theorem extracted_formal_statement_25 (ε a b : ℝ) (m : ℕ) (hm : m > 0) (hε : ε > 0) (hab : b > a) :
  let g := fun t => -↑t * ε + ↑t ^ 2 * ↑m * (b - a) ^ 2 / 8;
  let t_min :=
    ⟨4 * ε / ((b - a) ^ 2 * ↑m),
      div_pos
        (mul_pos
          (Mathlib.Meta.NormNum.isNat_lt_true (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_zero)
            (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 4)) (Eq.refl false))
          hε)
        (mul_pos (sq_pos_of_ne_zero (ne_of_gt (sub_pos.mpr hab))) (Nat.cast_pos.mpr hm))⟩;
  (∀ (t : { t // t > 0 }), g t = -↑t * ε + ↑t ^ 2 * ↑m * (b - a) ^ 2 / 8) →
    (∀ (t : { t // t > 0 }), g t = ↑m * (b - a) ^ 2 / 8 * (↑t - ↑t_min) ^ 2 - 2 * ε ^ 2 / ((b - a) ^ 2 * ↑m)) →
      ∀ (t : { t // t > 0 }), 0 ≤ (↑t - ↑t_min) ^ 2 := sorry

State before:
case hb
ε a b : ℝ
m : ℕ
hm : m > 0
hε : ε > 0
hab : b > a
g : { t // t > 0 } → ℝ := fun t => -↑t * ε + ↑t ^ 2 * ↑m * (b - a) ^ 2 / 8
t_min : { t // t > 0 } := ⟨4 * ε / ((b - a) ^ 2 * ↑m), ⋯⟩
gt : ∀ (t : { t // t > 0 }), g t = -↑t * ε + ↑t ^ 2 * ↑m * (b - a) ^ 2 / 8
h_rewrite : ∀ (t : { t // t > 0 }), g t = ↑m * (b - a) ^ 2 / 8 * (↑t - ↑t_min) ^ 2 - 2 * ε ^ 2 / ((b - a) ^ 2 * ↑m)
t : { t // t > 0 }
⊢ 0 ≤ (↑t - ↑t_min) ^ 2

Tactic: exact sq_nonneg (t.val - t_min)

State after:

Relevant lemmas:
sq_nonneg : ∀ {α : Type u} [inst : Semiring α] [inst_1 : LinearOrder α] [inst_2 : IsRightCancelAdd α]
  [inst_3 : ZeroLEOneClass α] [inst_4 : ExistsAddOfLE α] [inst_5 : PosMulMono α] [inst_6 : AddLeftStrictMono α] (a : α),
  0 ≤ a ^ 2