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

Goal state before:
pp: p q : ℝ
hp : 0 < p
hq : -1 < q
⊢ ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) =
    ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-x))

Tactic:
rw [← integral_comp_rpow_Ioi _ (one_div_ne_zero (ne_of_gt hp)),
        abs_eq_self.mpr (le_of_lt (one_div_pos.mpr hp))]
      refine setIntegral_congr_fun measurableSet_Ioi (fun _ hx => ?_)
      rw [← rpow_mul (le_of_lt hx) _ p, one_div_mul_cancel (ne_of_gt hp), rpow_one]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : SemilatticeSup β
inst✝² : Zero β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousSup β
ι : Type u_5
s : Finset ι
H : s.Nonempty
f : ι → α →C_c β
x✝ : α
⊢ (s.sup' H f) x✝ = s.sup' H (fun i ↦ ⇑(f i)) x✝

Tactic:
;

Goal state after:
α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : SemilatticeSup β
inst✝² : Zero β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousSup β
ι : Type u_5
s : Finset ι
H : s.Nonempty
f : ι → α →C_c β
x✝ : α
⊢ (s.sup' H f) x✝ = s.sup' H (fun i ↦ ⇑(f i)) x✝
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : SemilatticeInf β
inst✝² : Zero β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousInf β
ι : Type u_5
s : Finset ι
H : s.Nonempty
f : ι → α →C_c β
x✝ : α
⊢ (s.inf' H f) x✝ = s.inf' H (fun i ↦ ⇑(f i)) x✝

Tactic:
;

Goal state after:
α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : SemilatticeInf β
inst✝² : Zero β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousInf β
ι : Type u_5
s : Finset ι
H : s.Nonempty
f : ι → α →C_c β
x✝ : α
⊢ (s.inf' H f) x✝ = s.inf' H (fun i ↦ ⇑(f i)) x✝
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f g : α →C_c ℝ≥0
x✝ : α
⊢ (f + g).toReal x✝ = (f.toReal + g.toReal) x✝

Tactic:
;

Goal state after:
α : Type u_2
inst✝ : TopologicalSpace α
f g : α →C_c ℝ≥0
x✝ : α
⊢ (f + g).toReal x✝ = (f.toReal + g.toReal) x✝
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
r : ℝ≥0
f : α →C_c ℝ≥0
x✝ : α
⊢ (r • f).toReal x✝ = (r • f.toReal) x✝

Tactic:
;

Goal state after:
α : Type u_2
inst✝ : TopologicalSpace α
r : ℝ≥0
f : α →C_c ℝ≥0
x✝ : α
⊢ (r • f).toReal x✝ = (r • f.toReal) x✝
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ
x : α
⊢ (f.nnrealPart.toReal - (-f).nnrealPart.toReal) x = f x

Tactic:
;

Goal state after:
α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ
x : α
⊢ (f.nnrealPart.toReal - (-f).nnrealPart.toReal) x = f x
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
f g : α →C_c ℝ≥0
x : α
⊢ (f + g).toReal x = (f.toReal + g.toReal) x

Tactic:
;

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
f g : α →C_c ℝ≥0
x : α
⊢ (f + g).toReal x = (f.toReal + g.toReal) x
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
a : ℝ≥0
f : α →C_c ℝ≥0
x : α
⊢ (a • f).toReal x = ((RingHom.id ℝ≥0) a • f.toReal) x

Tactic:
;

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
a : ℝ≥0
f : α →C_c ℝ≥0
x : α
⊢ (a • f).toReal x = ((RingHom.id ℝ≥0) a • f.toReal) x
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ≥0
x : α
⊢ ↑(f.toReal.nnrealPart x) = ↑(f x)

Tactic:
;

Goal state after:
α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ≥0
x : α
⊢ ↑(f.toReal.nnrealPart x) = ↑(f x)
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ≥0
x : α
⊢ ↑((-f.toReal).nnrealPart x) = ↑(0 x)

Tactic:
;

Goal state after:
α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ≥0
x : α
⊢ ↑((-f.toReal).nnrealPart x) = ↑(0 x)
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ) →ₚ[ℝ] ℝ
f g : α →C_c ℝ≥0
⊢ NNReal.mk (Λ f.toReal + Λ g.toReal) ⋯ = NNReal.mk (Λ f.toReal) ⋯ + NNReal.mk (Λ g.toReal) ⋯

Tactic:
;

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ) →ₚ[ℝ] ℝ
f g : α →C_c ℝ≥0
⊢ NNReal.mk (Λ f.toReal + Λ g.toReal) ⋯ = NNReal.mk (Λ f.toReal) ⋯ + NNReal.mk (Λ g.toReal) ⋯
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ) →ₚ[ℝ] ℝ
a : ℝ≥0
f : α →C_c ℝ≥0
⊢ NNReal.mk (↑a * Λ f.toReal) ⋯ = a * NNReal.mk (Λ f.toReal) ⋯

Tactic:
;

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ) →ₚ[ℝ] ℝ
a : ℝ≥0
f : α →C_c ℝ≥0
⊢ NNReal.mk (↑a * Λ f.toReal) ⋯ = a * NNReal.mk (Λ f.toReal) ⋯
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: case e_a.e_a
p q b : ℝ
hp : 0 < p
hq : -1 < q
hb : 0 < b
⊢ p⁻¹ * -1 + -p⁻¹ * q = -(q + 1) / p

Tactic:
;

Goal state after:
case e_a.e_a
p q b : ℝ
hp : 0 < p
hq : -1 < q
hb : 0 < b
⊢ p⁻¹ * -1 + -p⁻¹ * q = -(q + 1) / p
Module: Mathlib.MeasureTheory.Integral.Indicator

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
A : Set α
ι : Type u_2
L : Filter ι
inst✝ : L.IsCountablyGenerated
As : ι → Set α
μ : Measure α
A_mble : MeasurableSet A
As_mble : ∀ (i : ι), MeasurableSet (As i)
B : Set α
B_mble : MeasurableSet B
B_finmeas : μ B ≠ ∞
As_le_B : ∀ᶠ (i : ι) in L, As i ⊆ B
h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
⊢ Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A))

Tactic:
← MeasureTheory.lintegral_indicator_one A_mble,

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
A : Set α
ι : Type u_2
L : Filter ι
inst✝ : L.IsCountablyGenerated
As : ι → Set α
μ : Measure α
A_mble : MeasurableSet A
As_mble : ∀ (i : ι), MeasurableSet (As i)
B : Set α
B_mble : MeasurableSet B
B_finmeas : μ B ≠ ∞
As_le_B : ∀ᶠ (i : ι) in L, As i ⊆ B
h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
⊢ Tendsto (fun i ↦ μ (As i)) L (𝓝 (∫⁻ (a : α), A.indicator 1 a ∂μ))
Module: Mathlib.MeasureTheory.Integral.Indicator

Goal state before:
pp: case inr
α : Type u_1
inst✝² : MeasurableSpace α
A : Set α
ι : Type u_2
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
As : ι → Set α
μ : Measure α
inst✝ : IsFiniteMeasure μ
As_mble : ∀ (i : ι), MeasurableSet (As i)
h_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
h✝ : L.NeBot
⊢ Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A))

Tactic:
apply tendsto_measure_of_ae_tendsto_indicator_of_isFiniteMeasure L ?_ As_mble (ae_of_all μ h_lim)

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
A : Set α
ι : Type u_2
L : Filter ι
inst✝¹ : L.IsCountablyGenerated
As : ι → Set α
μ : Measure α
inst✝ : IsFiniteMeasure μ
As_mble : ∀ (i : ι), MeasurableSet (As i)
h_lim : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
h✝ : L.NeBot
⊢ MeasurableSet A
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: case e'_7
p q : ℝ
hp : 1 ≤ p
hq : -2 < q
x✝ : ℝ × ℝ
⊢ x✝.1 * (|x✝.1| ^ q * rexp (-|x✝.1| ^ p)) = x✝.1 * |x✝.1| ^ q * rexp (-|x✝.1| ^ p)

Tactic:
;

Goal state after:
case e'_7
p q : ℝ
hp : 1 ≤ p
hq : -2 < q
x✝ : ℝ × ℝ
⊢ x✝.1 * (|x✝.1| ^ q * rexp (-|x✝.1| ^ p)) = x✝.1 * |x✝.1| ^ q * rexp (-|x✝.1| ^ p)
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: case e'_7
p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
x✝ : ℝ × ℝ
⊢ x✝.1 * (|x✝.1| ^ q * rexp (-b * |x✝.1| ^ p)) = x✝.1 * |x✝.1| ^ q * rexp (-b * |x✝.1| ^ p)

Tactic:
;

Goal state after:
case e'_7
p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
x✝ : ℝ × ℝ
⊢ x✝.1 * (|x✝.1| ^ q * rexp (-b * |x✝.1| ^ p)) = x✝.1 * |x✝.1| ^ q * rexp (-b * |x✝.1| ^ p)
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
inst✝ : NormedSpace ℝ E
C : ℝ
h : ∀ z ∈ [a -[ℝ] b], ‖ω z‖ ≤ C
this : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
⊢ ‖∫ᶜ (x : E) in Path.segment a b, ω x‖ ≤ C * ‖b - a‖ * |1 - 0|

