theorem extracted_formal_statement_145 {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 →
      ∀ i ∈ Finset.univ,
        xm i = 0 ∧ SubderivAt abs (xm i) = Set.Icc (-1) 1 ∨ ¬xm i = 0 ∧ SubderivAt abs (xm i) = {sign (xm i)} →
          ¬xm i = 0 → SubderivAt abs (xm i) = Set.Icc (-1) 1 → xm i ≠ 0 := sorry

State before:
case neg
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 : Fin ↑n
a✝ : i ∈ Finset.univ
abs_subg : xm i = 0 ∧ SubderivAt abs (xm i) = Set.Icc (-1) 1 ∨ ¬xm i = 0 ∧ SubderivAt abs (xm i) = {sign (xm i)}
hxm : ¬xm i = 0
hxm' : SubderivAt abs (xm i) = Set.Icc (-1) 1
⊢ xm i ≠ 0

Tactic: exact hxm

State after:

Relevant lemmas: