‖*‖a-b‖+‖a-b‖*‖ρ d‖ := (norm_sub_le _ _).trans
      (add_le_add (norm_mul_le _ _) (norm_mul_le _ _))
    _ ≤ ‖d‖*ε+ε*‖d‖ := add_le_add
      (mul_le_mul (NonUnitalStarAlgHom.norm_apply_le ρ d) h (norm_nonneg _) (norm_nonneg _))
      (mul_le_mul h (NonUnitalStarAlgHom.norm_apply_le ρ d) (norm_nonneg _) (le_trans (norm_nonneg _) h))
    _ = _ := by ring
lemma duplicate_difference (a : Mat P) (x y : Mat Q) :
    ‖sumBlock (a,block (fun _ : R => x))-sumBlock (a,block (fun _ : R => y))‖≤‖x-y‖ := by
  change ‖duplicate (a,x)-duplicate (a,y)‖≤_
  rw [←map_sub]
  exact (NonUnitalStarAlgHom.norm_apply_le (duplicate (P:=P) (Q:=Q) (R:=R)) _).trans
    (by simp [Prod.norm_def])
lemma leftRep_bound (h : 2≤Fintype.card (Unit ⊕ (Q×R))) (v : unitary (Mat (P×Q)))
    (a : Mat P) (x : Mat (P×Q)) (d : DimensionDrop23) {ε : ℝ}
    (hx : ‖x-conj v (MatrixTensor.leftMatrix (κ:=Q) a)‖≤ε) :
    ‖leftRep h v d*sumBlock (a,block (fun _ : R => x))-
      sumBlock (a,block (fun _ : R => x))*leftRep h v d‖≤2*ε*‖d‖ :=
  commutator_approx (leftRep h v) _ _ d (leftRep_commute h v a d)
    ((duplicate_difference a x _).trans hx)
lemma rightRep_bound (h : 2≤Fintype.card (Unit ⊕ (P×R))) (v : unitary (Mat (P×Q)))
    (a : Mat Q) (x : Mat (P×Q)) (d : DimensionDrop23) {ε : ℝ}
    (hx : ‖x-conj v (MatrixTensor.rightMatrix (ι:=P) a)‖≤ε) :
    ‖rightRep h v d*sumBlock (a,block (fun _ : R => x))-
      sumBlock (a,block (fun _ : R => x))*rightRep h v d‖≤2*ε*‖d‖ :=
  commutator_approx (rightRep h v) _ _ d (rightRep_commute h v a d)
    ((duplicate_difference a x _).trans hx)
end KirchbergRordam.EndpointMultiplicity
end

end OAI