Tactic:
curveIntegral_segment

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
inst✝ : NormedSpace ℝ E
C : ℝ
h : ∀ z ∈ [a -[ℝ] b], ‖ω z‖ ≤ C
this : NormedSpace ℝ F := NormedSpace.restrictScalars ℝ 𝕜 F
⊢ ‖∫ (t : ℝ) in 0..1, (ω ((lineMap a b) t)) (b - a)‖ ≤ C * ‖b - a‖ * |1 - 0|
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : E
ω₁ ω₂ : E → E →L[𝕜] F
γ : Path a b
x✝ : ℝ
⊢ curveIntegralFun (ω₁ + ω₂) γ x✝ = (curveIntegralFun ω₁ γ + curveIntegralFun ω₂ γ) x✝

Tactic:
;

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : E
ω₁ ω₂ : E → E →L[𝕜] F
γ : Path a b
x✝ : ℝ
⊢ curveIntegralFun (ω₁ + ω₂) γ x✝ = (curveIntegralFun ω₁ γ + curveIntegralFun ω₂ γ) x✝
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : E
γ : Path a b
x✝ : ℝ
⊢ curveIntegralFun 0 γ x✝ = 0 x✝

Tactic:
;

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : E
γ : Path a b
x✝ : ℝ
⊢ curveIntegralFun 0 γ x✝ = 0 x✝
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
γ : Path a b
x✝ : ℝ
⊢ curveIntegralFun (-ω) γ x✝ = (-curveIntegralFun ω γ) x✝

Tactic:
;

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
a b : E
ω : E → E →L[𝕜] F
γ : Path a b
x✝ : ℝ
⊢ curveIntegralFun (-ω) γ x✝ = (-curveIntegralFun ω γ) x✝
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
this : ν s = ⊤
⊢ False

Tactic:
;

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
this : ν s = ⊤
⊢ False
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := ⋯
ν : Measure α := ⋯
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
this : ν s = ⊤
⊢ (ν s).toReal = 0

Tactic:
;

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := ⋯
ν : Measure α := ⋯
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
this : ν s = ⊤
⊢ (ν s).toReal = 0
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
this : ν s = 0
⊢ False

Tactic:
;

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := fun x ↦ ENNReal.ofReal (g x)
ν : Measure α := μ.withDensity ρ
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
this : ν s = 0
⊢ False
Module: Mathlib.MeasureTheory.Integral.MeanValue

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := ⋯
ν : Measure α := ⋯
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
this : ν s = 0
⊢ (ν s).toReal = 0

Tactic:
;

Goal state after:
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
s : Set α
f g : α → ℝ
μ : Measure α
hs_conn : IsConnected s
hs_meas : MeasurableSet s
hf : ContinuousOn f s
hg : IntegrableOn g s μ
hfg : IntegrableOn (fun x ↦ f x * g x) s μ
hg0 : ∀ᵐ (x : α) ∂μ.restrict s, 0 ≤ g x
ρ : α → ENNReal := ⋯
ν : Measure α := ⋯
hρ_ae : AEMeasurable ρ (μ.restrict s)
hρ_top : ∀ᵐ (x : α) ∂μ.restrict s, ρ x < ⊤
h_toReal_ae : (fun x ↦ (ρ x).toReal) =ᵐ[μ.restrict s] g
heq : ∫ (x : α) in s, f x * g x ∂μ = ∫ (x : α) in s, f x ∂ν
hg1 : ∫ (x : α) in s, g x ∂μ = ∫ (x : α) in s, 1 ∂ν
hν0 : ¬∫ (x : α) in s, 1 ∂ν = 0
hν0' : (ν s).toReal ≠ 0
hνfin : ν s ≠ ⊤
this : ν s = 0
⊢ (ν s).toReal = 0
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral

Goal state before:
pp: case hfB
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C : ℝ
hab : a ≤ b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C
s : Set ℝ := toMeasurable volume {x | deriv f x ≠ 0}
hsm : MeasurableSet s
t : ℝ
ht : t ∈ Ioo a b → ‖deriv f t‖ ≤ C
ht_mem : t ∈ Ioo a b
⊢ ‖deriv f t‖ ≤ s.indicator (fun x ↦ C) t

