import OAI.Combinatorics.MatchingPSD.SequentialTests

namespace OAI

universe u_1

namespace PerfectMatchingPSD
open Matrix
open scoped MatrixOrder
variable {R : Type u_1} [Fintype R] [DecidableEq R]

lemma two_div_success {μ ε : ℝ} (hμ : 2 ≤ μ) (hε : 0 ≤ ε) (hub : μ ≤ 2+ε) :
    1-ε ≤ 2/μ := by
  rw [le_div_iff₀ (by linarith : 0 < μ)]
  have hh : ε ≤ ε*μ := by nlinarith
  nlinarith

lemma normalized_gauge_pair (G Q T : Matrix R R ℝ) (hu : IsUnit G) :
    (G*Q*G*(G⁻¹*T*G⁻¹)).trace = (Q*T).trace := by
  have hunit := (Matrix.isUnit_iff_isUnit_det G).mp hu
  have hl : G⁻¹*G=1 := Matrix.nonsing_inv_mul G hunit
  have hr : G*G⁻¹=1 := Matrix.mul_nonsing_inv G hunit
  have he : G*Q*G*(G⁻¹*T*G⁻¹) = G*(Q*T)*G⁻¹ := by
    calc
      _ = G*Q*(G*G⁻¹)*T*G⁻¹ := by simp only [mul_assoc]
      _ = _ := by rw [hr, mul_one]; simp only [mul_assoc]
  rw [he, Matrix.trace_mul_cycle, hl, one_mul]

lemma inverse_sqrt_congruence (D : Matrix R R ℝ) (hD : D.PosDef) :
    (CFC.sqrt D)⁻¹ * D * (CFC.sqrt D)⁻¹ = 1 := by
  let G := CFC.sqrt D
  have hGG : G*G=D := CFC.sqrt_mul_sqrt_self D hD.posSemidef.nonneg
  have hu : IsUnit G := (IsUnit.mul_iff.mp (hGG ▸ hD.isUnit)).1
  have hl : G⁻¹*G=1 := Matrix.nonsing_inv_mul G ((Matrix.isUnit_iff_isUnit_det G).mp hu)
  have hr : G*G⁻¹=1 := Matrix.mul_nonsing_inv G ((Matrix.isUnit_iff_isUnit_det G).mp hu)
  change G⁻¹*D*G⁻¹=1
  rw [← hGG, ← mul_assoc, hl, one_mul, hr]

lemma states_of_domination {q : ℕ}
    (Q : (Fin q → Bool) → Matrix R R ℝ) (T : Fin q → Bool → Matrix R R ℝ)
    (D : Matrix R R ℝ) (hD : D.PosDef)
    (hQ : ∀ y, (Q y).PosSemidef) (hT : ∀ i b, (T i b).PosSemidef)
    (hdom : ∀ i, T i false + T i true ≤ D)
    (ε : ℝ) (hε : 0 ≤ ε)
    (htr : ∀ y, 2 ≤ (Q y*D).trace ∧ (Q y*D).trace ≤ 2+ε)
    (hbit : ∀ y i, (Q y * T i (y i)).trace = 2) :
    ∃ (ρ : (Fin q → Bool) → Matrix R R ℝ) (F : Fin q → Bool → Matrix R R ℝ),
      (∀ y, (ρ y).PosSemidef) ∧ (∀ y, (ρ y).trace = 1) ∧
      (∀ i b, 0 ≤ F i b) ∧ (∀ i, F i false + F i true ≤ 1) ∧
      ∀ y i, 1-ε ≤ (ρ y*F i (y i)).trace := by
  let G := CFC.sqrt D
  have hG : G.PosSemidef := (CFC.sqrt_nonneg D).posSemidef
  have hGG : G*G=D := CFC.sqrt_mul_sqrt_self D hD.posSemidef.nonneg
  have hu : IsUnit G := (IsUnit.mul_iff.mp (hGG ▸ hD.isUnit)).1
  have hGi : G⁻¹.PosSemidef := hG.inv
  let μ := fun y => (Q y*D).trace
  have hμ (y : Fin q → Bool) : 0 < μ y := lt_of_lt_of_le (by norm_num) (htr y).1
  refine ⟨fun y => (μ y)⁻¹ • (G*Q y*G), fun i b => G⁻¹*T i b*G⁻¹,
    ?_, ?_, ?_, ?_, ?_⟩
  · intro y
    apply Matrix.PosSemidef.smul _ (le_of_lt (inv_pos.mpr (hμ y)))
    simpa only [hG.isHermitian.eq] using (hQ y).conjTranspose_mul_mul_same G
  · intro y
    rw [Matrix.trace_smul, Matrix.trace_mul_cycle, hGG, Matrix.trace_mul_comm]
    change (μ y)⁻¹ * μ y = 1
    exact inv_mul_cancel₀ (ne_of_gt (hμ y))
  · intro i b
    apply Matrix.PosSemidef.nonneg
    simpa only [hGi.isHermitian.eq] using (hT i b).conjTranspose_mul_mul_same G⁻¹
  · intro i
    have hg := congruence_mono (hdom i) G⁻¹
    have hGit : (G⁻¹)ᵀ=G⁻¹ := hGi.isHermitian.eq
    rw [hGit, inverse_sqrt_congruence D hD, mul_add, add_mul] at hg
    exact hg
  · intro y i
    rw [smul_mul_assoc, Matrix.trace_smul, normalized_gauge_pair G (Q y) (T i (y i)) hu,
      hbit y i]
    change 1-ε ≤ (μ y)⁻¹*2
    rw [mul_comm, ← div_eq_mul_inv]
    exact two_div_success (htr y).1 hε (htr y).2

end PerfectMatchingPSD

end OAI