l3lab/ntp-mathlib huggingface dataset:
theorem lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add {p q : ℝ}
    (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ) (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤
      ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
        (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
  calc
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤ ∫⁻ a, (f + g) a * (f + g) a ^ (p - 1) ∂μ := by
      gcongr with a
      by_cases h_zero : (f + g) a = 0
      · rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]
        exact zero_le _
      by_cases h_top : (f + g) a = ⊤
      · rw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top]
        exact le_top
      refine' le_of_eq _
      nth_rw 2 [← ENNReal.rpow_one ((f + g) a)]
      rw [← ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right]
    _ = (∫⁻ a : α, f a * (f + g) a ^ (p - 1) ∂μ) + ∫⁻ a : α, g a * (f + g) a ^ (p - 1) ∂μ := by
      have h_add_m : AEMeasurable (fun a : α => (f + g) a ^ (p - 1 : ℝ)) μ :=
        (hf.add hg).pow_const _
      

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)
⊢ ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ =
    ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ

Next tactic:
have h_add_apply :
        (∫⁻ a : α, (f + g) a * (f + g) a ^ (p - 1) ∂μ) =
          ∫⁻ a : α, (f a + g a) * (f + g) a ^ (p - 1) ∂μ :=
        rfl
theorem lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add {p q : ℝ}
    (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ) (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤
      ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
        (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
  calc
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤ ∫⁻ a, (f + g) a * (f + g) a ^ (p - 1) ∂μ := by
      gcongr with a
      by_cases h_zero : (f + g) a = 0
      · rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]
        exact zero_le _
      by_cases h_top : (f + g) a = ⊤
      · rw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top]
        exact le_top
      refine' le_of_eq _
      nth_rw 2 [← ENNReal.rpow_one ((f + g) a)]
      rw [← ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right]
    _ = (∫⁻ a : α, f a * (f + g) a ^ (p - 1) ∂μ) + ∫⁻ a : α, g a * (f + g) a ^ (p - 1) ∂μ := by
      have h_add_m : AEMeasurable (fun a : α => (f + g) a ^ (p - 1 : ℝ)) μ :=
        (hf.add hg).pow_const _
      have h_add_apply :
        (∫⁻ a : α, (f + g) a * (f + g) a ^ (p - 1) ∂μ) =
          ∫⁻ a : α, (f a + g a) * (f + g) a ^ (p - 1) ∂μ :=
        rfl
      

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)
h_add_apply : ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), (f a + g a) * (f + g) a ^ (p - 1) ∂μ
⊢ ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ =
    ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ

Next tactic:
simp_rw [h_add_apply, add_mul]
theorem lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add {p q : ℝ}
    (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ) (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤
      ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
        (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
  calc
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤ ∫⁻ a, (f + g) a * (f + g) a ^ (p - 1) ∂μ := by
      gcongr with a
      by_cases h_zero : (f + g) a = 0
      · rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]
        exact zero_le _
      by_cases h_top : (f + g) a = ⊤
      · rw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top]
        exact le_top
      refine' le_of_eq _
      nth_rw 2 [← ENNReal.rpow_one ((f + g) a)]
      rw [← ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right]
    _ = (∫⁻ a : α, f a * (f + g) a ^ (p - 1) ∂μ) + ∫⁻ a : α, g a * (f + g) a ^ (p - 1) ∂μ := by
      have h_add_m : AEMeasurable (fun a : α => (f + g) a ^ (p - 1 : ℝ)) μ :=
        (hf.add hg).pow_const _
      have h_add_apply :
        (∫⁻ a : α, (f + g) a * (f + g) a ^ (p - 1) ∂μ) =
          ∫⁻ a : α, (f a + g a) * (f + g) a ^ (p - 1) ∂μ :=
        rfl
      simp_rw [h_add_apply, add_mul]
      

