mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: D : Type u_4
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup D
inst✝ : NormedSpace ℝ D
g : E → F
f : D → E
t : Set E
ht : Set.range f ⊆ t
ht' : UniqueDiffOn ℝ t
hg₁ : ContDiffOn ℝ ∞ g t
hg₂ : ∀ (N : ℕ), ∃ k C, ∃ (_ : 0 ≤ C), ∀ n ≤ N, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n g t x‖ ≤ C * (1 + ‖x‖) ^ k
hf : HasTemperateGrowth f
n k₁ : ℕ
C₁ : ℝ
hC₁ : 0 ≤ C₁
h₁ : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
k₂ : ℕ
C₂ : ℝ
hC₂ : 0 ≤ C₂
h₂ : ∀ n_1 ≤ n, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n_1 g t x‖ ≤ C₂ * (1 + ‖x‖) ^ k₂
h₁' : ∀ (x : D), ‖f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
C₃ : ℝ := ∑ k ∈ Finset.range (k₂ + 1), C₂ * ↑(k₂.choose k) * C₁ ^ k
x : D
i✝ : ℕ
hi✝ : i✝ ≤ n
i : ℕ
hi : i ∈ Finset.range (k₂ + 1)
⊢ C₂ * ((C₁ * (1 + ‖x‖) ^ k₁) ^ i * ↑(k₂.choose i)) ≤ C₂ * ↑(k₂.choose i) * C₁ ^ i * (1 + ‖x‖) ^ (k₁ * k₂)

Tactic:
]

Goal state after:
D : Type u_4
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup D
inst✝ : NormedSpace ℝ D
g : E → F
f : D → E
t : Set E
ht : Set.range f ⊆ t
ht' : UniqueDiffOn ℝ t
hg₁ : ContDiffOn ℝ ∞ g t
hg₂ : ∀ (N : ℕ), ∃ k C, ∃ (_ : 0 ≤ C), ∀ n ≤ N, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n g t x‖ ≤ C * (1 + ‖x‖) ^ k
hf : HasTemperateGrowth f
n k₁ : ℕ
C₁ : ℝ
hC₁ : 0 ≤ C₁
h₁ : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
k₂ : ℕ
C₂ : ℝ
hC₂ : 0 ≤ C₂
h₂ : ∀ n_1 ≤ n, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n_1 g t x‖ ≤ C₂ * (1 + ‖x‖) ^ k₂
h₁' : ∀ (x : D), ‖f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
C₃ : ℝ := ∑ k ∈ Finset.range (k₂ + 1), C₂ * ↑(k₂.choose k) * C₁ ^ k
x : D
i✝ : ℕ
hi✝ : i✝ ≤ n
i : ℕ
hi : i ∈ Finset.range (k₂ + 1)
⊢ C₂ * ((C₁ * (1 + ‖x‖) ^ k₁) ^ i * ↑(k₂.choose i)) ≤ C₂ * ↑(k₂.choose i) * C₁ ^ i * (1 + ‖x‖) ^ (k₁ * k₂)
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
t : ℝ
t_pos : t > 0
⊢ IntegrableOn G ∅ volume

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
t : ℝ
t_pos : t > 0
⊢ IntegrableOn G ∅ volume
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
ω : α
hω : f ω = AEMeasurable.mk f f_mble ω
h'ω : 0 ω ≤ f ω
⊢ f ω = max (f ω) 0

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
ω : α
hω : f ω = AEMeasurable.mk f f_mble ω
h'ω : 0 ω ≤ f ω
⊢ f ω = max (f ω) 0
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
t : ℝ
ht : g t = G t
⊢ μ {a | t ≤ f a} * ENNReal.ofReal (G t) = μ {a | t ≤ F a} * ENNReal.ofReal (G t)

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
t : ℝ
ht : g t = G t
⊢ μ {a | t ≤ f a} * ENNReal.ofReal (G t) = μ {a | t ≤ F a} * ENNReal.ofReal (G t)
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
eq₁ :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ F a} * ENNReal.ofReal (G t)
ω : α
fω_nn : f ω = F ω
⊢ ENNReal.ofReal (∫ (t : ℝ) in 0..F ω, g t) = ENNReal.ofReal (∫ (t : ℝ) in 0..F ω, G t)

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
eq₁ :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ F a} * ENNReal.ofReal (G t)
ω : α
fω_nn : f ω = F ω
⊢ ENNReal.ofReal (∫ (t : ℝ) in 0..F ω, g t) = ENNReal.ofReal (∫ (t : ℝ) in 0..F ω, G t)
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
K_sub_Ω : ↑K ⊆ ↑Ω
⊢ coinduced (ofSupportedIn K_sub_Ω) ContDiffMapSupportedIn.topologicalSpace ≤ topologicalSpace Ω F n

Tactic:
]

Goal state after:
E : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
n : ℕ∞
K : Compacts E
K_sub_Ω : ↑K ⊆ ↑Ω
⊢ coinduced (ofSupportedIn K_sub_Ω) ContDiffMapSupportedIn.topologicalSpace ≤ topologicalSpace Ω F n
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
eq₁ :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ F a} * ENNReal.ofReal (G t)
ω : α
fω_nn✝ : f ω = F ω
fω_nn : 0 ≤ F ω
⊢ ∀ᵐ (x : ℝ) ∂volume.restrict (Ioc 0 (F ω)), g x = G x

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
G_intble : ∀ t > 0, IntervalIntegrable G volume 0 t
F : α → ℝ
F_mble : Measurable F
F_nn : 0 ≤ F
f_eq_F : f =ᵐ[μ] F
eq₁ :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ F a} * ENNReal.ofReal (G t)
ω : α
fω_nn✝ : f ω = F ω
fω_nn : 0 ≤ F ω
⊢ ∀ᵐ (x : ℝ) ∂volume.restrict (Ioc 0 (F ω)), g x = G x
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
inst✝⁹ : NormedSpace 𝕜 F
n : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
f : 𝓓^{n}(Ω, F) →ₗ[𝕜] V
⊢ topologicalSpace Ω F n ≤ induced (⇑(↑ℝ f)) t ↔
    ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (⇑(↑ℝ f) ∘ ofSupportedIn K_sub_Ω)

Tactic:
]

Goal state after:
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace ℝ F
inst✝⁹ : NormedSpace 𝕜 F
n : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
f : 𝓓^{n}(Ω, F) →ₗ[𝕜] V
⊢ topologicalSpace Ω F n ≤ induced (⇑(↑ℝ f)) t ↔
    ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (⇑(↑ℝ f) ∘ ofSupportedIn K_sub_Ω)
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: D : Type u_4
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup D
inst✝ : NormedSpace ℝ D
g : E → F
f : D → E
t : Set E
ht : Set.range f ⊆ t
ht' : UniqueDiffOn ℝ t
hg₁ : ContDiffOn ℝ ∞ g t
hg₂ : ∀ (N : ℕ), ∃ k C, ∃ (_ : 0 ≤ C), ∀ n ≤ N, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n g t x‖ ≤ C * (1 + ‖x‖) ^ k
hf : HasTemperateGrowth f
n k₁ : ℕ
C₁ : ℝ
hC₁ : 0 ≤ C₁
h₁ : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
k₂ : ℕ
C₂ : ℝ
hC₂ : 0 ≤ C₂
h₂ : ∀ n_1 ≤ n, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n_1 g t x‖ ≤ C₂ * (1 + ‖x‖) ^ k₂
h₁' : ∀ (x : D), ‖f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
C₃ : ℝ := ∑ k ∈ Finset.range (k₂ + 1), C₂ * ↑(k₂.choose k) * C₁ ^ k
x : D
hg' : ∀ i ≤ n, ‖iteratedFDerivWithin ℝ i g t (f x)‖ ≤ C₃ * (1 + ‖x‖) ^ (k₁ * k₂)
hf' : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDeriv ℝ i f x‖ ≤ ((1 + C₁) * (1 + ‖x‖) ^ k₁) ^ i
⊢ ↑n ! * (C₃ * (1 + ‖x‖) ^ (k₁ * k₂)) * ((1 + C₁) ^ n * (1 + ‖x‖) ^ (k₁ * n)) =
    ↑n ! * C₃ * (1 + C₁) ^ n * ((1 + ‖x‖) ^ (k₁ * k₂) * (1 + ‖x‖) ^ (k₁ * n))

Tactic:
]

Goal state after:
D : Type u_4
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup D
inst✝ : NormedSpace ℝ D
g : E → F
f : D → E
t : Set E
ht : Set.range f ⊆ t
ht' : UniqueDiffOn ℝ t
hg₁ : ContDiffOn ℝ ∞ g t
hg₂ : ∀ (N : ℕ), ∃ k C, ∃ (_ : 0 ≤ C), ∀ n ≤ N, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n g t x‖ ≤ C * (1 + ‖x‖) ^ k
hf : HasTemperateGrowth f
n k₁ : ℕ
C₁ : ℝ
hC₁ : 0 ≤ C₁
h₁ : ∀ N ≤ n, ∀ (x : D), ‖iteratedFDeriv ℝ N f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
k₂ : ℕ
C₂ : ℝ
hC₂ : 0 ≤ C₂
h₂ : ∀ n_1 ≤ n, ∀ x ∈ t, ‖iteratedFDerivWithin ℝ n_1 g t x‖ ≤ C₂ * (1 + ‖x‖) ^ k₂
h₁' : ∀ (x : D), ‖f x‖ ≤ C₁ * (1 + ‖x‖) ^ k₁
C₃ : ℝ := ∑ k ∈ Finset.range (k₂ + 1), C₂ * ↑(k₂.choose k) * C₁ ^ k
x : D
hg' : ∀ i ≤ n, ‖iteratedFDerivWithin ℝ i g t (f x)‖ ≤ C₃ * (1 + ‖x‖) ^ (k₁ * k₂)
hf' : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDeriv ℝ i f x‖ ≤ ((1 + C₁) * (1 + ‖x‖) ^ k₁) ^ i
⊢ ↑n ! * (C₃ * (1 + ‖x‖) ^ (k₁ * k₂)) * ((1 + C₁) ^ n * (1 + ‖x‖) ^ (k₁ * n)) =
    ↑n ! * C₃ * (1 + C₁) ^ n * ((1 + ‖x‖) ^ (k₁ * k₂) * (1 + ‖x‖) ^ (k₁ * n))
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
cst : ℝ → ℝ := fun x ↦ 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (x : ℝ) in 0..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (x : ℝ) in 0..f ω, 1) ∂μ

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
cst : ℝ → ℝ := fun x ↦ 1
cst_intble : ∀ t > 0, IntervalIntegrable cst volume 0 t
key : ∫⁻ (ω : α), ENNReal.ofReal (∫ (x : ℝ) in 0..f ω, 1) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a}
⊢ ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (x : ℝ) in 0..f ω, 1) ∂μ
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
𝕂 : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace ℝ E
Ω Ω₁ Ω₂ : Opens E
F : Type u_4
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace ℝ F
inst✝¹² : NormedSpace 𝕜 F
F' : Type u_5
inst✝¹¹ : NormedAddCommGroup F'
inst✝¹⁰ : NormedSpace ℝ F'
inst✝⁹ : NormedSpace 𝕜 F'
n n₁ n₂ k : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
toFun : 𝓓^{n}(Ω, F) → V
map_add : ∀ (f g : 𝓓^{n}(Ω, F)), toFun (f + g) = toFun f + toFun g
map_smul : ∀ (c : 𝕜) (f : 𝓓^{n}(Ω, F)), toFun (c • f) = c • toFun f
cont : ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (toFun ∘ ofSupportedIn K_sub_Ω)
Φ : 𝓓^{n}(Ω, F) →ₗ[𝕜] V := { toFun := toFun, map_add' := map_add, map_smul' := map_smul }
⊢ ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (⇑Φ ∘ ofSupportedIn K_sub_Ω)

Tactic:
]

Goal state after:
𝕜 : Type u_1
𝕂 : Type u_2
inst✝¹⁷ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹⁶ : NormedAddCommGroup E
inst✝¹⁵ : NormedSpace ℝ E
Ω Ω₁ Ω₂ : Opens E
F : Type u_4
inst✝¹⁴ : NormedAddCommGroup F
inst✝¹³ : NormedSpace ℝ F
inst✝¹² : NormedSpace 𝕜 F
F' : Type u_5
inst✝¹¹ : NormedAddCommGroup F'
inst✝¹⁰ : NormedSpace ℝ F'
inst✝⁹ : NormedSpace 𝕜 F'
n n₁ n₂ k : ℕ∞
V : Type u_6
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module ℝ V
t : TopologicalSpace V
inst✝⁶ : IsTopologicalAddGroup V
inst✝⁵ : ContinuousSMul ℝ V
inst✝⁴ : LocallyConvexSpace ℝ V
inst✝³ : Algebra ℝ 𝕜
inst✝² : IsScalarTower ℝ 𝕜 F
inst✝¹ : Module 𝕜 V
inst✝ : IsScalarTower ℝ 𝕜 V
toFun : 𝓓^{n}(Ω, F) → V
map_add : ∀ (f g : 𝓓^{n}(Ω, F)), toFun (f + g) = toFun f + toFun g
map_smul : ∀ (c : 𝕜) (f : 𝓓^{n}(Ω, F)), toFun (c • f) = c • toFun f
cont : ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (toFun ∘ ofSupportedIn K_sub_Ω)
Φ : 𝓓^{n}(Ω, F) →ₗ[𝕜] V := { toFun := toFun, map_add' := map_add, map_smul' := map_smul }
⊢ ∀ (K : Compacts E) (K_sub_Ω : ↑K ⊆ ↑Ω), Continuous[_, t] (⇑Φ ∘ ofSupportedIn K_sub_Ω)
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (g t)

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} * ENNReal.ofReal (g t)
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
t : ℝ
ht : μ {a | t ≤ f a} = μ {a | t < f a}
⊢ μ {a | t < f a} * ENNReal.ofReal (g t) = μ {a | t < f a} * ENNReal.ofReal (g t)

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case hg₁
D : Type u_4
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup D
inst✝ : NormedSpace ℝ D
g : E → F
f : D → E
hg : HasTemperateGrowth g
hf : HasTemperateGrowth f
⊢ ContDiff ℝ ∞ g

Tactic:
]

Goal state after:
case hg₁
D : Type u_4
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup D
inst✝ : NormedSpace ℝ D
g : E → F
f : D → E
hg : HasTemperateGrowth g
hf : HasTemperateGrowth f
⊢ ContDiff ℝ ∞ g
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: D : Type u_4
E : Type u_5
F : Type u_6
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup D
inst✝ : NormedSpace ℝ D
g : E → F
f : D → E
hg : HasTemperateGrowth g
hf : HasTemperateGrowth f
⊢ HasTemperateGrowth (g ∘ f)

Tactic:
by
  apply hf.comp' (t := Set.univ)
  · simp
  · simp
  · rw [contDiffOn_univ]
    exact hg.1
  · simpa [iteratedFDerivWithin_univ] using hg.norm_iteratedFDeriv_le_uniform

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
t : ℝ
ht : μ {a | t ≤ f a} = μ {a | t < f a}
⊢ μ {a | t < f a} = μ {a | t < f a}

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f g : E → F
hf : ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
hg : ContDiff ℝ ∞ g ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
⊢ ContDiff ℝ ∞ (f + g) ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n (f + g) =O[⊤] fun x ↦ (1 + ‖x‖) ^ k

Tactic:
]

Goal state after:
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f g : E → F
hf : ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
hg : ContDiff ℝ ∞ g ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
⊢ ContDiff ℝ ∞ (f + g) ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n (f + g) =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case h
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f g : E → F
hf : ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
hg : ContDiff ℝ ∞ g ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
n k₁ : ℕ
h₁ : iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k₁
k₂ : ℕ
h₂ : iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k₂
⊢ (iteratedFDeriv ℝ n f + iteratedFDeriv ℝ n g) =O[⊤] fun x ↦ (1 + ‖x‖) ^ max k₁ k₂

Tactic:
]

Goal state after:
case h
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f g : E → F
hf : ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
hg : ContDiff ℝ ∞ g ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
n k₁ : ℕ
h₁ : iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k₁
k₂ : ℕ
h₂ : iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k₂
⊢ (iteratedFDeriv ℝ n f + iteratedFDeriv ℝ n g) =O[⊤] fun x ↦ (1 + ‖x‖) ^ max k₁ k₂
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: case e'_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ∞
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ∞
⊢ ∫ (t : ℝ) in Ioi 0, μ.real {a | t < f a} = (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal

Tactic:
have aux := @integral_eq_lintegral_of_nonneg_ae _ _ ((volume : Measure ℝ).restrict (Set.Ioi 0))
      (fun t ↦ μ.real {a : α | t < f a}) ?_ ?_

Goal state after:
case e'_3.refine_3
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ∞
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ∞
aux :
  ∫ (a : ℝ) in Ioi 0, μ.real {a_1 | a < f a_1} = (∫⁻ (a : ℝ) in Ioi 0, ENNReal.ofReal (μ.real {a_1 | a < f a_1})).toReal
⊢ ∫ (t : ℝ) in Ioi 0, μ.real {a | t < f a} = (∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}).toReal

case e'_3.refine_1
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ∞
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ∞
⊢ 0 ≤ᵐ[volume.restrict (Ioi 0)] fun t ↦ μ.real {a | t < f a}

case e'_3.refine_2
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ∞
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ∞
⊢ AEStronglyMeasurable (fun t ↦ μ.real {a | t < f a}) (volume.restrict (Ioi 0))