Datasets:

roseking
/

openAI-math-dataset

import OAI.Analysis.CharacterCriterion.CubeField

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 [instSize : KirchbergRordam.GenericRW.Size]
local notation "rankNum" => KirchbergRordam.GenericRW.Size.n
variable [instPauliData : KirchbergRordam.GenericRW.PauliData]


namespace KirchbergRordam.GenericRW.ClosedRightRange
open scoped CStarAlgebra
variable {A B : Type*} [CStarAlgebra A] [CStarAlgebra B]
omit instSize instPauliData in
lemma map_mem.{contextLevel1, contextLevel2}
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {A : Type contextLevel1}
    {B : Type contextLevel2}
    [CStarAlgebra A]
    [CStarAlgebra B]
    (φ : A →⋆ₐ[ℂ] B) {a z : A} (hz : z ∈ space a) :
    φ z ∈ space (φ a) := by
  have hh (z : space a) : φ (z : A) ∈ space (φ a) := by
    refine (dense a).induction_on z ((closed (φ a)).preimage ((map_continuous φ).comp continuous_subtype_val)) ?_
    intro x
    change φ (a*x) ∈ space (φ a)
    rw [map_mul]
    exact mem_space _ _
  exact hh ⟨z,hz⟩
end KirchbergRordam.GenericRW.ClosedRightRange

namespace KirchbergRordam.GenericRW.CuntzBelow
open scoped CStarAlgebra
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]


omit instSize instPauliData in
lemma of_right_range.{contextLevel1}
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {A : Type contextLevel1}
    [CStarAlgebra A]
    [PartialOrder A]
    [StarOrderedRing A]
    {b z : A} (hb : 0≤b) (hz : z ∈ ClosedRightRange.space b) :
    CuntzBelow (star z*z) b := by
  have hh (z : ClosedRightRange.space (CFC.sqrt b)) : CuntzBelow (star (z:A)*(z:A)) b := by
    refine (ClosedRightRange.dense (CFC.sqrt b)).induction_on z
      (isClosed_closure.preimage (by fun_prop)) ?_
    intro x
    apply of_eq (x:=x)
    change star x*b*x=star (CFC.sqrt b*x)*(CFC.sqrt b*x)
    rw [star_mul,(IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg b)).star_eq]
    calc
      _ = star x*(CFC.sqrt b*CFC.sqrt b)*x := by rw [CFC.sqrt_mul_sqrt_self b hb]
      _ = _ := by noncomm_ring
  rw [← Domination.sqrt_right_range_eq b hb] at hz
  exact hh ⟨z,hz⟩
end KirchbergRordam.GenericRW.CuntzBelow

namespace KirchbergRordam.GenericRW.AmplifiedComparison
open scoped CStarAlgebra
open scoped BigOperators
variable {P R B : Type*} [CStarAlgebra P] [CStarAlgebra R] [CStarAlgebra B]
variable {J I : Type*} [Fintype J] [DecidableEq J] [Fintype I] [DecidableEq I]
variable (φ : P →⋆ₐ[ℂ] B) (ψ : R →⋆ₐ[ℂ] B)
variable (hc : ∀ a b, Commute (φ a) (ψ b))

def pair (a : P) (b : R) : B := φ a*ψ b
omit hc in
omit instSize instPauliData in
lemma pair_mul.{contextLevel1, contextLevel2, contextLevel3}
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type contextLevel1}
    {R : Type contextLevel2}
    {B : Type contextLevel3}
    [CStarAlgebra P]
    [CStarAlgebra R]
    [CStarAlgebra B]
    (φ : P →⋆ₐ[ℂ] B)
    (ψ : R →⋆ₐ[ℂ] B)
    (hc : ∀ (a : P) (b : R), Commute (φ a) (ψ b))
    (a c : P) (b d : R) : pair φ ψ a b*pair φ ψ c d=pair φ ψ (a*c) (b*d) := by
  simp only [pair,map_mul]
  calc
    (φ a*ψ b)*(φ c*ψ d)=φ a*(ψ b*φ c)*ψ d := by noncomm_ring
    _=φ a*(φ c*ψ b)*ψ d := by rw [← (hc c b).eq]
    _=_ := by noncomm_ring
omit hc in
omit instSize instPauliData in
lemma pair_star.{contextLevel1, contextLevel2, contextLevel3}
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type contextLevel1}
    {R : Type contextLevel2}
    {B : Type contextLevel3}
    [CStarAlgebra P]
    [CStarAlgebra R]
    [CStarAlgebra B]
    (φ : P →⋆ₐ[ℂ] B)
    (ψ : R →⋆ₐ[ℂ] B)
    (hc : ∀ (a : P) (b : R), Commute (φ a) (ψ b))
    (a : P) (b : R) : star (pair φ ψ a b)=pair φ ψ (star a) (star b) := by
  simp only [pair,star_mul,map_star]
  simpa only [map_star] using (hc (star a) (star b)).eq.symm
omit instSize instPauliData in
lemma pair_zero_left.{contextLevel1, contextLevel2, contextLevel3}
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type contextLevel1}
    {R : Type contextLevel2}
    {B : Type contextLevel3}
    [CStarAlgebra P]
    [CStarAlgebra R]
    [CStarAlgebra B]
    (φ : P →⋆ₐ[ℂ] B)
    (ψ : R →⋆ₐ[ℂ] B)
    (b : R) : pair φ ψ 0 b=0 := by simp [pair]
omit instSize instPauliData in
lemma pair_zero_right.{contextLevel1, contextLevel2, contextLevel3}
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type contextLevel1}
    {R : Type contextLevel2}
    {B : Type contextLevel3}
    [CStarAlgebra P]
    [CStarAlgebra R]
    [CStarAlgebra B]
    (φ : P →⋆ₐ[ℂ] B)
    (ψ : R →⋆ₐ[ℂ] B)
    (a : P) : pair φ ψ a 0=0 := by simp [pair]
omit [DecidableEq I] in
omit instSize instPauliData in
lemma pair_sum_left.{contextLevel1, contextLevel2, contextLevel3, contextLevel4}
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type contextLevel1}
    {R : Type contextLevel2}
    {B : Type contextLevel3}
    [CStarAlgebra P]
    [CStarAlgebra R]
    [CStarAlgebra B]
    {I : Type contextLevel4}
    [Fintype I]
    (φ : P →⋆ₐ[ℂ] B)
    (ψ : R →⋆ₐ[ℂ] B)
    (a : I→P) (b : R) : (∑ i,pair φ ψ (a i) b)=pair φ ψ (∑ i,a i) b := by
  simp only [pair,map_sum,Finset.sum_mul]
omit [DecidableEq J] in
omit instSize instPauliData in
lemma pair_sum_right.{contextLevel1, contextLevel2, contextLevel3, contextLevel4}
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type contextLevel1}
    {R : Type contextLevel2}
    {B : Type contextLevel3}
    [CStarAlgebra P]
    [CStarAlgebra R]
    [CStarAlgebra B]
    {J : Type contextLevel4}
    [Fintype J]
    (φ : P →⋆ₐ[ℂ] B)
    (ψ : R →⋆ₐ[ℂ] B)
    (a : P) (b : J→R) : (∑ j,pair φ ψ a (b j))=pair φ ψ a (∑ j,b j) := by
  simp only [pair,map_sum,Finset.mul_sum]

variable [PartialOrder P] [StarOrderedRing P]
variable (y : J→P) (hy : ∀ j,0≤y j) (horth : ∀ j k,j≠k → y j*y k=0)
variable (u : J→I→P) (z : I→R) (v : J→R) (e : R)
variable (hz : ∀ i l,star (z i)*z l=if i=l then e else 0)

def gramVector (j : J) (i : I) : B := pair φ ψ (CFC.sqrt (y j)*u j i) (z i*v j)
omit hy horth in
omit [Fintype J] [DecidableEq J] in
omit instSize instPauliData in
lemma sqrt_orthogonal.{contextLevel1, contextLevel2}
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type contextLevel1}
    [CStarAlgebra P]
    {J : Type contextLevel2}
    [PartialOrder P]
    [StarOrderedRing P]
    (y : J → P)
    (hy : ∀ (j : J), 0 ≤ y j)
    (horth : ∀ (j k : J), j ≠ k → y j * y k = 0)
    (j k : J) (hjk : j≠k) : CFC.sqrt (y j)*CFC.sqrt (y k)=0 := by
  exact Intertwining.mul_sqrt_eq_zero (hy k) (Intertwining.sqrt_mul_eq_zero (hy j) (horth j k hjk))

include hc hy horth hz in
omit [Fintype I] [Fintype J] in
lemma gram_inner (j k : J) (i l : I) :
    star (gramVector φ ψ y u z v j i)*gramVector φ ψ y u z v k l =
      if j=k ∧ i=l then pair φ ψ (star (u j i)*y j*u j i) (star (v j)*e*v j) else 0 := by
  simp only [gramVector,pair_star φ ψ hc,pair_mul φ ψ hc,star_mul,
    (IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg (y j))).star_eq]
  have hleft : (star (u j i)*CFC.sqrt (y j))*(CFC.sqrt (y k)*u k l)=
      star (u j i)*(CFC.sqrt (y j)*CFC.sqrt (y k))*u k l := by noncomm_ring
  have hright : (star (v j)*star (z i))*(z l*v k)=star (v j)*(star (z i)*z l)*v k := by noncomm_ring
  rw [hleft,hright,hz]
  by_cases hjk : j=k
  · subst k
    rw [CFC.sqrt_mul_sqrt_self (y j) (hy j)]
    by_cases hil : i=l
    · subst l; simp
    · simp [hil,pair_zero_right]
  · rw [sqrt_orthogonal y hy horth j k hjk]
    simp [hjk,pair_zero_left]

variable [PartialOrder B] [StarOrderedRing B]
include ψ hc hy horth z hz in


theorem compare (X : P)
    (hu : ∀ j,∑ i,star (u j i)*y j*u j i=X)
    (hv : ∑ j,star (v j)*e*v j=1) :
    CuntzBelow (φ X) (φ (∑ j,y j)) := by
  let w : B := ∑ j,∑ i,gramVector φ ψ y u z v j i
  have hww : star w*w=φ X := by
    dsimp only [w]
    simp only [star_sum,Finset.sum_mul,Finset.mul_sum]
    simp_rw [gram_inner φ ψ hc y hy horth u z v e hz]
    simp only [ite_and]
    simp only [Finset.sum_ite_irrel,Finset.sum_const_zero,Finset.sum_ite_eq',Finset.mem_univ,ite_true]
    simp_rw [pair_sum_left,hu]
    rw [pair_sum_right,hv]
    simp [pair]
  have hY : 0≤∑ j,y j := Finset.sum_nonneg (fun j _ => hy j)
  have hw : w ∈ ClosedRightRange.space (φ (∑ j,y j)) := by
    apply (ClosedRightRange.space _).sum_mem
    intro j _
    apply (ClosedRightRange.space _).sum_mem
    intro i _
    apply ClosedRightRange.mul_right_mem
    apply ClosedRightRange.map_mem
    apply ClosedRightRange.mul_right_mem
    apply Domination.sqrt_mem_right_range (hy j) hY (by norm_num : (0:ℝ)≤1)
    simp only [one_smul]
    exact Finset.single_le_sum (fun k _ => hy k) (Finset.mem_univ j)
  rw [← hww]
  exact CuntzBelow.of_right_range (map_nonneg φ hY) hw
end KirchbergRordam.GenericRW.AmplifiedComparison
namespace KirchbergRordam.GenericRW.MaxTensor
open scoped BigOperators
universe u
variable {P Q : Type u} [CStarAlgebra P] [CStarAlgebra Q]
def pure (a : P) (b : Q) : Algebra P Q := includeLeft a*includeRight b
lemma pure_mul (a c : P) (b d : Q) : pure a b*pure c d=pure (a*c) (b*d) :=
  AmplifiedComparison.pair_mul includeLeft includeRight legs_commute a c b d
lemma pure_star (a : P) (b : Q) : star (pure a b)=pure (star a) (star b) :=
  AmplifiedComparison.pair_star includeLeft includeRight legs_commute a b
omit instSize instPauliData in
@[simp] lemma pure_zero_left
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type u}
    {Q : Type u}
