theorem extracted_formal_statement_7 (g : { t // t > 0 } → ℝ) (t_min : { t // t > 0 })
  (h_exp : ∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)) : ⨅ t, rexp (g t) = rexp (g t_min) := 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✝)
h_g_tmin : g✝ t_min✝ = -2 * ε ^ 2 / ((b - a) ^ 2 * ↑m)
h_exp_tmin : rexp (g✝ t_min✝) = rexp (-2 * ε ^ 2 / ((b - a) ^ 2 * ↑m))
h_inf_rewrite : ⨅ t, rexp (-↑t * ε + ↑t ^ 2 * ↑m * (b - a) ^ 2 / 8) = ⨅ t, rexp (g✝ t)
g : { t // t > 0 } → ℝ
t_min : { t // t > 0 }
h_exp : ∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)
⊢ ⨅ t, rexp (g t) = rexp (g t_min)

Tactic: apply le_antisymm
    ·
  have bdd : BddBelow (Set.range (fun t => rexp (g t))) :=
    by
    use rexp (g t_min)
    intro y hy
    rcases hy with ⟨t, rfl⟩
    exact h_exp t
  apply ciInf_le bdd
    ·
  have positive_reals_nonempty : Nonempty { t : ℝ // t > 0 } := by exact ⟨⟨1, by norm_num⟩⟩
  to_theorem apply le_ciInf intro t exact (h_exp t)

State after:

Relevant lemmas:
error