theorem extracted_formal_statement_80.{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} (n : ℕ+) (x : E₂) :
  let c := ADMM.y E₁ E₂ n.natPred;
  ((fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂⟫_ℝ) =
      (fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x⟫_ℝ) + fun x =>
        ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ) →
    HasGradientAt (fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ) ((ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) c) x := 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
inst✝ : Setting E₁ E₂ F admm admm_kkt
n : ℕ+
x : E₂
c : F := ADMM.y E₁ E₂ n.natPred
c₁ : ℝ := ⟪c, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ
this :
  (fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) + (OptProblem.A₂ E₁) x - OptProblem.b E₁ E₂⟫_ℝ) =
    (fun x => ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₂ E₁) x⟫_ℝ) + fun x =>
      ⟪ADMM.y E₁ E₂ n.natPred, (OptProblem.A₁ E₂) (ADMM.x₁ E₂ F ↑n) - OptProblem.b E₁ E₂⟫_ℝ
⊢ HasGradientAt (fun x => ⟪c, (OptProblem.A₂ E₁) x⟫_ℝ) ((ContinuousLinearMap.adjoint (OptProblem.A₂ E₁)) c) x

Tactic: apply inner_gradient

State after:

Relevant lemmas: