import Mathlib
import OAI.Analysis.FreeWords.KronExpansion
import OAI.Analysis.FreeWords.MatrixCommutantTrace
import OAI.Analysis.FreeWords.MatrixProduct
import OAI.Analysis.FreeWords.MatrixReindex
import OAI.Analysis.FreeWords.MatrixSubfactor
import OAI.Analysis.FreeWords.MatrixTraceUnique

namespace OAI

noncomputable section

namespace KKVariableUltra

open KKTraceTransfer KKFreeLiftFrontier KKProjectionSupport KKMatrixQuantization

noncomputable section

variable {ι κ A : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ]
  [Ring A] [Algebra ℂ A] [StarRing A] [StarModule ℂ A]

theorem matrixProduct_kronecker_one
    (ρ : Matrix ι ι ℂ →⋆ₐ[ℂ] A) (σ : Matrix κ κ ℂ →⋆ₐ[ℂ] A)
    (hc : ∀ i j k l, Commute (matrixUnit ρ i j) (matrixUnit σ k l)) (a : Matrix ι ι ℂ) :
    matrixProduct ρ σ hc (Matrix.kronecker a 1)=ρ a := by
  rw [kronecker_one_expansion]
  simp only [map_sum,map_smul]
  change (∑ i,∑ j,∑ k, a i j • matrixUnit (matrixProduct ρ σ hc) (i,k) (j,k))=ρ a
  simp only [matrixProduct_unit,productUnits,← Finset.smul_sum,← Finset.mul_sum,
    matrixUnit_sum_diag,mul_one]
  exact (matrix_representation ρ a).symm

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

theorem matrix_extension_canonical (M : VonNeumannAlgebra H) (τ : FaithfulNormalTrace M)
    (hF : IsFactor M) (hD : IsDiffuse M) (n k : ℕ) (hn : 0<n) (hk : 0<k)
    (β : Matrix (Fin n) (Fin n) ℂ →⋆ₐ[ℂ] ConcreteAlgebra M) :
    ∃ ρ : Matrix (Fin (n*k)) (Fin (n*k)) ℂ →⋆ₐ[ℂ] ConcreteAlgebra M,
      Function.Injective ρ ∧
      (∀ a, ρ (matrixReindex finProdFinEquiv (Matrix.kronecker a (1 : Matrix (Fin k) (Fin k) ℂ)))=β a) ∧
      (∀ x, τ.map (ρ x)=((n*k:ℕ):ℂ)⁻¹*Matrix.trace x) := by
  classical
  have : NeZero n := ⟨Nat.ne_of_gt hn⟩
  have : NeZero k := ⟨Nat.ne_of_gt hk⟩
  obtain ⟨σ,hσ,_⟩ := exists_matrix_subfactor (matrixCommutant β) (commutantTrace β τ)
    (matrixCommutant_factor β hF) (matrixCommutant_diffuse β hD) k hk
  let σ' := (commutantInclusion β).comp σ
  have hc (i j : Fin n) (a b : Fin k) : Commute (matrixUnit β i j) (matrixUnit σ' a b) :=
    (inclusion_commute β (matrixUnit σ a b) i j).symm
  let P := matrixProduct β σ' hc
  let e : Fin n × Fin k ≃ Fin (n*k) := finProdFinEquiv
  let Q := P.comp (matrixReindex e.symm)
  have : Nontrivial (ConcreteAlgebra M) := ⟨⟨0,1,fun h => by
    have ht := congrArg τ.map h
    rw [map_zero,τ.normalized] at ht
    exact zero_ne_one ht⟩⟩
  have : NeZero (n*k) := ⟨Nat.ne_of_gt (Nat.mul_pos hn hk)⟩
  refine ⟨Q,Q.toAlgHom.toRingHom.injective,?_,?_⟩
  · intro a
    change P (matrixReindex e.symm (matrixReindex e (Matrix.kronecker a 1)))=β a
    have he (b : Matrix (Fin n × Fin k) (Fin n × Fin k) ℂ) :
        matrixReindex e.symm (matrixReindex e b)=b := by
      ext i j
      change b (e.symm (e i)) (e.symm (e j))=b i j
      simp only [Equiv.symm_apply_apply]
    rw [he]
    exact matrixProduct_kronecker_one β σ' hc a
  · intro x
    simpa only [Fintype.card_fin] using matrixHom_trace Q τ.map τ.normalized τ.tracial x

end

end KKVariableUltra
end

end OAI