theorem extracted_formal_statement_3 (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 }),
        g t ≥ g t_min →
          ↑m * (b - a) ^ 2 / 8 * (↑t - ↑t_min) ^ 2 - 2 * ε ^ 2 / ((b - a) ^ 2 * ↑m) ≥
            ↑m * (b - a) ^ 2 / 8 * 0 ^ 2 - 2 * ε ^ 2 / ((b - a) ^ 2 * ↑m) := sorry

State before:
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 }
⊢ g t ≥ g t_min

Tactic: rw [h_rewrite t, h_rewrite t_min]
    simp only [sub_self, mul_zero]
    have h_term_nonneg : ↑m * (b - a) ^ 2 / 8 * (t - t_min) ^ 2 ≥ 0 := by
  to_theorem
    apply mul_nonneg·
      apply div_nonneg
      · apply mul_nonneg
        · exact Nat.cast_nonneg m
        · exact sq_nonneg (b - a)
      · norm_num· exact sq_nonneg (t.val - t_min)

State after:
[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 }
 h_term_nonneg : ↑m * (b - a) ^ 2 / 8 * (↑t - ↑t_min) ^ 2 ≥ 0
 ⊢ ↑m * (b - a) ^ 2 / 8 * (↑t - ↑t_min) ^ 2 - 2 * ε ^ 2 / ((b - a) ^ 2 * ↑m) ≥
     ↑m * (b - a) ^ 2 / 8 * 0 ^ 2 - 2 * ε ^ 2 / ((b - a) ^ 2 * ↑m)]

Relevant lemmas:
error