theorem extracted_formal_statement_97 {n : ℕ+} {t μ : ℝ} {h : EuclideanSpace ℝ (Fin ↑n) → ℝ}
  (lasso : h = fun y => μ • ∑ i, ‖y i‖) (x xm : EuclideanSpace ℝ (Fin ↑n)) (tpos : 0 < t) (μpos : 0 < μ)
  (minpoint : ∀ (i : Fin ↑n), xm i = sign (x i) * ((|x i| - t * μ) ⊔ 0)) :
  let g := (t * μ) • fun x => ∑ i, ‖x i‖;
  g = t • h →
    0 < t * μ →
      (1 / (t * μ)) • (x - xm) ∈ SubderivAt (fun x => ∑ i, ‖x i‖) xm →
        ∀ (y : EuclideanSpace ℝ (Fin ↑n)), ∑ i, ‖y i‖ ≥ ∑ i, ‖xm i‖ + inner ((1 / (t * μ)) • (x - xm)) (y - xm) := sorry

State before:
n : ℕ+
t μ : ℝ
h : EuclideanSpace ℝ (Fin ↑n) → ℝ
lasso : h = fun y => μ • ∑ i, ‖y i‖
x xm : EuclideanSpace ℝ (Fin ↑n)
tpos : 0 < t
μpos : 0 < μ
minpoint : ∀ (i : Fin ↑n), xm i = sign (x i) * ((|x i| - t * μ) ⊔ 0)
g : EuclideanSpace ℝ (Fin ↑n) → ℝ := (t * μ) • fun x => ∑ i, ‖x i‖
geqth : g = t • h
tμpos : 0 < t * μ
⊢ (1 / (t * μ)) • (x - xm) ∈ SubderivAt (fun x => ∑ i, ‖x i‖) xm

Tactic: rw [← mem_SubderivAt, HasSubgradientAt]

State after:
[n : ℕ+
 t μ : ℝ
 h : EuclideanSpace ℝ (Fin ↑n) → ℝ
 lasso : h = fun y => μ • ∑ i, ‖y i‖
 x xm : EuclideanSpace ℝ (Fin ↑n)
 tpos : 0 < t
 μpos : 0 < μ
 minpoint : ∀ (i : Fin ↑n), xm i = sign (x i) * ((|x i| - t * μ) ⊔ 0)
 g : EuclideanSpace ℝ (Fin ↑n) → ℝ := (t * μ) • fun x => ∑ i, ‖x i‖
 geqth : g = t • h
 tμpos : 0 < t * μ
 ⊢ ∀ (y : EuclideanSpace ℝ (Fin ↑n)), ∑ i, ‖y i‖ ≥ ∑ i, ‖xm i‖ + inner ((1 / (t * μ)) • (x - xm)) (y - xm)]

Relevant lemmas:
mem_SubderivAt : ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : InnerProductSpace ℝ E] {f : E → ℝ} {g x : E},
  HasSubgradientAt f g x ↔ g ∈ SubderivAt f x