Tactic:
apply le_indicator_apply

Goal state after:
case hfB.hfg
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C : ℝ
hab : a ≤ b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C
s : Set ℝ := ⋯
hsm : MeasurableSet s
t : ℝ
ht : t ∈ Ioo a b → ‖deriv f t‖ ≤ C
ht_mem : t ∈ Ioo a b
⊢ t ∈ s → ‖deriv f t‖ ≤ C

case hfB.hf
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
a b C : ℝ
hab : a ≤ b
hfc : ContinuousOn f (Icc a b)
hfd : DifferentiableOn ℝ f (Ioo a b)
hnorm : ∀ᵐ (t : ℝ), t ∈ Ioo a b → ‖deriv f t‖ ≤ C
s : Set ℝ := ⋯
hsm : MeasurableSet s
t : ℝ
ht : t ∈ Ioo a b → ‖deriv f t‖ ≤ C
ht_mem : t ∈ Ioo a b
⊢ t ∉ s → ‖deriv f t‖ ≤ 0
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
this : ∀ k ∈ Finset.range N, |trapezoidal_error f 1 (ak k) (ak (k + 1))| ≤ ζ / 12 * h ^ 3
⊢ ↑N * (ζ / 12 * h ^ 3) = (a + ↑N * h - a) ^ 3 * ζ / (12 * ↑N ^ 2)

Tactic:
;

Goal state after:
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
this : ∀ k ∈ Finset.range N, |trapezoidal_error f 1 (ak k) (ak (k + 1))| ≤ ζ / 12 * h ^ 3
⊢ ↑N * (ζ / 12 * h ^ 3) = (a + ↑N * h - a) ^ 3 * ζ / (12 * ↑N ^ 2)
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
⊢ 0 ≤ ↑k * h

Tactic:
;

Goal state after:
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
⊢ 0 ≤ ↑k * h
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
⊢ a + ↑(k + 1) * h ≤ a + ↑N * h

Tactic:
;

Goal state after:
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
⊢ a + ↑(k + 1) * h ≤ a + ↑N * h
Module: Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule

Goal state before:
pp: f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
x : ℝ
hx : x ∉ Set.Icc (ak k) (ak (k + 1))
⊢ x ∉ closure (Set.Icc (ak k) (ak (k + 1)))

Tactic:
closure_Icc

Goal state after:
f : ℝ → ℝ
ζ a b : ℝ
a_lt_b : a < b
h_df : DifferentiableOn ℝ f (Set.Icc a b)
h_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)
fpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ
N : ℕ
N_nonzero : 0 < N
h : ℝ := (b - a) / ↑N
ak : ℕ → ℝ := fun k ↦ a + ↑k * h
h0 : ∀ (k : ℕ), ak (k + 1) - ak k = h
hab : 0 < b - a
hpos : 0 < h
hb : b = a + ↑N * h
k : ℕ
hk : k < N
h1 : a ≤ ak k
h2 : ak (k + 1) ≤ b
h3 : Set.Icc (ak k) (ak (k + 1)) ⊆ Set.Icc a b
h4 : ak k < ak (k + 1)
h5 :
  EqOn (_root_.derivWithin f (Set.Icc a b)) (_root_.derivWithin f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
h6 :
  EqOn (iteratedDerivWithin 2 f (Set.Icc a b)) (iteratedDerivWithin 2 f (Set.Icc (ak k) (ak (k + 1))))
    (Set.Icc (ak k) (ak (k + 1)))
x : ℝ
hx : x ∉ Set.Icc (ak k) (ak (k + 1))
⊢ x ∉ Set.Icc (ak k) (ak (k + 1))
Module: Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.NNReal

Goal state before:
pp: X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f : X →C_c ℝ≥0
a : X
⊢ 0 a ≤ (fun a ↦ ↑(f a)) a

Tactic:
;

Goal state after:
X : Type u_1
inst✝⁴ : TopologicalSpace X
inst✝³ : T2Space X
inst✝² : LocallyCompactSpace X
inst✝¹ : MeasurableSpace X
inst✝ : BorelSpace X
Λ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f : X →C_c ℝ≥0
a : X
⊢ 0 a ≤ (fun a ↦ ↑(f a)) a
Module: Mathlib.MeasureTheory.Integral.TorusIntegral

Goal state before:
pp: n : ℕ
c : Fin n → ℂ
R θ : Fin n → ℝ
i : Fin n
⊢ (torusMap c R θ - c) i = torusMap 0 R θ i

Tactic:
;

Goal state after:
n : ℕ
c : Fin n → ℂ
R θ : Fin n → ℝ
i : Fin n
⊢ (torusMap c R θ - c) i = torusMap 0 R θ i