theorem extracted_formal_statement_5 (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) →
        (∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)) →
          g t_min = -2 * ε ^ 2 / ((b - a) ^ 2 * ↑m) →
            (-(4 * ε * ε * (((b - a) ^ 2 * ↑m) ^ 2 * 8)) + (4 * ε) ^ 2 * ↑m * (b - a) ^ 2 * ((b - a) ^ 2 * ↑m)) *
                ((b - a) ^ 2 * ↑m) =
              -(2 * ε ^ 2 * ((b - a) ^ 2 * ↑m * (((b - a) ^ 2 * ↑m) ^ 2 * 8))) := 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)
h_inf : ∀ (t : { t // t > 0 }), g t ≥ g t_min
h_exp : ∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)
⊢ g t_min = -2 * ε ^ 2 / ((b - a) ^ 2 * ↑m)

Tactic: simp only [g, t_min]
    have h : m ≠ 0 ∧ (b - a) ≠ 0 := by
  constructor
  · exact ne_of_gt hm
  · exact ne_of_gt (sub_pos.mpr hab)
    field_simp [h.left, h.right]

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)
 h_inf : ∀ (t : { t // t > 0 }), g t ≥ g t_min
 h_exp : ∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)
 h : m ≠ 0 ∧ b - a ≠ 0
 ⊢ (-(4 * ε * ε * (((b - a) ^ 2 * ↑m) ^ 2 * 8)) + (4 * ε) ^ 2 * ↑m * (b - a) ^ 2 * ((b - a) ^ 2 * ↑m)) *
       ((b - a) ^ 2 * ↑m) =
     -(2 * ε ^ 2 * ((b - a) ^ 2 * ↑m * (((b - a) ^ 2 * ↑m) ^ 2 * 8)))]

Relevant lemmas:
error