mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
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
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
aux₂ :
  (Function.uncurry fun x y ↦ (Ioc 0 (f x)).indicator (fun t ↦ ENNReal.ofReal (g t)) y) =
    {p | p.2 ∈ Ioc 0 (f p.1)}.indicator fun p ↦ ENNReal.ofReal (g p.2)
⊢ MeasurableSet {p | p.2 ∈ Ioc 0 (f p.1)}

Tactic:
by

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
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
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t)

Tactic:
by

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
⊢ ∀ (ω : α), ∫ (t : ℝ) in 0..f ω, g t = ∫ (t : ℝ) in 0..f ω, 0

Tactic:
by

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
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
hr : 0 < r
h_meas : Measurable fun ω ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int : ∀ (t : ℝ), 0 < t → μ {a | t ≤ (1 + ‖a‖) ^ (-r)} = μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
f : ℝ → ℝ≥0∞ := fun t ↦ μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := μ (Metric.ball 0 1)
t : ℝ
ht : t ∈ Ioc 0 1
⊢ -r⁻¹ ≤ 0

Tactic:
by

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
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = 0
t : ℝ
ht : g t = 0 t
⊢ μ {a | t ≤ f a} * ENNReal.ofReal (g t) = 0 t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
hr : 0 < r
h_meas : Measurable fun ω ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int : ∀ (t : ℝ), 0 < t → μ {a | t ≤ (1 + ‖a‖) ^ (-r)} = μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
f : ℝ → ℝ≥0∞ := fun t ↦ μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := μ (Metric.ball 0 1)
t : ℝ
ht : t ∈ Ioc 0 1
⊢ f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB

Tactic:
by

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
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = 0
⊢ (fun t ↦ μ {a | t ≤ f a} * ENNReal.ofReal (g t)) =ᵐ[volume.restrict (Ioi 0)] 0

Tactic:
by

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
A : ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = 0
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
hr : 0 < r
h_meas : Measurable fun ω ↦ (1 + ‖ω‖) ^ (-r)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int : ∀ (t : ℝ), 0 < t → μ {a | t ≤ (1 + ‖a‖) ^ (-r)} = μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
f : ℝ → ℝ≥0∞ := fun t ↦ μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := μ (Metric.ball 0 1)
t : ℝ
ht : t ∈ Ioi 1
⊢ f t = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
⊢ ∫⁻ (x : E), ENNReal.ofReal ((1 + ‖x‖) ^ (-r)) ∂μ < ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
⊢ Measurable fun x ↦ (1 + ‖x‖) ^ (-r)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
x✝ : E
⊢ 0 ≤ 1 + ‖x✝‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
⊢ Integrable (fun x ↦ (1 + ‖x‖) ^ (-r)) μ

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
hr : 0 < r
⊢ Measurable fun x ↦ (1 + ‖x‖ ^ 2) ^ (-r / 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
⊢ Integrable (fun x ↦ (1 + ‖x‖ ^ 2) ^ (-r / 2)) μ

Tactic:
by

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
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 : ℝ) in Ioc 0 s, ENNReal.ofReal (g a) ≠ 0

Tactic:
by

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
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
⊢ ∞ = ∫⁻ (t : ℝ) in Ioc 0 s, ∞ * ENNReal.ofReal (g t)

Tactic:
by

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
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
⊢ ∫⁻ (t : ℝ) in Ioc 0 s, ∞ * ENNReal.ofReal (g t) ≤ ∫⁻ (t : ℝ) in Ioc 0 s, μ {a | t ≤ f a} * ENNReal.ofReal (g t)

Tactic:
by

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
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
⊢ ∫⁻ (t : ℝ) in Ioi 0, μ {a | t ≤ f a} * ENNReal.ofReal (g t) = ∞

Tactic:
by

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
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) = ∞
⊢ ∞ = ∫⁻ (x : α) in {a | s < f a}, ENNReal.ofReal (∫ (t : ℝ) in 0..s, g t) ∂μ

