theorem extracted_formal_statement_118 {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) :
  let g := (t * μ) • fun x => ∑ i, ‖x i‖;
  g = t • h →
    0 < t * μ →
      ∀ i ∈ Finset.univ,
        let abs_subg := SubderivAt_abs (xm i);
        xm i = 0 → x i = 0 ∨ |x i| ≤ t * μ → |x i| ≤ t * μ → 0 ≤ t⁻¹ → 0 ≤ t := sorry

State before:
case a0.hb
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
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 : SubderivAt abs (xm i) = if xm i = 0 then Set.Icc (-1) 1 else {sign (xm i)} := SubderivAt_abs (xm i)
hxm : xm i = 0
minpoint : x i = 0 ∨ |x i| ≤ t * μ
aux : |x i| ≤ t * μ
⊢ 0 ≤ t⁻¹

Tactic: simp

State after:
[case a0.hb
 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
 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 : SubderivAt abs (xm i) = if xm i = 0 then Set.Icc (-1) 1 else {sign (xm i)} := SubderivAt_abs (xm i)
 hxm : xm i = 0
 minpoint : x i = 0 ∨ |x i| ≤ t * μ
 aux : |x i| ≤ t * μ
 ⊢ 0 ≤ t]

Relevant lemmas: