-- File: optlib/Optlib/Algorithm/ADMM/Theroem_converge.lean
Theorem statement:
theorem extracted_formal_statement_450.{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)] :
  Tendsto (fun n => ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * Φ n) atTop (𝓝 (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * 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)
⊢ Tendsto (fun n => ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * Φ n) atTop (𝓝 (ADMM.τ E₁ E₂ F * ADMM.ρ E₁ E₂ F * 0))

Tactic: apply Filter.Tendsto.const_mul (τ * ρ) (Φ_converge_zero fullrank₁ fullrank₂)

Tactic state after:

Premises used:
Filter.Tendsto.const_mul : ∀ {α : Type u_2} {M : Type u_3} [inst : TopologicalSpace M] [inst_10 : Mul M]
  [inst_11 : ContinuousMul M] (b : M) {c : M} {f : α → M} {l : Filter α},
  Tendsto (fun k => f k) l (𝓝 c) → Tendsto (fun k => b * f k) l (𝓝 (b * c))