    [CStarAlgebra P]
    [CStarAlgebra Q]
    (b : Q) : pure (0:P) b=0 := by simp [pure]
omit instSize instPauliData in
@[simp] lemma pure_zero_right
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type u}
    {Q : Type u}
    [CStarAlgebra P]
    [CStarAlgebra Q]
    (a : P) : pure a (0:Q)=0 := by simp [pure]
omit instSize instPauliData in
@[simp] lemma pure_one
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type u}
    {Q : Type u}
    [CStarAlgebra P]
    [CStarAlgebra Q]
    : pure (1:P) (1:Q)=1 := by simp [pure]
omit instSize instPauliData in
lemma pure_sum_left
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type u}
    {Q : Type u}
    [CStarAlgebra P]
    [CStarAlgebra Q]
    {I : Type*} [Fintype I] (f : I→P) (b : Q) :
    pure (∑ i,f i) b=∑ i,pure (f i) b := by simp only [pure,map_sum,Finset.sum_mul]
omit instSize instPauliData in
lemma pure_sum_right
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {P : Type u}
    {Q : Type u}
    [CStarAlgebra P]
    [CStarAlgebra Q]
    {I : Type*} [Fintype I] (a : P) (f : I→Q) :
    pure a (∑ i,f i)=∑ i,pure a (f i) := by simp only [pure,map_sum,Finset.mul_sum]
def word : (n : ℕ) → (Fin (n+1)→P) → powerPack P n
  | 0,f => f 0
  | n+1,f => pure (f 0) (word n (fun i => f i.succ))
lemma word_mul (n : ℕ) (f g : Fin (n+1)→P) :
    word n f*word n g=word n (fun i => f i*g i) := by
  induction n with
  | zero => rfl
  | succ n ih => exact (pure_mul _ _ _ _).trans (congrArg (pure (f 0*g 0)) (ih _ _))
lemma word_star (n : ℕ) (f : Fin (n+1)→P) : star (word n f)=word n (fun i => star (f i)) := by
  induction n with
  | zero => rfl
  | succ n ih => exact (pure_star _ _).trans (congrArg (pure (star (f 0))) (ih _))
lemma word_one (n : ℕ) : word n (fun _ => (1:P))=1 := by
  induction n with
  | zero => rfl
  | succ n ih => change pure 1 (word n fun _ => (1:P))=1; rw [ih,pure_one]
lemma word_zero_coordinate (n : ℕ) (f : Fin (n+1)→P) (i : Fin (n+1)) (hi : f i=0) : word n f=0 := by
  induction n with
  | zero =>
    have he : i=0 := by omega
    change f 0=0
    simpa only [he] using hi
  | succ n ih =>
    change pure (f 0) (word n (fun j => f j.succ))=0
    rcases Fin.eq_zero_or_eq_succ i with rfl | ⟨j,rfl⟩
    · rw [hi,pure_zero_left]
    · rw [ih _ j hi,pure_zero_right]
variable {I : Type*} [Fintype I]
lemma sum_word (n : ℕ) (f : I→P) :
    (∑ σ : Fin (n+1)→I,word n (fun i => f (σ i)))=word n (fun _ => ∑ a,f a) := by
  induction n with
  | zero =>
    exact (Equiv.funUnique (Fin 1) I).sum_comp f
  | succ n ih =>
    rw [← (Fin.consEquiv (fun _ : Fin (n+1+1) => I)).sum_comp]
    rw [Fintype.sum_prod_type]
    change (∑ a,∑ σ : Fin (n+1)→I,pure (f a) (word n (fun i => f (σ i))))=
      pure (∑ a,f a) (word n (fun _ => ∑ a,f a))
    simp_rw [← pure_sum_right,ih]
    rw [pure_sum_left]
lemma word_full_comparison (n : ℕ) (c : P) (v : I→P)
    (hv : ∑ i,star (v i)*c*v i=1) :
    ∑ σ : Fin (n+1)→I,star (word n (fun i => v (σ i)))*word n (fun _ => c)*
      word n (fun i => v (σ i))=1 := by
  simp_rw [word_star,word_mul]
  rw [sum_word n (fun i => star (v i)*c*v i),hv,word_one]
variable [DecidableEq I]
omit [Fintype I] in
lemma word_gram (n : ℕ) (c : P) (z : I→P)
    (hz : ∀ i j,star (z i)*z j=if i=j then c else 0)
    (σ τ : Fin (n+1)→I) :
    star (word n (fun i => z (σ i)))*word n (fun i => z (τ i))=
      if σ=τ then word n (fun _ => c) else 0 := by
  rw [word_star,word_mul]
  simp_rw [hz]
  by_cases h : σ=τ
  · subst τ; simp
  · rw [ite_eq_right h]
    obtain ⟨i,hi⟩ := Function.ne_iff.mp h
    exact word_zero_coordinate n _ i (ite_eq_right hi)
theorem auxiliary {N : ℕ} (c : P) (v : Fin N→P)
    (hv : ∑ i,star (v i)*c*v i=1) (z : Fin 2→P)
    (hz : ∀ i j,star (z i)*z j=if i=j then c else 0) :
    ∃ L : ℕ, ∃ e : powerPack P L,
      ∃ a : Fin N→powerPack P L,
      ∃ b : (Fin (L+1)→Fin N)→powerPack P L,
        (∀ i j,star (a i)*a j=if i=j then e else 0) ∧
        (∑ j,star (b j)*e*b j=1) := by
  classical
  let L := Nat.log 2 N
  have hcard : Fintype.card (Fin N) ≤ Fintype.card (Fin (L+1)→Fin 2) := by
    simp only [Fintype.card_fin,Fintype.card_fun]
    exact (Nat.lt_pow_succ_log_self (by norm_num : 1<2) N).le
  obtain ⟨ι⟩ := Function.Embedding.nonempty_of_card_le hcard
  refine ⟨L,word L (fun _ => c),fun i => word L (fun j => z (ι i j)),
    fun σ => word L (fun i => v (σ i)),?_,word_full_comparison L c v hv⟩
  intro i j
  rw [word_gram L c z hz]
  simp only [EmbeddingLike.apply_eq_iff_eq]
end KirchbergRordam.GenericRW.MaxTensor
namespace KirchbergRordam.GenericRW.MaxTensor
open scoped BigOperators CStarAlgebra
variable {P : Type*} [CStarAlgebra P]
lemma word_single (n : ℕ) (j : Fin (n+1)) (a : P) :
    word n (fun i => if i=j then a else 1)=leg P n j a := by
  induction n with
  | zero => rw [Fin.eq_zero j]; rfl
  | succ n ih =>
    induction j using Fin.cases with
    | zero =>
      have he : (fun i : Fin (n+1) => if i.succ=(0:Fin (n+2)) then a else 1)=fun _=>1 := by simp
      change pure a (word n _)=includeLeft a
      rw [he,word_one]
      simp [pure]
    | succ j =>
      have he : (fun i : Fin (n+1) => if i.succ=j.succ then a else 1)=fun i=>if i=j then a else 1 := by simp
      change pure 1 (word n _)=includeRight (leg P n j a)
      rw [he,ih]
      simp [pure]
lemma leg_word_mul (n : ℕ) (j : Fin (n+1)) (a : P) (f : Fin (n+1)→P) :
    leg P n j a*word n f=word n (fun i => if i=j then a*f i else f i) := by
  rw [← word_single,word_mul]
  congr 1
  funext i
  split_ifs <;> simp
end KirchbergRordam.GenericRW.MaxTensor
namespace KirchbergRordam.GenericRW.ConeCube
open scoped CStarAlgebra BigOperators ComplexOrder
open _root_.OAI.KirchbergRordam.GenericRW.MaxTensor ConeModel
omit instSize instPauliData in
lemma left_fixed
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    (n : ℕ) (a b : M2) (c : Fiber n) (hab : a*b=b) :
    MatrixTensor.leftMatrix a * (MatrixTensor.leftMatrix b * MatrixTensor.rightMatrix c)=
      MatrixTensor.leftMatrix b * MatrixTensor.rightMatrix c := by
  rw [← mul_assoc,← map_mul,hab]
lemma right_fixed (n : ℕ) (a : M2) (b c : Fiber n) (hbc : b*c=c) :
    MatrixTensor.rightMatrix b * (MatrixTensor.leftMatrix a * MatrixTensor.rightMatrix c)=
      MatrixTensor.leftMatrix a * MatrixTensor.rightMatrix c := by
