-- Mathlib/Analysis/FunctionalSpaces/SobolevInequality.lean
theorem T_insert_le_T_lmarginal_singleton [∀ i, SigmaFinite (μ i)] (hp₀ : 0 ≤ p) (s : Finset ι)
    (hp : (s.card : ℝ) * p ≤ 1)
    (i : ι) (hi : i ∉ s) {f : (∀ i, A i) → ℝ≥0∞} (hf : Measurable f) :
    T μ p f (insert i s) ≤ T μ p (∫⋯∫⁻_{i}, f ∂μ) s

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ Measurable fun x => (f ^ (1 - ↑(#s) * p) * ∏ j ∈ insert i s, (∫⋯∫⁻_{j}, f ∂μ) ^ p) x

Tactic:
simp only [Pi.mul_apply, Pi.pow_apply, Finset.prod_apply]

Result:
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ Measurable fun x => f x ^ (1 - ↑(#s) * p) * ∏ x_1 ∈ insert i s, (∫⋯∫⁻_{x_1}, f ∂μ) x ^ p

Proof state:
case h.e'_4.h
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
x : (i : ι) → A i
k : ℝ := ↑(#s)
hk' : 0 ≤ 1 - k * p
X : A i → (a : ι) → A a
hF₁ : ∀ {j : ι}, Measurable fun t => (∫⋯∫⁻_{j}, f ∂μ) (X t)
hF₀ : Measurable fun t => f (X t)
t : A i
⊢ f (X t) ^ (1 - k * p) * ((∫⋯∫⁻_{i}, f ∂μ) (X t) ^ p * ∏ j ∈ s, (∫⋯∫⁻_{j}, f ∂μ) (X t) ^ p) =
    (∫⋯∫⁻_{i}, f ∂μ) (X t) ^ p * (f (X t) ^ (1 - k * p) * ∏ j ∈ s, (∫⋯∫⁻_{j}, f ∂μ) (X t) ^ p)

Tactic:
ring_nf

Result:
no goals

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ Measurable (f ^ (1 - ↑(#s) * p) * ∏ j ∈ insert i s, (∫⋯∫⁻_{j}, f ∂μ) ^ p)

Tactic:
change Measurable (fun x ↦ _)

Result:
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ Measurable fun x => (f ^ (1 - ↑(#s) * p) * ∏ j ∈ insert i s, (∫⋯∫⁻_{j}, f ∂μ) ^ p) x

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
x : (i : ι) → A i
⊢ ∫⁻ (t : A i), f (update x i t) ^ (1 - ↑(#s) * p) * ∏ j ∈ insert i s, (∫⋯∫⁻_{j}, f ∂μ) (update x i t) ^ p ∂μ i ≤
    (∫⋯∫⁻_{i}, f ∂μ) x ^ (1 - (↑(#s) - 1) * p) * ∏ x_1 ∈ s, (∫⋯∫⁻_{x_1}, ∫⋯∫⁻_{i}, f ∂μ ∂μ) x ^ p

Tactic:
set X := update x i

Result:
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
x : (i : ι) → A i
X : A i → (a : ι) → A a := update x i
⊢ ∫⁻ (t : A i), f (X t) ^ (1 - ↑(#s) * p) * ∏ j ∈ insert i s, (∫⋯∫⁻_{j}, f ∂μ) (X t) ^ p ∂μ i ≤
    (∫⋯∫⁻_{i}, f ∂μ) x ^ (1 - (↑(#s) - 1) * p) * ∏ x_1 ∈ s, (∫⋯∫⁻_{x_1}, ∫⋯∫⁻_{i}, f ∂μ ∂μ) x ^ p

Proof state:
case h.e'_6.h.e'_6
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
x : (i : ι) → A i
X : A i → (a : ι) → A a := update x i
hF₁ : ∀ {j : ι}, Measurable fun t => (∫⋯∫⁻_{j}, f ∂μ) (X t)
hF₀ : Measurable fun t => f (X t)
k : ℝ := ↑(#s)
hk' : 0 ≤ 1 - k * p
⊢ ∏ j ∈ s, (∫⁻ (t : A i), (∫⋯∫⁻_{j}, f ∂μ) (X t) ∂μ i) ^ p = ∏ j ∈ s, (∫⋯∫⁻_{i, j}, f ∂μ) x ^ p

Tactic:
refine prod_congr rfl fun j hj => ?_

Result:
case h.e'_6.h.e'_6
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
x : (i : ι) → A i
X : A i → (a : ι) → A a := update x i
hF₁ : ∀ {j : ι}, Measurable fun t => (∫⋯∫⁻_{j}, f ∂μ) (X t)
hF₀ : Measurable fun t => f (X t)
k : ℝ := ↑(#s)
hk' : 0 ≤ 1 - k * p
j : ι
hj : j ∈ s
⊢ (∫⁻ (t : A i), (∫⋯∫⁻_{j}, f ∂μ) (X t) ∂μ i) ^ p = (∫⋯∫⁻_{i, j}, f ∂μ) x ^ p

Proof state:
case h.e'_7.h.e'_5.h.e'_6.h.e'_6.h.e'_5
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ ↑(#s + 1) - 1 = ↑(#s)

Tactic:
push_cast

Result:
case h.e'_7.h.e'_5.h.e'_6.h.e'_6.h.e'_5
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ ↑(#s) + 1 - 1 = ↑(#s)

Proof state:
case bc.hq
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
x : (i : ι) → A i
X : A i → (a : ι) → A a := update x i
hF₁ : ∀ {j : ι}, Measurable fun t => (∫⋯∫⁻_{j}, f ∂μ) (X t)
hF₀ : Measurable fun t => f (X t)
k : ℝ := ↑(#s)
hk' : 0 ≤ 1 - k * p
⊢ 0 ≤ 1 - k * p

Tactic:
exact hk'

Result:
no goals

Proof state:
case bc.hg
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
x : (i : ι) → A i
X : A i → (a : ι) → A a := update x i
hF₁ : ∀ {j : ι}, Measurable fun t => (∫⋯∫⁻_{j}, f ∂μ) (X t)
hF₀ : Measurable fun t => f (X t)
k : ℝ := ↑(#s)
hk' : 0 ≤ 1 - k * p
⊢ AEMeasurable (fun a => f (X a)) (μ i)

Tactic:
exact hF₀.aemeasurable

Result:
no goals

Proof state:
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ ∫⋯∫⁻_insert i s, f ^ (1 - (↑(#s + 1) - 1) * p) * ∏ i ∈ insert i s, (∫⋯∫⁻_{i}, f ∂μ) ^ p ∂μ =
    ∫⋯∫⁻_insert i s, f ^ (1 - ↑(#s) * p) * ∏ i ∈ insert i s, (∫⋯∫⁻_{i}, f ∂μ) ^ p ∂μ

Tactic:
congr!

Result:
case h.e'_7.h.e'_5.h.e'_6.h.e'_6.h.e'_5
ι : Type u_1
A : ι → Type u_2
inst✝² : (i : ι) → MeasurableSpace (A i)
μ : (i : ι) → Measure (A i)
inst✝¹ : DecidableEq ι
p : ℝ
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
hp₀ : 0 ≤ p
s : Finset ι
hp : ↑(#s) * p ≤ 1
i : ι
hi : i ∉ s
f : ((i : ι) → A i) → ℝ≥0∞
hf : Measurable f
⊢ ↑(#s + 1) - 1 = ↑(#s)