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_m : AEMeasurable fun a => (f + g) a ^ (p - 1)
h_add_apply : ∫⁻ (a : α), (f + g) a * (f + g) a ^ (p - 1) ∂μ = ∫⁻ (a : α), (f a + g a) * (f + g) a ^ (p - 1) ∂μ
⊢ ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) + g a * (f + g) a ^ (p - 1) ∂μ =
    ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ

Next tactic:
rw [lintegral_add_left' (hf.mul h_add_m)]
theorem lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add {p q : ℝ}
    (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ) (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤
      ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
        (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
  calc
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤ ∫⁻ a, (f + g) a * (f + g) a ^ (p - 1) ∂μ := by
      gcongr with a
      by_cases h_zero : (f + g) a = 0
      · rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]
        exact zero_le _
      by_cases h_top : (f + g) a = ⊤
      · rw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top]
        exact le_top
      refine' le_of_eq _
      nth_rw 2 [← ENNReal.rpow_one ((f + g) a)]
      rw [← ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right]
    _ = (∫⁻ a : α, f a * (f + g) a ^ (p - 1) ∂μ) + ∫⁻ a : α, g a * (f + g) a ^ (p - 1) ∂μ := by
      have h_add_m : AEMeasurable (fun a : α => (f + g) a ^ (p - 1 : ℝ)) μ :=
        (hf.add hg).pow_const _
      have h_add_apply :
        (∫⁻ a : α, (f + g) a * (f + g) a ^ (p - 1) ∂μ) =
          ∫⁻ a : α, (f a + g a) * (f + g) a ^ (p - 1) ∂μ :=
        rfl
      simp_rw [h_add_apply, add_mul]
      rw [lintegral_add_left' (hf.mul h_add_m)]
    _ ≤
        ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
          (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
      

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ ≤
    ((∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)) *
      (∫⁻ (a : α), (f a + g a) ^ p ∂μ) ^ (1 / q)

Next tactic:
rw [add_mul]
theorem lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add {p q : ℝ}
    (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ) (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤
      ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
        (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
  calc
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤ ∫⁻ a, (f + g) a * (f + g) a ^ (p - 1) ∂μ := by
      gcongr with a
      by_cases h_zero : (f + g) a = 0
      · rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]
        exact zero_le _
      by_cases h_top : (f + g) a = ⊤
      · rw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top]
        exact le_top
      refine' le_of_eq _
      nth_rw 2 [← ENNReal.rpow_one ((f + g) a)]
      rw [← ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right]
    _ = (∫⁻ a : α, f a * (f + g) a ^ (p - 1) ∂μ) + ∫⁻ a : α, g a * (f + g) a ^ (p - 1) ∂μ := by
      have h_add_m : AEMeasurable (fun a : α => (f + g) a ^ (p - 1 : ℝ)) μ :=
        (hf.add hg).pow_const _
      have h_add_apply :
        (∫⁻ a : α, (f + g) a * (f + g) a ^ (p - 1) ∂μ) =
          ∫⁻ a : α, (f a + g a) * (f + g) a ^ (p - 1) ∂μ :=
        rfl
      simp_rw [h_add_apply, add_mul]
      rw [lintegral_add_left' (hf.mul h_add_m)]
    _ ≤
        ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
          (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
      rw [add_mul]
      

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ + ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ ≤
    (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (f a + g a) ^ p ∂μ) ^ (1 / q) +
      (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (f a + g a) ^ p ∂μ) ^ (1 / q)

Next tactic:
gcongr
theorem lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add {p q : ℝ}
    (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ) (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤
      ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
        (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
  calc
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤ ∫⁻ a, (f + g) a * (f + g) a ^ (p - 1) ∂μ := by
      gcongr with a
      by_cases h_zero : (f + g) a = 0
      · rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]
        exact zero_le _
      by_cases h_top : (f + g) a = ⊤
      · rw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top]
        exact le_top
      refine' le_of_eq _
      nth_rw 2 [← ENNReal.rpow_one ((f + g) a)]
      rw [← ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right]
    _ = (∫⁻ a : α, f a * (f + g) a ^ (p - 1) ∂μ) + ∫⁻ a : α, g a * (f + g) a ^ (p - 1) ∂μ := by
      have h_add_m : AEMeasurable (fun a : α => (f + g) a ^ (p - 1 : ℝ)) μ :=
        (hf.add hg).pow_const _
      have h_add_apply :
        (∫⁻ a : α, (f + g) a * (f + g) a ^ (p - 1) ∂μ) =
          ∫⁻ a : α, (f a + g a) * (f + g) a ^ (p - 1) ∂μ :=
        rfl
      simp_rw [h_add_apply, add_mul]
      rw [lintegral_add_left' (hf.mul h_add_m)]
    _ ≤
        ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
          (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
      rw [add_mul]
      gcongr
      · 

Proof state:
case h₁
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), f a * (f + g) a ^ (p - 1) ∂μ ≤
    (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (f a + g a) ^ p ∂μ) ^ (1 / q)

Next tactic:
exact lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hf (hf.add hg) hf_top
theorem lintegral_rpow_add_le_add_snorm_mul_lintegral_rpow_add {p q : ℝ}
    (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞} (hf : AEMeasurable f μ)
    (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ) (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤
      ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
        (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
  calc
    (∫⁻ a, (f + g) a ^ p ∂μ) ≤ ∫⁻ a, (f + g) a * (f + g) a ^ (p - 1) ∂μ := by
      gcongr with a
      by_cases h_zero : (f + g) a = 0
      · rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]
        exact zero_le _
      by_cases h_top : (f + g) a = ⊤
      · rw [h_top, ENNReal.top_rpow_of_pos hpq.sub_one_pos, ENNReal.top_mul_top]
        exact le_top
      refine' le_of_eq _
      nth_rw 2 [← ENNReal.rpow_one ((f + g) a)]
      rw [← ENNReal.rpow_add _ _ h_zero h_top, add_sub_cancel'_right]
    _ = (∫⁻ a : α, f a * (f + g) a ^ (p - 1) ∂μ) + ∫⁻ a : α, g a * (f + g) a ^ (p - 1) ∂μ := by
      have h_add_m : AEMeasurable (fun a : α => (f + g) a ^ (p - 1 : ℝ)) μ :=
        (hf.add hg).pow_const _
      have h_add_apply :
        (∫⁻ a : α, (f + g) a * (f + g) a ^ (p - 1) ∂μ) =
          ∫⁻ a : α, (f a + g a) * (f + g) a ^ (p - 1) ∂μ :=
        rfl
      simp_rw [h_add_apply, add_mul]
      rw [lintegral_add_left' (hf.mul h_add_m)]
    _ ≤
        ((∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p)) *
          (∫⁻ a, (f a + g a) ^ p ∂μ) ^ (1 / q) := by
      rw [add_mul]
      gcongr
      · exact lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hf (hf.add hg) hf_top
      · 

Proof state:
case h₂
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
⊢ ∫⁻ (a : α), g a * (f + g) a ^ (p - 1) ∂μ ≤
    (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (f a + g a) ^ p ∂μ) ^ (1 / q)

Next tactic:
exact lintegral_mul_rpow_le_lintegral_rpow_mul_lintegral_rpow hpq hg (hf.add hg) hg_top
private theorem lintegral_Lp_add_le_aux {p q : ℝ} (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ)
    (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) (h_add_zero : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ 0)
    (h_add_top : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) := by
  

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Next tactic:
have hp_not_nonpos : ¬p ≤ 0 := by simp [hpq.pos]
private theorem lintegral_Lp_add_le_aux {p q : ℝ} (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ)
    (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) (h_add_zero : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ 0)
    (h_add_top : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) := by
  have hp_not_nonpos : ¬p ≤ 0 := by 

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤
⊢ ¬p ≤ 0

Next tactic:
simp [hpq.pos]
private theorem lintegral_Lp_add_le_aux {p q : ℝ} (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ)
    (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) (h_add_zero : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ 0)
    (h_add_top : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) := by
  have hp_not_nonpos : ¬p ≤ 0 := by simp [hpq.pos]
  

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Next tactic:
have htop_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤ := by
    by_contra h
    exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)
private theorem lintegral_Lp_add_le_aux {p q : ℝ} (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ)
    (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) (h_add_zero : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ 0)
    (h_add_top : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) := by
  have hp_not_nonpos : ¬p ≤ 0 := by simp [hpq.pos]
  have htop_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤ := by
    

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤

Next tactic:
by_contra h
private theorem lintegral_Lp_add_le_aux {p q : ℝ} (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ)
    (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) (h_add_zero : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ 0)
    (h_add_top : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) := by
  have hp_not_nonpos : ¬p ≤ 0 := by simp [hpq.pos]
  have htop_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤ := by
    by_contra h
    

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
h : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) = ⊤
⊢ False

Next tactic:
exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)
private theorem lintegral_Lp_add_le_aux {p q : ℝ} (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ)
    (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) (h_add_zero : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ 0)
    (h_add_top : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) := by
  have hp_not_nonpos : ¬p ≤ 0 := by simp [hpq.pos]
  have htop_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤ := by
    by_contra h
    exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by 

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
h : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) = ⊤
⊢ 0 ≤ 1 / p

Next tactic:
simp [hpq.nonneg]
private theorem lintegral_Lp_add_le_aux {p q : ℝ} (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ)
    (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) (h_add_zero : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ 0)
    (h_add_top : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) := by
  have hp_not_nonpos : ¬p ≤ 0 := by simp [hpq.pos]
  have htop_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤ := by
    by_contra h
    exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)
  

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Next tactic:
have h0_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0 := by
    simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]
private theorem lintegral_Lp_add_le_aux {p q : ℝ} (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ)
    (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) (h_add_zero : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ 0)
    (h_add_top : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) := by
  have hp_not_nonpos : ¬p ≤ 0 := by simp [hpq.pos]
  have htop_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤ := by
    by_contra h
    exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)
  have h0_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0 := by
    

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0

Next tactic:
simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]
private theorem lintegral_Lp_add_le_aux {p q : ℝ} (hpq : p.IsConjugateExponent q) {f g : α → ℝ≥0∞}
    (hf : AEMeasurable f μ) (hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) (hg : AEMeasurable g μ)
    (hg_top : (∫⁻ a, g a ^ p ∂μ) ≠ ⊤) (h_add_zero : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ 0)
    (h_add_top : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤) :
    (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≤
      (∫⁻ a, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a, g a ^ p ∂μ) ^ (1 / p) := by
  have hp_not_nonpos : ¬p ≤ 0 := by simp [hpq.pos]
  have htop_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤ := by
    by_contra h
    exact h_add_top (@ENNReal.rpow_eq_top_of_nonneg _ (1 / p) (by simp [hpq.nonneg]) h)
  have h0_rpow : (∫⁻ a, (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0 := by
    simp [h_add_zero, h_add_top, hpq.nonneg, hp_not_nonpos, -Pi.add_apply]
  

Proof state:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : Real.IsConjugateExponent p q
f g : α → ℝ≥0∞
hf : AEMeasurable f
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ⊤
hg : AEMeasurable g
hg_top : ∫⁻ (a : α), g a ^ p ∂μ ≠ ⊤
h_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0
h_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ⊤
hp_not_nonpos : ¬p ≤ 0
htop_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ ⊤
h0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Next tactic:
suffices h :
    1 ≤
      (∫⁻ a : α, (f + g) a ^ p ∂μ) ^ (-(1 / p)) *
        ((∫⁻ a : α, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a : α, g a ^ p ∂μ) ^ (1 / p))