mathlib-initiative/mathlib-tactics huggingface dataset:
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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
A : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) = ∞
⊢ ∞ = ∞

pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
A : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) = ∞
⊢ ENNReal.ofReal (∫ (t : ℝ) in 0..s, g t) ≠ 0

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
A : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) = ∞
⊢ ENNReal.ofReal (∫ (t : ℝ) in 0..s, g t) ≠ 0
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: case hfi
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
A : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) = ∞
a : α
ha : a ∈ {a | s < f a}
⊢ IntervalIntegrable g volume 0 (f a)

Tactic:
· exact g_intble _ (s_pos.trans ha)

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

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
A : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) = ∞
B : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∞
⊢ ∞ = ∞

Tactic:
]

Goal state after:
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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
H1 : ¬BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
this✝ : ∀ (n : ℕ), g =ᵐ[volume.restrict (Ioc 0 ↑n)] 0
Hg : g =ᵐ[volume.restrict (Ioi 0)] 0
this : ⋃ n, Ioc 0 ↑n = Ioi 0
⊢ g =ᵐ[volume.restrict (Ioi 0)] 0

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
H1 : ¬BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
this✝ : ∀ (n : ℕ), g =ᵐ[volume.restrict (Ioc 0 ↑n)] 0
Hg : g =ᵐ[volume.restrict (Ioi 0)] 0
this : ⋃ n, Ioc 0 ↑n = Ioi 0
⊢ g =ᵐ[volume.restrict (Ioi 0)] 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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
⊢ g =ᵐ[volume.restrict (Ioo 0 M)] 0

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
⊢ g =ᵐ[volume.restrict (Ioo 0 M)] 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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
⊢ μ ({a | f a ≤ M} ∩ {a | M < f a}) = 0

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
⊢ μ ({a | f a ≤ M} ∩ {a | M < f a}) = 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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
⊢ Disjoint {a | f a ≤ M} {a | M < f a}

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
⊢ Disjoint {a | f a ≤ M} {a | M < f a}
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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
⊢ μ {a | u n < f a} ≠ ∞

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
⊢ μ {a | u n < f a} ≠ ∞
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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
H3 : ∫ (x : ℝ) in Ioc 0 (u n), g x ∂volume ≤ 0
⊢ False

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
H3 : ∫ (x : ℝ) in Ioc 0 (u n), g x ∂volume ≤ 0
⊢ False
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 : E → F
n k : ℕ
C : ℝ
x : E
⊢ (1 + ‖x‖) ^ k = (1 + ‖x‖) ^ k

Tactic:
]

Goal state after:
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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
J : μ {a | u n < f a} < ∞
⊢ ν {a | u n < f a} < ∞

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
u : ℕ → ℝ
uM : ∀ (n : ℕ), M < u n
ulim : Tendsto u atTop (𝓝 M)
n : ℕ
I : ν {a | f a ≤ M} = 0
J : μ {a | u n < f a} < ∞
⊢ ν {a | u n < f a} < ∞
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 : E → F
hf_temperate : HasTemperateGrowth f
N : ℕ
k : ℕ → ℕ
hk : ∀ (n : ℕ), iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k n
n : ℕ
hn : n ≤ N
x : E
⊢ (1 + ‖x‖) ^ k n ≤ (1 + ‖x‖) ^ (Finset.range (N + 1)).sup k

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 : E → F
hf_temperate : HasTemperateGrowth f
N : ℕ
k : ℕ → ℕ
hk : ∀ (n : ℕ), iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k n
n : ℕ
hn : n ≤ N
x : E
⊢ (1 + ‖x‖) ^ k n ≤ (1 + ‖x‖) ^ (Finset.range (N + 1)).sup k
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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
meas : MeasurableSet {a | M < f a}
I : ∫⁻ (ω : α) in {a | M < f a}ᶜ, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = 0
⊢ ∫⁻ (x : α) in {a | M < f a}, ENNReal.ofReal (∫ (t : ℝ) in 0..f x, g t) ∂μ =
    ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν

Tactic:
]

Goal state after:
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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B1 :
  ∫⁻ (t : ℝ) in Ioc 0 M, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioc 0 M, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
t : ℝ
ht : t ∈ Ioi M
⊢ μ {a | t ≤ f a} * ENNReal.ofReal (g t) = μ ({a | t ≤ f a} ∩ {a | M < 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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B1 :
  ∫⁻ (t : ℝ) in Ioc 0 M, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioc 0 M, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
t : ℝ
ht : t ∈ Ioi M
⊢ μ {a | t ≤ f a} * ENNReal.ofReal (g t) = μ ({a | t ≤ f a} ∩ {a | M < f a}) * ENNReal.ofReal (g t)
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 ∈ Finset.range (k₂ + 1), C₂ * (‖f x‖ ^ i * 1 ^ (k₂ - i) * ↑(k₂.choose i)) =
    ∑ i ∈ Finset.range (k₂ + 1), C₂ * (‖f x‖ ^ i * ↑(k₂.choose i))

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 ∈ Finset.range (k₂ + 1), C₂ * (‖f x‖ ^ i * 1 ^ (k₂ - i) * ↑(k₂.choose i)) =
    ∑ i ∈ Finset.range (k₂ + 1), C₂ * (‖f x‖ ^ i * ↑(k₂.choose i))
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 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B1 :
  ∫⁻ (t : ℝ) in Ioc 0 M, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioc 0 M, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
B2 :
  ∫⁻ (t : ℝ) in Ioi M, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi M, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
I : Ioi 0 = Ioc 0 M ∪ Ioi M
J : Disjoint (Ioc 0 M) (Ioi M)
⊢ (∫⁻ (t : ℝ) in Ioc 0 M, ν {a | t ≤ f a} * ENNReal.ofReal (g t)) +
      ∫⁻ (t : ℝ) in Ioi M, ν {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (a : ℝ) in Ioc 0 M, ν {a_1 | a ≤ f a_1} * ENNReal.ofReal (g a) ∂volume +
      ∫⁻ (a : ℝ) in Ioi M, ν {a_1 | a ≤ f a_1} * ENNReal.ofReal (g a) ∂volume

Tactic:
]

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

Goal state before:
pp: case neg
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν = ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)

Tactic:
]

Goal state after:
case neg
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
H2 : ∀ s > 0, 0 < ∫ (t : ℝ) in 0..s, g t → μ {a | s < f a} ≠ ∞
M_bdd : BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M : ℝ := sSup {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
zero_mem : 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}
M_nonneg : 0 ≤ M
hgM : g =ᵐ[volume.restrict (Ioc 0 M)] 0
ν : Measure α := μ.restrict {a | M < f a}
this : SigmaFinite ν
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν
B :
  ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) =
    ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂ν = ∫⁻ (t : ℝ) in Ioi 0, ν {a | t ≤ f a} * ENNReal.ofReal (g t)