  rw [← mul_assoc,← (MatrixTensor.legs_commute _ _).eq,mul_assoc,← map_mul,hbc]
lemma repr_word_succ (n : ℕ) (f : Fin (n+2) → ConeModel.Cone) (t : Cube (n+1)) :
    repr (n+1) (word (n+1) f) t=
      MatrixTensor.leftMatrix ((f 0:C(unitInterval,M2)) (t 0))*
        MatrixTensor.rightMatrix (repr n (word n (fun i=>f i.succ)) (tail n t)) := by
  have product_eq := congrArg (fun field : C(Cube (n+1),Fiber (n+1)) => field t)
    ((repr (n+1)).map_mul (includeLeft (f 0)) (includeRight (word n (fun i=>f i.succ))))
  exact product_eq.trans (congrArg₂ (· * ·) (repr_left n (f 0) t)
    (repr_right n (word n (fun i=>f i.succ)) t))
lemma repr_word_zero (n : ℕ) (f : Fin (n+1) → ConeModel.Cone) (t : Cube n) (j : Fin (n+1))
    (hj : (f j:C(unitInterval,M2)) (t j)=0) : repr n (word n f) t=0 := by
  induction n with
  | zero =>
    rw [Fin.eq_zero j] at hj
    exact hj
  | succ n ih =>
    have product_zero :
        MatrixTensor.leftMatrix (κ := PauliPower.Row n) ((f 0 : C(unitInterval,M2)) (t 0)) *
          MatrixTensor.rightMatrix (ι := Fin Size.n)
            (repr n (word n (fun index => f index.succ)) (tail n t)) = 0 := by
      induction j using Fin.cases with
      | zero => rw [hj,map_zero,zero_mul]
      | succ j => rw [ih _ (tail n t) j hj,map_zero,mul_zero]
    exact (repr_word_succ n f t).trans product_zero
lemma repr_word_fix (n : ℕ) (f : Fin (n+1) → ConeModel.Cone) (t : Cube n) (j : Fin (n+1)) (b : M2)
    (hj : b*(f j:C(unitInterval,M2)) (t j)=(f j:C(unitInterval,M2)) (t j)) :
    matLeg n j b*repr n (word n f) t=repr n (word n f) t := by
  induction n with
  | zero =>
    rw [Fin.eq_zero j] at hj
    exact hj
  | succ n ih =>
    simp only [repr_word_succ]
    induction j using Fin.cases with
    | zero => exact left_fixed n _ _ _ hj
    | succ j => exact right_fixed n _ _ _ (ih _ (tail n t) j hj)
end KirchbergRordam.GenericRW.ConeCube
namespace KirchbergRordam.GenericRW.ConeCut
open scoped CStarAlgebra ComplexOrder BigOperators
open ConeModel MatrixMix
local instance : PartialOrder Cone := CStarAlgebra.spectralOrder _
local instance : LE Cone := (CStarAlgebra.spectralOrder Cone).toLE
local instance : StarOrderedRing Cone := CStarAlgebra.spectralOrderedRing _
def unit (i j : Fin Size.n) : M2 := CStarMatrix.ofMatrix (e i j)
lemma unit_star (i j : Fin Size.n) : star (unit i j)=unit j i := by
  exact (map_star CStarMatrix.ofMatrixStarAlgEquiv (e i j)).symm.trans (congrArg CStarMatrix.ofMatrix (e_star i j))
lemma unit_mul (i j k l : Fin Size.n) : unit i j*unit k l=if j=k then unit i l else 0 := by
  change CStarMatrix.ofMatrixStarAlgEquiv (e i j)*CStarMatrix.ofMatrixStarAlgEquiv (e k l)=_
  rw [← map_mul,e_mul]
  split_ifs
  · rfl
  · exact map_zero CStarMatrix.ofMatrixStarAlgEquiv
def coeff (δ : ℝ) (t : unitInterval) : ℝ := max ((t:ℝ)-δ) 0
omit instSize instPauliData in
lemma coeff_continuous
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    (δ : ℝ) : Continuous (coeff δ) := by unfold coeff; fun_prop
omit instSize instPauliData in
lemma coeff_zero