Tactic:
by

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
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) = ∞
⊢ ∫⁻ (x : α) in {a | s < f a}, ENNReal.ofReal (∫ (t : ℝ) in 0..s, g t) ∂μ ≤
    ∫⁻ (ω : α) in {a | s < f a}, ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ

Tactic:
by

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
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) = ∞
⊢ ∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) ∂μ = ∞

Tactic:
by

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}
⊢ ∀ (n : ℕ), g =ᵐ[volume.restrict (Ioc 0 ↑n)] 0

Tactic:
by

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} ≠ ∞
⊢ BddAbove {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}

Tactic:
by

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}
⊢ 0 ∈ {s | g =ᵐ[volume.restrict (Ioc 0 s)] 0}

Tactic:
by

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
u : ℕ → ℝ
uM : ∀ (n : ℕ), u n < M
ulim : Tendsto u atTop (𝓝 M)
⊢ ∀ (n : ℕ), g =ᵐ[volume.restrict (Ioc 0 (u n))] 0

Tactic:
by

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
⊢ g =ᵐ[volume.restrict (Ioc 0 M)] 0

Tactic:
by

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
a : α
ha : a ∈ {a | f a ≤ M}
⊢ a ∉ {a | M < f a}

Tactic:
by

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
⊢ ν {a | f a ≤ M} = 0

Tactic:
by

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
H3 : ∫ (x : ℝ) in Ioc 0 (u n), g x ∂volume ≤ 0
⊢ ∀ᵐ (t : ℝ) ∂volume.restrict (Ioc 0 (u n)), 0 ≤ g t

Tactic:
by

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
⊢ μ {a | u n < f a} < ∞

Tactic:
by

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 : E → F
n k : ℕ
C : ℝ
x : E
⊢ 0 ≤ 1 + ‖x‖

Tactic:
by

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 : E → F
⊢ HasTemperateGrowth f ↔ ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k

Tactic:
by

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 : 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
⊢ 0 ≤ (1 + ‖x‖) ^ k n

Tactic:
by

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 : 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
⊢ 0 ≤ (1 + ‖x‖) ^ (Finset.range (N + 1)).sup k

Tactic:
by

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)
⊢ ∀ (i : ℕ), ν ({a | f a ≤ M} ∪ {a | u i < f a}) < ∞

Tactic:
by

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 : 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
⊢ n ∈ Finset.range (N + 1)

Tactic:
by

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 : E → F
hf_temperate : HasTemperateGrowth f
N : ℕ
⊢ ∃ k, ∀ n ≤ N, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k

Tactic:
by

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 : E → F✝
hf_temperate : HasTemperateGrowth f
n k : ℕ
hk : ∀ n_1 ≤ n, iteratedFDeriv ℝ n_1 f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
F : E → (N : Fin (n + 1)) → E [×↑N]→L[ℝ] F✝ := fun x N ↦ iteratedFDeriv ℝ (↑N) f x
⊢ F =O[⊤] fun x ↦ (1 + ‖x‖) ^ k

Tactic:
by

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 : E → F
hf_temperate : HasTemperateGrowth f
n : ℕ
⊢ ∃ k C, 0 ≤ C ∧ ∀ N ≤ n, ∀ (x : E), ‖iteratedFDeriv ℝ N f x‖ ≤ C * (1 + ‖x‖) ^ k

Tactic:
by

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 : E → F
h'f : HasTemperateGrowth (fderiv ℝ f)
hf : Differentiable ℝ f
k : ℕ
C : ℝ
h : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k
⊢ ∞ = ω → AnalyticOnNhd ℝ f Set.univ

Tactic:
by

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)
⊢ ⋃ i, {a | f a ≤ M} ∪ {a | u i < f a} = univ

Tactic:
by

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 : E → F
h'f : HasTemperateGrowth (fderiv ℝ f)
hf : Differentiable ℝ f
k : ℕ
C : ℝ
h : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k
x : E
⊢ ‖iteratedFDeriv ℝ 0 f x‖ ≤ C * (1 + ‖x‖) ^ k

Tactic:
by

Goal state after: