-- Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
theorem lintegral_prod_lintegral_pow_le [Fintype ι] [∀ i, SigmaFinite (μ i)]
    {p : ℝ} (hp : Real.IsConjExponent #ι p)
    {f} (hf : Measurable f) :
    ∫⁻ x, ∏ i, (∫⁻ xᵢ, f (update x i xᵢ) ∂μ i) ^ ((1 : ℝ) / (#ι - 1 : ℝ)) ∂.pi μ
    ≤ (∫⁻ x, f x ∂.pi μ) ^ p

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤
    (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p

Tactic:
have h2 : 0 ≤ ((1 : ℝ) / (#ι - 1 : ℝ)) := by positivity

Result:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤
    (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤
    (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p

Tactic:
have h4 : p = 1 + 1 / (↑#ι - 1) := by field_simp; rw [mul_comm, hp.sub_one_mul_conj]

Result:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
h4 : p = 1 + 1 / (↑#ι - 1)
⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤
    (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
⊢ (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1

Tactic:
field_simp

Result:
no goals

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
⊢ 0 ≤ 1 / (↑#ι - 1)

Tactic:
positivity

Result:
no goals

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
⊢ p * (↑#ι - 1) = ↑#ι

Tactic:
rw [mul_comm, hp.sub_one_mul_conj]

Result:
no goals

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
⊢ 1 < #ι

Tactic:
exact_mod_cast hp.one_lt

Result:
no goals

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
⊢ 0 < ↑#ι - 1

Tactic:
linarith

Result:
no goals

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
h4 : p = 1 + 1 / (↑#ι - 1)
⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤
    (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ (1 + 1 / (↑#ι - 1))

Tactic:
convert lintegral_mul_prod_lintegral_pow_le μ h2 h3 hf using 2

Result:
case h.e'_3.h.e'_4
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
h0 : 1 < ↑#ι
h1 : 0 < ↑#ι - 1
h2 : 0 ≤ 1 / (↑#ι - 1)
h3 : (↑#ι - 1) * (1 / (↑#ι - 1)) ≤ 1
h4 : p = 1 + 1 / (↑#ι - 1)
⊢ (fun x => ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1))) = fun x =>
    f x ^ (1 - (↑#ι - 1) * (1 / (↑#ι - 1))) * ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1))

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
⊢ 1 < ↑#ι

Tactic:
norm_cast

Result:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
⊢ 1 < #ι

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤
    (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p

Tactic:
have : Nontrivial ι :=
  Fintype.one_lt_card_iff_nontrivial.mp (by exact_mod_cast hp.one_lt)

Result:
ι : Type u_1
A : ι → Type u_2
inst✝³ : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝² : DecidableEq ι
inst✝¹ : Fintype ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
p : ℝ
hp : (↑#ι).IsConjExponent p
f : ((a : ι) → A a) → ℝ≥0∞
hf : Measurable f
this : Nontrivial ι
⊢ ∫⁻ (x : (i : ι) → A i), ∏ i : ι, (∫⁻ (xᵢ : A i), f (update x i xᵢ) ∂μ i) ^ (1 / (↑#ι - 1)) ∂Measure.pi μ ≤
    (∫⁻ (x : (i : ι) → A i), f x ∂Measure.pi μ) ^ p