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    {δ : ℝ} (hδ : 0≤δ) : coeff δ 0=0 := by simp [coeff,max_eq_right (neg_nonpos.mpr hδ)]
omit instSize instPauliData in
lemma coeff_nonneg
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    (δ : ℝ) (t : unitInterval) : 0≤coeff δ t := le_max_right _ _
def q (δ : ℝ) (hδ : 0≤δ) : Cone :=
  ⟨⟨fun t=>coeff δ t • unit 0 0,(coeff_continuous δ).smul continuous_const⟩,by
    change coeff δ 0 • unit 0 0∈(⊥:StarSubalgebra ℂ M2)
    rw [coeff_zero hδ,zero_smul]
    exact zero_mem _⟩
def z (δ : ℝ) (hδ : 0≤δ) (i : Fin Size.n) : Cone :=
  ⟨⟨fun t=>Real.sqrt (coeff δ t) • unit i 0,(coeff_continuous δ).sqrt.smul continuous_const⟩,by
    change Real.sqrt (coeff δ 0) • unit i 0∈(⊥:StarSubalgebra ℂ M2)
    rw [coeff_zero hδ,Real.sqrt_zero,zero_smul]
    exact zero_mem _⟩
lemma z_gram (δ : ℝ) (hδ : 0≤δ) (i j : Fin Size.n) :
    star (z δ hδ i)*z δ hδ j=if i=j then q δ hδ else 0 := by
  apply Subtype.ext
  apply ContinuousMap.ext
  intro t
  change star (Real.sqrt (coeff δ t) • unit i 0)*(Real.sqrt (coeff δ t) • unit j 0)=_
  rw [star_smul,star_trivial,unit_star,smul_mul_smul_comm,Real.mul_self_sqrt (coeff_nonneg δ t),unit_mul]
  split_ifs <;> simp [q]
lemma q_nonneg (δ : ℝ) (hδ : 0≤δ) : 0≤q δ hδ := by
  have hz := z_gram δ hδ 0 0
  rw [ite_eq_left rfl] at hz
  rw [← hz]
  exact star_mul_self_nonneg (z δ hδ (0:Fin Size.n))
def g (δ : ℝ) : Cone := scalar ⟨fun t => (CubeWeights.g δ t:ℂ),Complex.continuous_ofReal.comp (CubeWeights.g_continuous δ)⟩
lemma g_nonneg (δ : ℝ) : 0≤g δ := by
  let f : C(unitInterval,ℂ) := ⟨fun t => (CubeWeights.g δ t:ℂ),Complex.continuous_ofReal.comp (CubeWeights.g_continuous δ)⟩
  have hf : 0≤f := fun t => Complex.zero_le_real.mpr (CubeWeights.g_nonneg δ t)
  have he : star (scalar (CFC.sqrt f))*scalar (CFC.sqrt f)=g δ := by
    rw [← map_star,← map_mul,(IsSelfAdjoint.of_nonneg (CFC.sqrt_nonneg f)).star_eq,
      CFC.sqrt_mul_sqrt_self f hf]
    rfl
  rw [← he]
  exact star_mul_self_nonneg _
lemma qg_zero {δ : ℝ} (hδ : 0<δ) : q δ hδ.le*g δ=0 := by
  apply Subtype.ext
  apply ContinuousMap.ext
  intro t
  change (coeff δ t • unit 0 0)*(algebraMap ℂ M2 (↑(CubeWeights.g δ t)))=0
  by_cases ht : (t:ℝ)≤δ
  · rw [show coeff δ t=0 by simp [coeff,max_eq_right (sub_nonpos.mpr ht)],zero_smul,zero_mul]
  · rw [CubeWeights.g_of_large hδ t (le_of_not_ge ht),Complex.ofReal_zero,map_zero,mul_zero]
lemma q_support {δ : ℝ} (hδ : 0≤δ) (t : unitInterval)
    (hq : (q δ hδ:C(unitInterval,M2)) t≠0) : δ<(t:ℝ) := by
  by_contra ht
  apply hq
  change coeff δ t • unit 0 0=0
  rw [show coeff δ t=0 by simp [coeff,max_eq_right (sub_nonpos.mpr (le_of_not_gt ht))],zero_smul]
lemma g_support {δ : ℝ} (hδ : 0<δ) (t : unitInterval)
    (hg : (g δ:C(unitInterval,M2)) t≠0) : (t:ℝ)<δ/2 := by
  apply CubeWeights.g_pos hδ
  apply lt_of_le_of_ne (CubeWeights.g_nonneg δ t)
  intro hz
  apply hg
  change algebraMap ℂ M2 (↑(CubeWeights.g δ t))=0
  rw [← hz,Complex.ofReal_zero,map_zero]
lemma q_fix (δ : ℝ) (hδ : 0≤δ) (t : unitInterval) :
    unit 0 0*(q δ hδ:C(unitInterval,M2)) t=(q δ hδ:C(unitInterval,M2)) t := by
  change unit 0 0*(coeff δ t • unit 0 0)=coeff δ t • unit 0 0
