theorem extracted_formal_statement_30 {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 →
      ∀ (y : EuclideanSpace ℝ (Fin ↑n)),
        ∀ i ∈ Finset.univ,
          ¬xm i = 0 →
            SubderivAt abs (xm i) = {sign (xm i)} →
              let sgnxm := sign (xm i);
              |y i| ≥ |xm i| + inner sgnxm (y i - xm i) →
                x i ≠ 0 →
                  t * μ < |x i| →
                    (|x i| - t * μ) ⊔ 0 = |x i| - t * μ →
                      ¬0 < x i → x i < 0 → (sign (x i) * (|x i| - t * μ)).sign = -1 → 0 < |x i| - t * μ := 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
y : EuclideanSpace ℝ (Fin ↑n)
i : Fin ↑n
a✝ : i ∈ Finset.univ
left✝ : ¬xm i = 0
abs_subg : SubderivAt abs (xm i) = {sign (xm i)}
sgnxm : ℝ := sign (xm i)
aux : |y i| ≥ |xm i| + inner sgnxm (y i - xm i)
xiieq0 : x i ≠ 0
ieq : t * μ < |x i|
eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
hx : ¬0 < x i
xneg : x i < 0
⊢ (sign (x i) * (|x i| - t * μ)).sign = -1

Tactic: apply Real.sign_of_neg
    apply mul_neg_of_neg_of_pos
    calc
  sign (x i) = -1 := by apply Real.sign_of_neg xneg
  _ < 0 := by linarith

State after:
[case hr.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
 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
 left✝ : ¬xm i = 0
 abs_subg : SubderivAt abs (xm i) = {sign (xm i)}
 sgnxm : ℝ := sign (xm i)
 aux : |y i| ≥ |xm i| + inner sgnxm (y i - xm i)
 xiieq0 : x i ≠ 0
 ieq : t * μ < |x i|
 eq1 : (|x i| - t * μ) ⊔ 0 = |x i| - t * μ
 hx : ¬0 < x i
 xneg : x i < 0
 ⊢ 0 < |x i| - t * μ]

Relevant lemmas:
Real.sign_of_neg : ∀ {r : ℝ}, r < 0 → r.sign = -1
mul_neg_of_neg_of_pos : ∀ {α : Type u_3} {a b : α} [inst : MulZeroClass α] [inst_1 : Preorder α]
  [inst_2 : MulPosStrictMono α], a < 0 → 0 < b → a * b < 0