lean/OAI/Analysis/CharacterCriterion/CubeField.lean

import OAI.Analysis.CharacterCriterion.MatrixSimplex

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]

namespace KirchbergRordam.GenericRW.CubeField
open scoped BigOperators Topology
open Filter MatrixMix MatrixSimplex CubeWeights
variable {A : Type*} [CStarAlgebra A]
variable {n : ℕ} {φ : Fin (n+1) → Mat (Fin Size.n) →⋆ₐ[ℂ] A}
variable (F : Family φ)

lemma norm_matrix_unit (ψ : Mat (Fin Size.n) →⋆ₐ[ℂ] A) (i j : Fin Size.n) : ‖ψ (e i j)‖≤1 := by
  have hp : IsStarProjection (ψ (e j j)) := ⟨by change ψ _*ψ _=ψ _; rw [← map_mul,e_mul,ite_eq_left rfl],by change star (ψ _) = ψ _; rw [← map_star,e_star]⟩
  have hn := hp.norm_le _
  have he : ‖ψ (e i j)‖*‖ψ (e i j)‖=‖ψ (e j j)‖ := by
    rw [← CStarRing.norm_star_mul_self,← map_star,← map_mul,e_star,e_mul,ite_eq_left rfl]
  nlinarith [norm_nonneg (ψ (e i j))]

def dampedFun (f : C(Cube n,ℝ)) (i j : Fin Size.n) (t : Cube n) : A :=
  if ht : t≠0 then f t • F.hom (normalized ⟨t,ht⟩) (e i j) else 0
lemma damped_zero (f : C(Cube n,ℝ)) (i j : Fin Size.n) : dampedFun F f i j 0=0 := by simp [dampedFun]
lemma damped_norm (f : C(Cube n,ℝ)) (i j : Fin Size.n) (t : Cube n) : ‖dampedFun F f i j t‖≤‖f t‖ := by
  unfold dampedFun
  split_ifs
  · rw [norm_smul]
    exact mul_le_of_le_one_right (norm_nonneg _) (norm_matrix_unit _ _ _)
  · simp only [norm_zero]; exact norm_nonneg (f t)
lemma damped_continuous (f : C(Cube n,ℝ)) (hf : f 0=0) (i j : Fin Size.n) :
    Continuous (dampedFun F f i j) := by
  have ho : ContinuousOn (dampedFun F f i j) {t | t≠0} := by
    rw [continuousOn_iff_continuous_domRestrict]
    change Continuous (fun t : Punctured n => dampedFun F f i j t.1)
    have hh := ((f.continuous.comp continuous_subtype_val).smul
      ((F.continuous (e i j)).comp normalized_continuous))
    exact hh.congr (fun t => by simp [dampedFun,t.2])
  rw [continuous_iff_continuousAt]
  intro t
  by_cases ht : t=0
  · subst t
    change Tendsto (dampedFun F f i j) _ (𝓝 (dampedFun F f i j 0))
    rw [damped_zero]
    apply squeeze_zero_norm (damped_norm F f i j)
    simpa [ContinuousAt,hf] using f.continuous.norm.continuousAt (x:=0)
  · exact ho.continuousAt (isOpen_ne.mem_nhds ht)

def damped (f : C(Cube n,ℝ)) (hf : f 0=0) (i j : Fin Size.n) : C(Cube n,A) :=
  ⟨dampedFun F f i j,damped_continuous F f hf i j⟩

def scalarH (δ : ℝ) : C(Cube n,ℝ) := ⟨H δ,H_continuous δ⟩
def scalarSqrt (δ : ℝ) : C(Cube n,ℝ) := ⟨fun t => Real.sqrt (H δ t),(H_continuous δ).sqrt⟩
def x (δ : ℝ) (i j : Fin Size.n) : C(Cube n,A) := damped F (scalarH δ) (H_zero δ) i j
def v (δ : ℝ) : C(Cube n,A) := damped F (scalarSqrt δ) (by simp [scalarSqrt,H_zero]) 1 0

def h (δ : ℝ) : C(Cube n,A) := ⟨fun t => H δ t • (1:A),(H_continuous δ).smul continuous_const⟩
lemma x_apply (δ : ℝ) (i j : Fin Size.n) (t : Punctured n) :
    x F δ i j t.1=H δ t.1 • F.hom (normalized t) (e i j) := by simp [x,damped,dampedFun,t.2,scalarH]
lemma x_zero (δ : ℝ) (i j : Fin Size.n) : x F δ i j 0=0 := damped_zero F _ _ _
lemma v_apply (δ : ℝ) (t : Punctured n) :
    v F δ t.1=Real.sqrt (H δ t.1) • F.hom (normalized t) (e 1 0) := by simp [v,damped,dampedFun,t.2,scalarSqrt]
lemma v_zero (δ : ℝ) : v F δ 0=0 := damped_zero F _ _ _

lemma x_star (δ : ℝ) (i j : Fin Size.n) : star (x F δ i j)=x F δ j i := by
  ext t
  by_cases ht : t=0
  · subst t; simp [x_zero]
  · change star (x F δ i j t)=_
    rw [x_apply F δ i j ⟨t,ht⟩,x_apply F δ j i ⟨t,ht⟩,star_smul,star_trivial,← map_star,e_star]
lemma x_mul (δ : ℝ) (i j k l : Fin Size.n) :
    x F δ i j*x F δ k l=if j=k then h δ*x F δ i l else 0 := by
  ext t
  by_cases ht : t=0
  · subst t; by_cases hjk : j=k <;> simp [hjk,x_zero,h,H_zero]
  · change x F δ i j t*x F δ k l t=_
    rw [x_apply F δ i j ⟨t,ht⟩,x_apply F δ k l ⟨t,ht⟩,smul_mul_smul_comm,← map_mul,e_mul]
    by_cases hjk : j=k
    · rw [ite_eq_left hjk,ite_eq_left hjk]
      change _=H δ t • (1:A)*x F δ i l t
      rw [x_apply F δ i l ⟨t,ht⟩,smul_mul_smul_comm,one_mul]
    · simp [hjk]
lemma x_diagonal (δ : ℝ) : ∑ i : Fin Size.n,x F δ i i=h δ := by
  ext t
  by_cases ht : t=0
  · subst t; simp [x_zero,h,H_zero]
  · simp only [ContinuousMap.sum_apply]
    simp_rw [x_apply F δ _ _ ⟨t,ht⟩]
    rw [← Finset.smul_sum,← map_sum,e_sum,map_one]
    rfl
lemma v_star_mul {δ : ℝ} (hδ : 0<δ) : star (v F δ)*v F δ=x F δ 0 0 := by
  ext t
  by_cases ht : t=0
  · subst t; simp [v_zero,x_zero]
  · change star (v F δ t)*v F δ t=_
    rw [v_apply F δ ⟨t,ht⟩,x_apply F δ 0 0 ⟨t,ht⟩,star_smul,star_trivial,← map_star,
      e_star,smul_mul_smul_comm,Real.mul_self_sqrt (H_nonneg hδ t),← map_mul,e_mul,ite_eq_left rfl]
lemma v_mul_star {δ : ℝ} (hδ : 0<δ) : v F δ*star (v F δ)=x F δ 1 1 := by
  ext t
  by_cases ht : t=0
  · subst t; simp [v_zero,x_zero]
  · change v F δ t*star (v F δ t)=_
    rw [v_apply F δ ⟨t,ht⟩,x_apply F δ 1 1 ⟨t,ht⟩,star_smul,star_trivial,← map_star,
      e_star,smul_mul_smul_comm,Real.mul_self_sqrt (H_nonneg hδ t),← map_mul,e_mul,ite_eq_left rfl]

lemma x_support (δ : ℝ) (t : Cube n) (S : StarSubalgebra ℂ A)
    (hs : ∀ k,0<(t k:ℝ) → ∀ a,φ k a∈S) (i j : Fin Size.n) : x F δ i j t∈S := by
  by_cases ht : t=0
  · subst t; rw [x_zero]; exact S.zero_mem
  · rw [x_apply F δ i j ⟨t,ht⟩]
    exact S.smul_mem (F.support _ S (fun k hk => hs k (normalized_support ⟨t,ht⟩ k hk)) _) _

lemma x_localized {δ : ℝ} (hδ : 0<δ) (t : Punctured n) (j : Fin (n+1))
    (hj : δ<(t.1 j:ℝ)) (hi : ∀ i,i≠j → (t.1 i:ℝ)<δ/2) (a b : Fin Size.n) :
    x F δ a b t.1=φ j (e a b) := by
  rw [x_apply,localized_normalized t hδ j hj hi,F.vertex,H_of_large hδ t.1 j hj.le,one_smul]
end KirchbergRordam.GenericRW.CubeField

namespace KirchbergRordam.GenericRW.CubeField
open scoped BigOperators CStarAlgebra
open MatrixMix MatrixSimplex CubeWeights
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
variable {n : ℕ} {φ : Fin (n+1) → Mat (Fin Size.n) →⋆ₐ[ℂ] A} (F : Family φ)
lemma h_nonneg {δ : ℝ} (hδ : 0<δ) : 0≤(h δ : C(Cube n,A)) := by
  intro t
  exact smul_nonneg (H_nonneg hδ t) zero_le_one
omit [PartialOrder A] [StarOrderedRing A] in
lemma h_norm {δ : ℝ} (hδ : 0<δ) : ‖(h δ : C(Cube n,A))‖≤1 := by
  apply (ContinuousMap.norm_le _ (by norm_num)).mpr
  intro t
  change ‖H δ t • (1:A)‖≤1
  rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (H_nonneg hδ t)]
  have hn : ‖(1:A)‖≤1 := (IsStarProjection.one (R:=A)).norm_le _
  exact (mul_le_of_le_one_right (H_nonneg hδ t) hn).trans (H_le_one δ t)

def weighted {δ : ℝ} (hδ : 0<δ) : OrderZero.WeightedUnits (Fin Size.n) C(Cube n,A) where
  h := h δ
  x := x F δ
  nonneg := h_nonneg hδ
  norm_le := h_norm hδ
  star_x := x_star F δ
  mul_x := x_mul F δ
  diagonal := x_diagonal F δ

lemma first_nonneg {δ : ℝ} (hδ : 0<δ) : 0≤x F δ 0 0 := by
  rw [← v_star_mul F hδ]
  intro t
  exact star_mul_self_nonneg (v F δ t)
lemma second_nonneg {δ : ℝ} (hδ : 0<δ) : 0≤x F δ 1 1 := by
  rw [← v_mul_star F hδ]
  intro t
  exact mul_star_self_nonneg (v F δ t)
omit [PartialOrder A] [StarOrderedRing A] in
lemma pair_orthogonal (δ : ℝ) : x F δ 0 0*x F δ 1 1=0 := by
  have hn : 1 < Size.n := by have := PauliData.two_le; omega
  have h01 : (0:Fin Size.n) ≠ 1 := by
    intro he
    have hv := congrArg Fin.val he
    simp only [Fin.val_zero,Fin.val_one',Nat.mod_eq_of_lt hn] at hv
    omega
  rw [x_mul,ite_eq_right h01]
omit [PartialOrder A] [StarOrderedRing A] in
lemma entry_norm {δ : ℝ} (hδ : 0<δ) (i j : Fin Size.n) : ‖x F δ i j‖≤1 := by
  apply (ContinuousMap.norm_le _ (by norm_num)).mpr
  intro t
  exact (damped_norm F (scalarH δ) i j t).trans (by
    change ‖H δ t‖≤1
    rw [Real.norm_eq_abs,abs_of_nonneg (H_nonneg hδ t)]
    exact H_le_one δ t)
end KirchbergRordam.GenericRW.CubeField

namespace KirchbergRordam.GenericRW.ConeCube
open scoped CStarAlgebra ComplexOrder BigOperators
open ConeModel MatrixSimplex
local instance (n : ℕ) : PartialOrder (Tensor n) := CStarAlgebra.spectralOrder _
local instance (n : ℕ) : StarOrderedRing (Tensor n) := CStarAlgebra.spectralOrderedRing _

def family (n : ℕ) : Family (matLeg n) := (exists_family n (matLeg n) (matLeg_commute n)).some

lemma x_face_commute (n : ℕ) (δ : ℝ) (i l : Fin Size.n) (t : Cube n) (j : Fin (n+1))
    (hj : t j=0) (b : M2) : Commute (matLeg n j b) (CubeField.x (family n) δ i l t) := by
  let S := StarSubalgebra.centralizer ℂ (Set.range (matLeg n j))
  have hm : CubeField.x (family n) δ i l t∈S := CubeField.x_support _ _ _ S (by
    intro k hk a
    have hkj : k≠j := by intro h; subst k; rw [hj] at hk; exact (lt_irrefl (0:ℝ)) hk
    rw [StarSubalgebra.mem_centralizer_iff]
    rintro _ ⟨c,rfl⟩
    constructor
    · exact (matLeg_commute n j k hkj.symm c a).eq
    · rw [← map_star]
      exact (matLeg_commute n j k hkj.symm (star c) a).eq) i l
  exact (((StarSubalgebra.mem_centralizer_iff ℂ).mp hm) _ ⟨b,rfl⟩).1

lemma x_liftable (n : ℕ) (δ : ℝ) (i j : Fin Size.n) :
    ∃! b : Tensor n,repr n b=CubeField.x (family n) δ i j ∧
      ‖b‖=‖CubeField.x (family n) δ i j‖ := by
  apply exists_unique_lift
  intro a t ha
  obtain ⟨k,hk,hne⟩ := p_zero n a t ha
  exact ccoeff_zero n a k hne _ (x_face_commute n δ i j t k hk)

def x (n : ℕ) (δ : ℝ) (i j : Fin Size.n) : Tensor n := (x_liftable n δ i j).exists.choose
lemma repr_x (n : ℕ) (δ : ℝ) (i j : Fin Size.n) :
    repr n (x n δ i j)=CubeField.x (family n) δ i j := ((x_liftable n δ i j).exists.choose_spec).1

def hScalar (n : ℕ) (δ : ℝ) : C(Cube n,ℂ) :=
  ⟨fun t => ((CubeWeights.H δ t:ℝ):ℂ),Complex.continuous_ofReal.comp (CubeWeights.H_continuous δ)⟩
def h (n : ℕ) (δ : ℝ) : Tensor n := κ n (hScalar n δ)
lemma repr_h (n : ℕ) (δ : ℝ) : repr n (h n δ)=CubeField.h δ := by
  apply ContinuousMap.ext
  intro t
  rw [h,repr_κ]
  change algebraMap ℂ (Fiber n) (((CubeWeights.H δ t):ℝ):ℂ)=CubeWeights.H δ t • (1:Fiber n)
  rw [Algebra.algebraMap_eq_smul_one]
  exact Complex.coe_smul _ _
lemma h_nonneg (n : ℕ) {δ : ℝ} (hδ : 0<δ) : 0≤h n δ := by
  apply map_nonneg (κ n)
  intro t
  exact Complex.zero_le_real.mpr (CubeWeights.H_nonneg hδ t)
lemma h_norm (n : ℕ) {δ : ℝ} (hδ : 0<δ) : ‖h n δ‖≤1 := by
  have hn : ‖h n δ‖=‖repr n (h n δ)‖ := by
    exact (NonUnitalStarAlgHom.norm_map (repr n) (faithful n) _).symm
  rw [hn,repr_h]
  exact CubeField.h_norm hδ


def weighted (n : ℕ) {δ : ℝ} (hδ : 0<δ) : OrderZero.WeightedUnits (Fin Size.n) (Tensor n) where
  h := h n δ
  x := x n δ
  nonneg := h_nonneg n hδ
  norm_le := h_norm n hδ
  star_x i j := by apply faithful n; rw [map_star,repr_x,repr_x]; exact CubeField.x_star _ _ _ _
  mul_x i j k l := by
    have product_eq := CubeField.x_mul (family n) δ i j k l
    apply faithful n
    rw [map_mul,repr_x,repr_x]
    refine product_eq.trans ?_
    split_ifs <;> simp [map_mul,repr_h,repr_x]
  diagonal := by
    apply faithful n
    rw [map_sum,repr_h]
    simp_rw [repr_x]
    exact CubeField.x_diagonal _ _
end KirchbergRordam.GenericRW.ConeCube
end

end OAI