theorem extracted_formal_statement_7.{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]
  (h_original_goal :
    Tendsto
      (fun n =>
        (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑(φ₁ n)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n)) - OptProblem.b E₁ E₂)
      atTop (𝓝 0)) :
  StrictMono fun n => φ (n + 1) := sorry

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
⊢ Tendsto
    (fun n =>
      (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑(φ₁ n)) + (OptProblem.A₂ E₁) (ADMM.x₂ E₁ F ↑(φ₁ n)) - OptProblem.b E₁ E₂)
    atTop (𝓝 0)

Tactic: apply Filter.tendsto_iff_seq_tendsto.1 Seq_converge_zero₂
    apply StrictMono.tendsto_atTop
    have : (fun n => (Int.toNat (φ₁ n))) = (fun n => (φ (n + 1))) := by ext n; rw [φ₁_equ fullrank₁ fullrank₂ n]; simp
    simp at this
    rw [this]

State after:
[case a.h
 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 => ↑(φ₁ n)) = fun n => φ (n + 1)
 ⊢ StrictMono fun n => φ (n + 1)]

Relevant lemmas:
Filter.tendsto_iff_seq_tendsto : ∀ {α : Type u_1} {β : Type u_2} {f : α → β} {k : Filter α} {l : Filter β}
  [inst : k.IsCountablyGenerated], Tendsto f k l ↔ ∀ (x : ℕ → α), Tendsto x atTop k → Tendsto (f ∘ x) atTop l
StrictMono.tendsto_atTop : ∀ {φ : ℕ → ℕ}, StrictMono φ → Tendsto φ atTop atTop