import Mathlib
import OAI.Probability.SKGap.Matrix.OpNorm

namespace OAI

section
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory Real Set
open scoped NNReal ENNReal
open Filter
open scoped Topology
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius NNReal
noncomputable section
open MeasureTheory ProbabilityTheory Real Set Filter
open MeasureTheory.Measure
open scoped ENNReal NNReal MeasureTheory Topology
open MeasureTheory
noncomputable section
noncomputable section
open MeasureTheory Set NormedSpace
open scoped Topology
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius
noncomputable section
open Set Real
open scoped Topology
noncomputable section
open Matrix Set Filter
open scoped Topology Matrix.Norms.Frobenius
noncomputable section
open Matrix NormedSpace ContinuousLinearMap
open scoped Matrix.Norms.Frobenius
noncomputable section
open Matrix
namespace SKGap.RealComplex
open scoped Matrix.Norms.Frobenius
variable {ι : Type*} [Fintype ι] [DecidableEq ι]

def liftMatrix (M : Matrix ι ι ℝ) : Matrix ι ι ℂ := M.map Complex.ofReal
abbrev clin (M : Matrix ι ι ℂ) := M.toEuclideanLin.toContinuousLinearMap
abbrev rlin (M : Matrix ι ι ℝ) := M.toEuclideanLin.toContinuousLinearMap

def reVec (x : EuclideanSpace ℂ ι) : EuclideanSpace ℝ ι :=
  WithLp.toLp 2 (fun i => (x i).re)
def imVec (x : EuclideanSpace ℂ ι) : EuclideanSpace ℝ ι :=
  WithLp.toLp 2 (fun i => (x i).im)
def liftVec (x : EuclideanSpace ℝ ι) : EuclideanSpace ℂ ι :=
  WithLp.toLp 2 (fun i => (x i : ℂ))

omit [DecidableEq ι] in
lemma vec_norm_split (x : EuclideanSpace ℂ ι) :
    ‖x‖^2 = ‖reVec x‖^2+‖imVec x‖^2 := by
  simp only [EuclideanSpace.norm_sq_eq,← Finset.sum_add_distrib]
  apply Finset.sum_congr rfl
  intro i _
  change ‖x i‖^2 = ‖(x i).re‖^2+‖(x i).im‖^2
  rw [Complex.sq_norm,Complex.normSq_apply]
  simp only [Real.norm_eq_abs]
  rw [sq_abs,sq_abs]
  ring

omit [DecidableEq ι] in
lemma norm_liftVec (x : EuclideanSpace ℝ ι) : ‖liftVec x‖ = ‖x‖ := by
  have hh : ‖liftVec x‖^2 = ‖x‖^2 := by simp [EuclideanSpace.norm_sq_eq,liftVec]
  nlinarith [norm_nonneg (liftVec x),norm_nonneg x]

lemma reVec_clin (M : Matrix ι ι ℝ) (x : EuclideanSpace ℂ ι) :
    reVec (clin (liftMatrix M) x) = rlin M (reVec x) := by
  ext i
  change (∑ j, (M i j : ℂ)*x j).re = ∑ j, M i j*(x j).re
  simp

lemma imVec_clin (M : Matrix ι ι ℝ) (x : EuclideanSpace ℂ ι) :
    imVec (clin (liftMatrix M) x) = rlin M (imVec x) := by
  ext i
  change (∑ j, (M i j : ℂ)*x j).im = ∑ j, M i j*(x j).im
  simp

lemma clin_liftVec (M : Matrix ι ι ℝ) (x : EuclideanSpace ℝ ι) :
    clin (liftMatrix M) (liftVec x) = liftVec (rlin M x) := by
  ext i
  change (∑ j, (M i j : ℂ)*(x j : ℂ)) = ((∑ j, M i j*x j : ℝ) : ℂ)
  simp

lemma operator_norm_lift (M : Matrix ι ι ℝ) : ‖clin (liftMatrix M)‖ = opNorm M := by
  apply le_antisymm
  · apply ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _)
    intro x
    apply nonneg_le_nonneg_of_sq_le_sq (mul_nonneg (norm_nonneg _) (norm_nonneg _))
    simp only [← sq]
    change ‖clin (liftMatrix M) x‖^2 ≤ (opNorm M*‖x‖)^2
    rw [vec_norm_split,reVec_clin,imVec_clin,mul_pow,vec_norm_split,mul_add]
    have hr := (rlin M).le_opNorm (reVec x)
    have hi := (rlin M).le_opNorm (imVec x)
    change ‖rlin M (reVec x)‖ ≤ opNorm M*‖reVec x‖ at hr
    change ‖rlin M (imVec x)‖ ≤ opNorm M*‖imVec x‖ at hi
    nlinarith [norm_nonneg (rlin M (reVec x)),norm_nonneg (rlin M (imVec x)),
      mul_nonneg (norm_nonneg (rlin M)) (norm_nonneg (reVec x)),
      mul_nonneg (norm_nonneg (rlin M)) (norm_nonneg (imVec x))]
  · apply ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _)
    intro x
    have hh := (clin (liftMatrix M)).le_opNorm (liftVec x)
    rwa [clin_liftVec,norm_liftVec,norm_liftVec] at hh

lemma norm_liftMatrix (M : Matrix ι ι ℝ) : ‖liftMatrix M‖ = ‖M‖ :=
  Matrix.frobenius_norm_map_eq M Complex.ofReal (fun x => by simp)

omit [Fintype ι] [DecidableEq ι] in
lemma liftMatrix_sub (M N : Matrix ι ι ℝ) :
    liftMatrix (M-N) = liftMatrix M-liftMatrix N := by ext i j; simp [liftMatrix]
omit [Fintype ι] [DecidableEq ι] in
lemma liftMatrix_add (M N : Matrix ι ι ℝ) :
    liftMatrix (M+N) = liftMatrix M+liftMatrix N := by ext i j; simp [liftMatrix]
omit [DecidableEq ι] in
lemma liftMatrix_mul (M N : Matrix ι ι ℝ) :
    liftMatrix (M*N) = liftMatrix M*liftMatrix N := by
  exact Matrix.map_mul (f := Complex.ofRealHom)
omit [Fintype ι] in
lemma liftMatrix_one : liftMatrix (1 : Matrix ι ι ℝ) = 1 :=
  Matrix.map_one Complex.ofReal (by simp) (by simp)
omit [Fintype ι] [DecidableEq ι] in
lemma liftMatrix_smul (s : ℝ) (M : Matrix ι ι ℝ) :
    liftMatrix (s • M) = s • liftMatrix M := by ext i j; simp [liftMatrix]
omit [Fintype ι] [DecidableEq ι] in
lemma liftMatrix_conjTranspose (M : Matrix ι ι ℝ) :
    (liftMatrix M)ᴴ = liftMatrix Mᵀ := by ext i j; simp [liftMatrix]
omit [Fintype ι] [DecidableEq ι] in
lemma liftMatrix_hermitian (M : Matrix ι ι ℝ) (hM : Mᵀ = M) :
    (liftMatrix M)ᴴ = liftMatrix M := by rw [liftMatrix_conjTranspose,hM]
omit [Fintype ι] [DecidableEq ι] in
lemma liftMatrix_injective : Function.Injective (liftMatrix (ι := ι)) := by
  intro M N h
  ext i j
  exact Complex.ofReal_injective (congrFun (congrFun h i) j)

end SKGap.RealComplex

end
end
end
end
end
end
end
end
end
end
end
end

end OAI