import OAI.Analysis.CStarAlgebra.PureInfiniteness.Signal

namespace OAI

noncomputable section
open Filter
open scoped Topology CStarAlgebra ComplexOrder InnerProductSpace

namespace MainGap
variable {A : Type*} [NonUnitalCStarAlgebra A]
local instance boundedDiagonalizationPartialOrder : PartialOrder A := CStarAlgebra.spectralOrder A
local instance boundedDiagonalizationStarOrderedRing : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A

theorem bounded_diagonalization (hP : PropertyP A) {a b : A} (ha : 0 ≤ a) (hb : 0 ≤ b)
    (han : ‖a‖ ≤ 1) (hbn : ‖b‖ ≤ 1) (c : A) {δ ρ ζ : ℝ}
    (hδ : 0 < δ) (hδhalf : δ < 1/2) (hρ : 0 < ρ) (hρδ : ρ ≤ 2*δ) (hζ : 0 < ζ) :
    ∃ s t : A, ‖star s*a*s-a‖ < 2*δ+ρ ∧ ‖star t*b*t-b‖ < 2*δ+ρ ∧
      ‖star s*c*t‖ < ζ ∧ ‖s‖ ≤ Real.sqrt δ⁻¹ ∧ ‖t‖ ≤ Real.sqrt δ⁻¹ := by
  let M := max 1 ‖c‖
  have hM1 : 1 ≤ M := le_max_left _ _
  have hM : 0 < M := lt_of_lt_of_le zero_lt_one hM1
  have hcn : ‖M⁻¹ • c‖ ≤ 1 := by
    rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hM)]
    calc M⁻¹ * ‖c‖ ≤ M⁻¹ * M := mul_le_mul_of_nonneg_left (le_max_right _ _) (inv_nonneg.mpr hM.le)
         _ = 1 := inv_mul_cancel₀ hM.ne'
  obtain ⟨p,q,κ,hp0,hpn,hpH,hq0,hqn,hqH,hκ0,hκζ,hcomp,hps,hqs⟩ :=
    iterated_separation hP (ramp_nonneg a (by linarith : δ ≤ 2*δ))
      (norm_ramp_le_one a (by linarith : δ ≤ 2*δ))
      (ramp_nonneg b (by linarith : δ ≤ 2*δ))
      (norm_ramp_le_one b (by linarith : δ ≤ 2*δ)) hcn
      (by positivity : 0 < δ*ζ/M)
  obtain ⟨w,hw,hsn⟩ := signal_coefficient hP ha han hδ hδhalf hρ hρδ hpH hps
  obtain ⟨z,hz,htn⟩ := signal_coefficient hP hb hbn hδ hδhalf hρ hρδ hqH hqs
  refine ⟨p*w,q*z,hw,hz,?_,hsn,htn⟩
  have hc : HerCompression p q c (M*κ) := by
    intro x hx y hy
    have hh := hcomp x hx y hy
    rw [mul_smul_comm, smul_mul_assoc, norm_smul, Real.norm_eq_abs,
      abs_of_pos (inv_pos.mpr hM)] at hh
    have hh' := mul_le_mul_of_nonneg_left hh hM.le
    calc ‖x*c*y‖ = M * (M⁻¹ * ‖x*c*y‖) := by rw [← mul_assoc, mul_inv_cancel₀ hM.ne', one_mul]
         _ ≤ M*(κ*‖x‖*‖y‖) := hh'
         _ = (M*κ)*‖x‖*‖y‖ := by ring
  have hprod : ‖p*w‖*‖q*z‖ ≤ δ⁻¹ := by
    calc ‖p*w‖*‖q*z‖ ≤ Real.sqrt δ⁻¹ * Real.sqrt δ⁻¹ :=
            mul_le_mul hsn htn (norm_nonneg _) (Real.sqrt_nonneg _)
         _ = δ⁻¹ := Real.mul_self_sqrt (inv_nonneg.mpr hδ.le)
  calc
    ‖star (p*w)*c*(q*z)‖ ≤ (M*κ)*‖p*w‖*‖q*z‖ :=
      HerCompression.right_products (mul_nonneg hM.le hκ0) hp0 hq0 hc w z
    _ = (M*κ)*(‖p*w‖*‖q*z‖) := by ring
    _ ≤ (M*κ)*δ⁻¹ := mul_le_mul_of_nonneg_left hprod (mul_nonneg hM.le hκ0)
    _ < ζ := by
      rw [← div_eq_mul_inv]
      apply (div_lt_iff₀ hδ).mpr
      have hh := (lt_div_iff₀ hM).mp hκζ
      nlinarith

end MainGap

namespace MainGap
variable {A : Type*} [NonUnitalCStarAlgebra A]
local instance diagonalizationPartialOrder : PartialOrder A := CStarAlgebra.spectralOrder A
local instance diagonalizationStarOrderedRing : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A

theorem individual_to_strong (hP : PropertyP A) : UnrestrictedDiagonalization A := by
  intro a b ha hb c ε hε
  let M := max 1 (max ‖a‖ ‖b‖)
  have hM1 : 1 ≤ M := le_max_left _ _
  have hM : 0 < M := lt_of_lt_of_le zero_lt_one hM1
  have haM : ‖a‖ ≤ M := (le_max_left _ _).trans (le_max_right _ _)
  have hbM : ‖b‖ ≤ M := (le_max_right _ _).trans (le_max_right _ _)
  let δ := min (1/4 : ℝ) (ε/(8*M))
  have hδ : 0 < δ := lt_min (by norm_num) (div_pos hε (mul_pos (by norm_num) hM))
  have hδhalf : δ < 1/2 := (min_le_left _ _).trans_lt (by norm_num)
  have hsmall : M*(2*δ+δ) < ε := by
    have hh : δ ≤ ε/(8*M) := min_le_right _ _
    have hh' := (le_div_iff₀ (by positivity : 0 < 8*M)).mp hh
    nlinarith
  have hn {x : A} (hx : ‖x‖ ≤ M) : ‖M⁻¹ • x‖ ≤ 1 := by
    rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hM)]
    calc M⁻¹*‖x‖ ≤ M⁻¹*M := mul_le_mul_of_nonneg_left hx (inv_nonneg.mpr hM.le)
         _ = 1 := inv_mul_cancel₀ hM.ne'
  obtain ⟨s,t,hs,ht,hc,hsn,htn⟩ := bounded_diagonalization hP
    (smul_nonneg (inv_nonneg.mpr hM.le) ha) (smul_nonneg (inv_nonneg.mpr hM.le) hb)
    (hn haM) (hn hbM) c hδ hδhalf hδ (by linarith : δ ≤ 2*δ) hε
  have hscale (x r : A) : M • (star r*(M⁻¹ • x)*r-(M⁻¹ • x)) = star r*x*r-x := by
    simp only [mul_smul_comm, smul_mul_assoc, smul_sub, smul_smul,
      mul_inv_cancel₀ hM.ne', one_smul]
  refine ⟨s,t,?_,?_,hc⟩
  · rw [← hscale a s, norm_smul, Real.norm_eq_abs, abs_of_pos hM]
    exact (mul_lt_mul_of_pos_left hs hM).trans hsmall
  · rw [← hscale b t, norm_smul, Real.norm_eq_abs, abs_of_pos hM]
    exact (mul_lt_mul_of_pos_left ht hM).trans hsmall

theorem individual_to_strong_spi (hP : PropertyP A) : StronglyPurelyInfinite A := by
  intro x y hx hy ε hε
  exact individual_to_strong hP (x*x) (y*y)
    (by simpa only [hx.isSelfAdjoint.star_eq] using star_mul_self_nonneg x)
    (by simpa only [hy.isSelfAdjoint.star_eq] using star_mul_self_nonneg y) (x*y) ε hε

end MainGap

end

end OAI