import Mathlib
import Optlib


open Set InnerProductSpace Topology Filter Bornology Metric Real

theorem extracted_formal_statement_426.{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_kkt₁ fullrank₁ fullrank₂)]
  (h :
    ∀ (n : ℕ),
      ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤
        ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖)
  (h' :
    ∀ (n : ℕ),
      ‖‖(OptProblem.A₁ E₂) (e₁ n)‖‖ ≤
        ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖)
  (h'' :
    Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖) atTop
      (𝓝 (0 + 0)))
  (h_original_goal :
    Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖) atTop
      (𝓝 0)) :
  Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖) atTop
    (𝓝 (0 + 0)) := 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_kkt₁ fullrank₁ fullrank₂)
h :
  ∀ (n : ℕ),
    ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖
h' :
  ∀ (n : ℕ),
    ‖‖(OptProblem.A₁ E₂) (e₁ n)‖‖ ≤
      ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖
h'' :
  Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖) atTop
    (𝓝 (0 + 0))
this : 0 = 0 + 0
⊢ Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖) atTop (𝓝 0)

Tactic:
rw [this]

Tactic state after:
[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_kkt₁ fullrank₁ fullrank₂)
 h :
   ∀ (n : ℕ),
     ‖(OptProblem.A₁ E₂) (e₁ n)‖ ≤ ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖
 h' :
   ∀ (n : ℕ),
     ‖‖(OptProblem.A₁ E₂) (e₁ n)‖‖ ≤
       ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖
 h'' :
   Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖) atTop
     (𝓝 (0 + 0))
 this : 0 = 0 + 0
 ⊢ Tendsto (fun n => ‖(OptProblem.A₁ E₂) (e₁ n) + (OptProblem.A₂ E₁) (e₂ n)‖ + ‖(OptProblem.A₂ E₁) (e₂ n)‖) atTop
     (𝓝 (0 + 0))]