  rw [mul_smul_comm,unit_mul,ite_eq_left rfl]
end KirchbergRordam.GenericRW.ConeCut
namespace KirchbergRordam.GenericRW.ConeCube
open scoped CStarAlgebra ComplexOrder BigOperators
open ConeModel _root_.OAI.KirchbergRordam.GenericRW.MaxTensor
def y (n : ℕ) (δ : ℝ) (hδ : 0≤δ) (j : Fin (n+1)) : Tensor n :=
  word n (fun i => if i=j then ConeCut.q δ hδ else ConeCut.g δ)
local instance (n : ℕ) : PartialOrder (Tensor n) := CStarAlgebra.spectralOrder _
local instance (n : ℕ) : StarOrderedRing (Tensor n) := CStarAlgebra.spectralOrderedRing _
lemma x_fixes_y (n : ℕ) {δ : ℝ} (hδ : 0<δ) (j : Fin (n+1)) :
    x n δ 0 0*y n δ hδ.le j=y n δ hδ.le j := by
  apply faithful n
  rw [map_mul,repr_x]
  apply ContinuousMap.ext
  intro t
  rw [ContinuousMap.mul_apply]
  by_cases hy : repr n (y n δ hδ.le j) t=0
  · rw [hy,mul_zero]
  · have hf (i : Fin (n+1)) :
        ((if i=j then ConeCut.q δ hδ.le else ConeCut.g δ : Cone) : C(unitInterval,M2)) (t i)≠0 := by
      intro hi
      exact hy (repr_word_zero n _ t i hi)
    have hj : δ<(t j:ℝ) := ConeCut.q_support hδ.le _ (by simpa using hf j)
    have hi (i : Fin (n+1)) (hij : i≠j) : (t i:ℝ)<δ/2 :=
      ConeCut.g_support hδ _ (by simpa [hij] using hf i)
    have ht : t≠0 := by
      intro hz
      have he : (t j:ℝ)=0 := by rw [hz]; rfl
      linarith
    rw [CubeField.x_localized (family n) hδ ⟨t,ht⟩ j hj hi]
    apply repr_word_fix n _ t j (ConeCut.unit 0 0)
    simpa only [ite_true] using ConeCut.q_fix δ hδ.le (t j)
omit instSize instPauliData in
lemma matrix_scalar_product
    [KirchbergRordam.GenericRW.Size]
    [KirchbergRordam.GenericRW.PauliData]
    (n : ℕ) (a b : ℂ) :
    MatrixTensor.leftMatrix (κ:=PauliPower.Row n) (algebraMap ℂ M2 a) *
      MatrixTensor.rightMatrix (ι:=Fin Size.n) (algebraMap ℂ (Fiber n) b)=
        algebraMap ℂ (Fiber (n+1)) (a*b) := by
  rw [AlgHomClass.commutes,AlgHomClass.commutes,← map_mul]
  rfl
lemma repr_one (n : ℕ) : repr n (1:Tensor n)=1 :=
  (repr n).toAlgHom.map_one
lemma repr_scalar_word (n : ℕ) (f : Fin (n+1) → C(unitInterval,ℂ)) (t : Cube n) :
    repr n (word n (fun i => scalar (f i))) t = algebraMap ℂ (Fiber n) (∏ i,f i (t i)) := by
  induction n with
  | zero =>
    rw [Fin.prod_univ_succ]
    simp only [Finset.univ_eq_empty,Finset.prod_empty,mul_one]
    rfl
  | succ n ih =>
    rw [repr_word_succ,ih]
    rw [scalar_apply]
    have product_eq := matrix_scalar_product n (f 0 (t 0))
      (∏ index : Fin (n+1), f index.succ (t index.succ))
    exact product_eq.trans (congrArg (algebraMap ℂ (Fiber (n+1)))
      (Fin.prod_univ_succ (fun i => f i (t i))).symm)
lemma defect_word (n : ℕ) (δ : ℝ) : 1-h n δ=word n (fun _=>ConeCut.g δ) := by
  apply faithful n
  rw [map_sub,repr_one,repr_h]
  apply ContinuousMap.ext
  intro t
  rw [ContinuousMap.sub_apply]
  change 1-CubeWeights.H δ t • (1:Fiber n)=repr n (word n (fun _=>ConeCut.g δ)) t
  rw [ConeCut.g,repr_scalar_word]
  simp only [ContinuousMap.coe_mk,← Complex.ofReal_prod]
  rw [CubeWeights.H]
  rw [Algebra.algebraMap_eq_smul_one,Complex.coe_smul]
  module
end KirchbergRordam.GenericRW.ConeCube
end
end OAI