import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_9.{u_3, u_2, u_1} {E₁ : Type u_1} {E₂ : Type u_2} {F : Type u_3}
  [inst : NormedAddCommGroup E₁] [inst_1 : InnerProductSpace ℝ E₁] [inst_2 : FiniteDimensional ℝ E₁]
  [inst_3 : NormedAddCommGroup E₂] [inst_4 : InnerProductSpace ℝ E₂] [inst_5 : FiniteDimensional ℝ E₂]
  [inst_6 : NormedAddCommGroup F] [inst_7 : InnerProductSpace ℝ F] [inst_8 : FiniteDimensional ℝ F]
  {admm : ADMM E₁ E₂ F} {admm_kkt : Existance_of_kkt admm} (fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂))
  (fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)) [inst_9 : Setting E₁ E₂ F admm admm_kkt]
  (this :
    (fun n => u (φ₁ n)) = fun n =>
      -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
          (ADMM.y E₁ E₂ ↑(φ₁ n) +
              ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
                ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))) +
            ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n))))
  (h_original_goal : Tendsto (fun n => u (φ₁ n)) atTop (𝓝 (-(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) y''))) :
  Tendsto
    (fun n =>
      ADMM.y E₁ E₂ ↑(φ₁ n) +
          ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
            ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))) +
        ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n)))
    atTop (𝓝 y'') := sorry

Tactic state before:
E₁ : Type u_1
E₂ : Type u_2
F : Type u_3
inst✝⁹ : NormedAddCommGroup E₁
inst✝⁸ : InnerProductSpace ℝ E₁
inst✝⁷ : FiniteDimensional ℝ E₁
inst✝⁶ : NormedAddCommGroup E₂
inst✝⁵ : InnerProductSpace ℝ E₂
inst✝⁴ : FiniteDimensional ℝ E₂
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace ℝ F
inst✝¹ : FiniteDimensional ℝ F
admm : ADMM E₁ E₂ F
admm_kkt : Existance_of_kkt admm
fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
inst✝ : Setting E₁ E₂ F admm admm_kkt
this :
  (fun n => u (φ₁ n)) = fun n =>
    -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
        (ADMM.y E₁ E₂ ↑(φ₁ n) +
            ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
              ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))) +
          ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n)))
⊢ Tendsto (fun n => u (φ₁ n)) atTop (𝓝 (-(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) y''))

Tactic:
rw [this]
    have : Tendsto (fun n => (y (φ₁ n)) + ((1 - τ) * ρ) • (A₁ (e₁ (φ₁ n)) + A₂ (e₂ (φ₁ n)))) atTop (𝓝 y'') :=
  by
  rw [← add_zero y'']
  apply Filter.Tendsto.add (y_subseq_converge' fullrank₁ fullrank₂) (const_smul_subseq₁ fullrank₁ fullrank₂)
    have h :
  Tendsto
    (fun n => (y (φ₁ n)) + ((1 - τ) * ρ) • (A₁ (e₁ (φ₁ n)) + A₂ (e₂ (φ₁ n))) + ρ • (A₂ (x₂ ((φ₁ n) - 1) - x₂ (φ₁ n))))
    atTop (𝓝 y'') :=
  by
  rw [← add_zero y'']
  apply Filter.Tendsto.add (this) (const_smul_subseq₂ fullrank₁ fullrank₂)
    have : Tendsto (-A₁†) (𝓝 y'') (𝓝 (-A₁† y'')) := by apply Continuous.tendsto; apply ContinuousLinearMap.continuous
    apply Filter.tendsto_iff_seq_tendsto.1 this

Tactic state after:
[case a
 E₁ : Type u_1
 E₂ : Type u_2
 F : Type u_3
 inst✝⁹ : NormedAddCommGroup E₁
 inst✝⁸ : InnerProductSpace ℝ E₁
 inst✝⁷ : FiniteDimensional ℝ E₁
 inst✝⁶ : NormedAddCommGroup E₂
 inst✝⁵ : InnerProductSpace ℝ E₂
 inst✝⁴ : FiniteDimensional ℝ E₂
 inst✝³ : NormedAddCommGroup F
 inst✝² : InnerProductSpace ℝ F
 inst✝¹ : FiniteDimensional ℝ F
 admm : ADMM E₁ E₂ F
 admm_kkt : Existance_of_kkt admm
 fullrank₁ : Function.Injective ⇑(OptProblem.A₁ E₂)
 fullrank₂ : Function.Injective ⇑(OptProblem.A₂ E₁)
 inst✝ : Setting E₁ E₂ F admm admm_kkt
 this✝¹ :
   (fun n => u (φ₁ n)) = fun n =>
     -(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))
         (ADMM.y E₁ E₂ ↑(φ₁ n) +
             ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
               ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))) +
           ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n)))
 this✝ :
   Tendsto
     (fun n =>
       ADMM.y E₁ E₂ ↑(φ₁ n) +
         ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) • ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))))
     atTop (𝓝 y'')
 h :
   Tendsto
     (fun n =>
       ADMM.y E₁ E₂ ↑(φ₁ n) +
           ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
             ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))) +
         ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n)))
     atTop (𝓝 y'')
 this :
   Tendsto (⇑(-ContinuousLinearMap.adjoint (OptProblem.A₁ E₂))) (𝓝 y'')
     (𝓝 (-(ContinuousLinearMap.adjoint (OptProblem.A₁ E₂)) y''))
 ⊢ Tendsto
     (fun n =>
       ADMM.y E₁ E₂ ↑(φ₁ n) +
           ((1 - ADMM.τ E₁ E₂ F) * ADMM.ρ E₁ E₂ F) •
             ((OptProblem.A₁ E₂) (e₁ ↑(φ₁ n)) + (OptProblem.A₂ E₁) (e₂ ↑(φ₁ n))) +
         ADMM.ρ E₁ E₂ F • (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F (↑(φ₁ n) - 1) - ADMM.x₂ E₁ F ↑(φ₁ n)))
     atTop (𝓝 y'')]