zzhisthebest/LeanBenchmark huggingface dataset:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_33 (g : { t // t > 0 } → ℝ) (t_min : { t // t > 0 })
  (h_exp : ∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)) (h_original_goal : ⨅ t, rexp (g t) = rexp (g t_min)) :
  ⨅ t, rexp (g t) ≤ rexp (g t_min) := sorry

Tactic 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

Tactic state after:
[case a
 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),
 case a
 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)
 ⊢ rexp (g t_min) ≤ ⨅ t, rexp (g t)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_34 (g : { t // t > 0 } → ℝ) (t_min : { t // t > 0 })
  (h_exp : ∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)) (h_original_goal : ⨅ t, rexp (g t) = rexp (g t_min)) :
  rexp (g t_min) ≤ ⨅ t, rexp (g t) := sorry

Tactic 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

Tactic state after:
[case a
 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),
 case a
 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)
 ⊢ rexp (g t_min) ≤ ⨅ t, rexp (g t)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_35 (g : { t // t > 0 } → ℝ) (t_min : { t // t > 0 })
  (h_exp : ∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)) (t : { t // t > 0 }) :
  rexp (g t_min) ≤ (fun t => rexp (g t)) t := sorry

Tactic state before:
case h.intro
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 : { t // t > 0 }
⊢ rexp (g t_min) ≤ (fun t => rexp (g t)) t

Tactic:
exact h_exp t

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_36 (g : { t // t > 0 } → ℝ) (t_min : { t // t > 0 })
  (h_exp : ∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)) (bdd : BddBelow (Set.range fun t => rexp (g t))) :
  ⨅ t, rexp (g t) ≤ rexp (g t_min) := sorry

Tactic state before:
case a
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)
bdd : BddBelow (Set.range fun t => rexp (g t))
⊢ ⨅ t, rexp (g t) ≤ rexp (g t_min)

Tactic:
apply ciInf_le bdd

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_37 : Nonempty { t // t > 0 } := sorry

Tactic 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)
⊢ Nonempty { t // t > 0 }

Tactic:
exact ⟨⟨1, by norm_num⟩⟩

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_38 (g : { t // t > 0 } → ℝ) (t_min : { t // t > 0 })
  (h_exp : ∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)) (h_original_goal : rexp (g t_min) ≤ ⨅ t, rexp (g t))
  (x : { t // t > 0 }) : rexp (g t_min) ≤ rexp (g x) := sorry

Tactic state before:
case a
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)
positive_reals_nonempty : Nonempty { t // t > 0 }
⊢ rexp (g t_min) ≤ ⨅ t, rexp (g t)

Tactic:
apply le_ciInf

Tactic state after:
[case a.H
 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)
 positive_reals_nonempty : Nonempty { t // t > 0 }
 ⊢ ∀ (x : { t // t > 0 }), rexp (g t_min) ≤ rexp (g x)]
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_39 (g : { t // t > 0 } → ℝ) (t_min : { t // t > 0 })
  (h_exp : ∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)) (t : { t // t > 0 }) :
  rexp (g t_min) ≤ rexp (g t) := sorry

Tactic state before:
case a.H
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)
positive_reals_nonempty : Nonempty { t // t > 0 }
t : { t // t > 0 }
⊢ rexp (g t_min) ≤ rexp (g t)

Tactic:
exact (h_exp t)

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_40 (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) →
            rexp (g t_min) = rexp (-2 * ε ^ 2 / ((b - a) ^ 2 * ↑m)) →
              ⨅ t, rexp (-↑t * ε + ↑t ^ 2 * ↑m * (b - a) ^ 2 / 8) = ⨅ t, rexp (g t) →
                ⨅ t, rexp (g t) = rexp (g t_min) := sorry

Tactic 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)
inf_eq_at_min :
  ∀ (g : { t // t > 0 } → ℝ) (t_min : { t // t > 0 }),
    (∀ (t : { t // t > 0 }), rexp (g t) ≥ rexp (g t_min)) → ⨅ t, rexp (g t) = rexp (g t_min)
⊢ ⨅ t, rexp (g t) = rexp (g t_min)

Tactic:
exact inf_eq_at_min g t_min h_exp

Tactic state after:
import Mathlib
import FoML


open MeasureTheory ProbabilityTheory Real

theorem extracted_formal_statement_41 : Nonempty { t // t > 0 } := sorry

Tactic state before:
Ω : Type u
inst✝¹ : MeasurableSpace Ω
μ : autoParam (Measure Ω) _auto✝
inst✝ : IsProbabilityMeasure μ
a b : ℝ
m : ℕ
X : Fin m → Ω → ℝ
hXm : ∀ (i : Fin m), Measurable (X i)
hXi : iIndepFun X μ
hab : b > a
h : ∀ (i : Fin m), ∀ᵐ (ω : Ω) ∂μ, X i ω ∈ Set.Icc a b
ε : ℝ
hm : m > 0
hε : ε > 0
Y : Fin m → Ω → ℝ := fun i ω => X i ω - ∫ (x : Ω), (fun ω => X i ω) x ∂μ
hXint : ∀ (i : Fin m), Integrable (X i) μ
hY_b : ∀ (i : Fin m), ∀ᵐ (ω : Ω) ∂μ, Y i ω ∈ Set.Icc (a - ∫ (x : Ω), X i x ∂μ) (b - ∫ (x : Ω), X i x ∂μ)
hYmeas_ae : ∀ (i : Fin m), AEMeasurable (Y i) μ
hYmeas : ∀ (i : Fin m), Measurable (Y i)
h_hoeffding_bound : ∀ (t : ℝ) (i : Fin m), mgf (Y i) μ t ≤ rexp (t ^ 2 * (b - a) ^ 2 / 8)
h_ysum : {ω | ε ≤ ∑ i, Y i ω} = {ω | ε ≤ ∑ i, X i ω - ∫ (x : Ω), (fun ω => ∑ i, X i ω) x ∂μ}
h_ysum_mgf : ∀ (t : ℝ), mgf (fun ω => ∑ i, Y i ω) μ t = ∏ i, mgf (Y i) μ t
h_markov :
  ∀ t > 0,
    (μ {ω | rexp (t * ε) ≤ rexp (t * ∑ i, Y i ω)}).toReal ≤
      (∫ (x : Ω), (fun ω => rexp (t * ∑ i, Y i ω)) x ∂μ) / rexp (t * ε)
h_t_bound : ∀ t > 0, (μ {ω | ε ≤ ∑ i, Y i ω}).toReal ≤ rexp (-t * ε + t ^ 2 * ↑m * (b - a) ^ 2 / 8)
⊢ Nonempty { t // t > 0 }

Tactic:
exact ⟨⟨1, by norm_num⟩⟩

Tactic state after: