-- Mathlib/MeasureTheory/Measure/Hausdorff.lean
theorem hausdorffMeasure_pi_real {ι : Type*} [Fintype ι] :
    (μH[Fintype.card ι] : Measure (ι → ℝ)) = volume

Proof state:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
C : ∀ᶠ (n : ℕ) in atTop, (univ.pi fun i => Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
a✝ : ℕ
hn : ∀ (i : γ a✝), diam (t a✝ i) ≤ 1 / ↑a✝
⊢ ∀ i ∈ Finset.univ, diam (t a✝ i) ^ Fintype.card ι ≤ (1 / ↑a✝) ^ Fintype.card ι

Tactic:
intros i _

Result:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
C : ∀ᶠ (n : ℕ) in atTop, (univ.pi fun i => Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
a✝ : ℕ
hn : ∀ (i : γ a✝), diam (t a✝ i) ≤ 1 / ↑a✝
i : γ a✝
x✝ : i ∈ Finset.univ
⊢ diam (t a✝ i) ^ Fintype.card ι ≤ (1 / ↑a✝) ^ Fintype.card ι

Proof state:
case h
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
n : ℕ
hn : n ≥ 1
npos : 0 < ↑n
x : ι → ℝ
hx : ∀ (i : ι), ↑(a i) < x i ∧ x i < ↑(b i)
i : ι
⊢ (x i - ↑(a i)) * ↑n < (↑(b i) - ↑(a i)) * ↑n

Tactic:
refine (mul_lt_mul_right npos).2 ?_

Result:
case h
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
n : ℕ
hn : n ≥ 1
npos : 0 < ↑n
x : ι → ℝ
hx : ∀ (i : ι), ↑(a i) < x i ∧ x i < ↑(b i)
i : ι
⊢ x i - ↑(a i) < ↑(b i) - ↑(a i)

Proof state:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
C : ∀ᶠ (n : ℕ) in atTop, (univ.pi fun i => Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
⊢ liminf (fun n => ∏ i : ι, ↑⌈(↑(b i) - ↑(a i)) * ↑n⌉₊ / ↑n) atTop = ∏ x : ι, ENNReal.ofReal (↑(b x) - ↑(a x))

Tactic:
apply Tendsto.liminf_eq

Result:
case h
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
C : ∀ᶠ (n : ℕ) in atTop, (univ.pi fun i => Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
⊢ Tendsto (fun n => ∏ i : ι, ↑⌈(↑(b i) - ↑(a i)) * ↑n⌉₊ / ↑n) atTop (𝓝 (∏ x : ι, ENNReal.ofReal (↑(b x) - ↑(a x))))

Proof state:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
C : ∀ᶠ (n : ℕ) in atTop, (univ.pi fun i => Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
i : ι
x✝ : i ∈ Finset.univ
⊢ ∀ n ≥ 1,
    (ENNReal.ofReal ∘ (fun x => ↑⌈(↑(b i) - ↑(a i)) * x⌉₊ / x) ∘ Nat.cast) n =
      (fun n => ↑⌈(↑(b i) - ↑(a i)) * ↑n⌉₊ / ↑n) n

Tactic:
intros n hn

Result:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
C : ∀ᶠ (n : ℕ) in atTop, (univ.pi fun i => Ioo ↑(a i) ↑(b i)) ⊆ ⋃ i, t n i
i : ι
x✝ : i ∈ Finset.univ
n : ℕ
hn : n ≥ 1
⊢ (ENNReal.ofReal ∘ (fun x => ↑⌈(↑(b i) - ↑(a i)) * x⌉₊ / x) ∘ Nat.cast) n =
    (fun n => ↑⌈(↑(b i) - ↑(a i)) * ↑n⌉₊ / ↑n) n

Proof state:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
Hle : volume ≤ μH[↑(Fintype.card ι)]
⊢ volume (univ.pi fun i => Ioo ↑(a i) ↑(b i)) ≤ μH[↑(Fintype.card ι)] (univ.pi fun i => Ioo ↑(a i) ↑(b i))

Tactic:
exact Measure.le_iff'.1 Hle _

Result:
no goals

Proof state:
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
n : ℕ
hn : n ≥ 1
npos : 0 < ↑n
x : ι → ℝ
hx : ∀ (i : ι), ↑(a i) < x i ∧ x i < ↑(b i)
i : ι
⊢ ⌊(x i - ↑(a i)) * ↑n⌋₊ < ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊

Tactic:
apply Nat.floor_lt_ceil_of_lt_of_pos

Result:
case h
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
n : ℕ
hn : n ≥ 1
npos : 0 < ↑n
x : ι → ℝ
hx : ∀ (i : ι), ↑(a i) < x i ∧ x i < ↑(b i)
i : ι
⊢ (x i - ↑(a i)) * ↑n < (↑(b i) - ↑(a i)) * ↑n

case h'
ι : Type u_4
inst✝ : Fintype ι
a b : ι → ℚ
H : ∀ (i : ι), a i < b i
I : ∀ (i : ι), 0 ≤ ↑(b i) - ↑(a i)
γ : ℕ → Type u_4 := fun n => (i : ι) → Fin ⌈(↑(b i) - ↑(a i)) * ↑n⌉₊
t : (n : ℕ) → γ n → Set (ι → ℝ) := fun n f => univ.pi fun i => Icc (↑(a i) + ↑↑(f i) / ↑n) (↑(a i) + (↑↑(f i) + 1) / ↑n)
A : Tendsto (fun n => 1 / ↑n) atTop (𝓝 0)
B : ∀ᶠ (n : ℕ) in atTop, ∀ (i : γ n), diam (t n i) ≤ 1 / ↑n
n : ℕ
hn : n ≥ 1
npos : 0 < ↑n
x : ι → ℝ
hx : ∀ (i : ι), ↑(a i) < x i ∧ x i < ↑(b i)
i : ι
⊢ 0 < (↑(b i) - ↑(a i)) * ↑n