mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: case refine_1
α : Type u_2
inst✝ : TopologicalSpace α
f₁ f₂ : α →C_c ℝ≥0
h : f₁ ≤ f₂
x : α
hx : x ∈ Function.support (f₂.toContinuousMap - f₁.toContinuousMap).toFun
⊢ x ∈ Function.support f₁.toFun ∪ Function.support f₂.toFun

Tactic:
contrapose hx

Goal state after:
case refine_1
α : Type u_2
inst✝ : TopologicalSpace α
f₁ f₂ : α →C_c ℝ≥0
h : f₁ ≤ f₂
x : α
hx : x ∉ Function.support f₁.toFun ∪ Function.support f₂.toFun
⊢ x ∉ Function.support (f₂.toContinuousMap - f₁.toContinuousMap).toFun
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f₁ f₂ : α →C_c ℝ≥0
h : f₁ ≤ f₂
⊢ ∃ g, f₁ + g = f₂

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ
hf : 0 ≤ f
⊢ (-f).nnrealPart = 0

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ
a : ℝ
ha : 0 ≤ a
⊢ (a • f).nnrealPart = a.toNNReal • f.nnrealPart

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ
a : ℝ
ha : a ≤ 0
⊢ (a • f).nnrealPart = (-a).toNNReal • (-f).nnrealPart

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f g : α →C_c ℝ
⊢ (f + g).nnrealPart ≤ f.nnrealPart + g.nnrealPart

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f g : α →C_c ℝ
⊢ ∃ h,
    (f + g).nnrealPart + h = f.nnrealPart + g.nnrealPart ∧ (-f + -g).nnrealPart + h = (-f).nnrealPart + (-g).nnrealPart

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
f : α →C_c ℝ≥0
x✝ : α
⊢ 0 x✝ ≤ f.toReal x✝

Tactic:
by

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

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

Tactic:
by

Goal state after:
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
⊢ (r • f).toReal = r • f.toReal

Tactic:
by

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

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

Tactic:
by

Goal state after:
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
⊢ (f + g).toReal = f.toReal + g.toReal

Tactic:
by

Goal state after:
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
⊢ (a • f).toReal = (RingHom.id ℝ≥0) a • f.toReal

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

Goal state after:
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 : α →C_c ℝ≥0
⊢ 0 ≤ toRealLinearMap f

Tactic:
by

Goal state after:
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 (toRealLinearMap (f + g))) ⋯ = NNReal.mk (Λ (toRealLinearMap f)) ⋯ + NNReal.mk (Λ (toRealLinearMap g)) ⋯

Tactic:
by

Goal state after:
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 (Λ (toRealLinearMap (a • f))) ⋯ = (RingHom.id ℝ≥0) a • NNReal.mk (Λ (toRealLinearMap f)) ⋯

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
Λ₁ Λ₂ : (α →C_c ℝ) →ₚ[ℝ] ℝ
h : Λ₁ = Λ₂
⊢ toNNRealLinear Λ₁ = toNNRealLinear Λ₂

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
Λ₁ Λ₂ : (α →C_c ℝ) →ₚ[ℝ] ℝ
⊢ toNNRealLinear Λ₁ = toNNRealLinear Λ₂ ↔ Λ₁ = Λ₂

Tactic:
by

Goal state after:
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≥0) →ₗ[ℝ≥0] ℝ≥0
f g : α →C_c ℝ
⊢ ↑(Λ (f + g).nnrealPart) - ↑(Λ (-(f + g)).nnrealPart) =
    ↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart) + (↑(Λ g.nnrealPart) - ↑(Λ (-g).nnrealPart))

Tactic:
by

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

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

Tactic:
by

Goal state after:
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≥0) →ₗ[ℝ≥0] ℝ≥0
a : ℝ
f : α →C_c ℝ
⊢ ↑(Λ (a • f).nnrealPart) - ↑(Λ (-(a • f)).nnrealPart) = (RingHom.id ℝ) a • (↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart))

Tactic:
by

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

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

Tactic:
by

Goal state after:
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≥0) →ₗ[ℝ≥0] ℝ≥0
g : α →C_c ℝ
hg : 0 ≤ g
⊢ 0 ≤ { toFun := fun f ↦ ↑(Λ f.nnrealPart) - ↑(Λ (-f).nnrealPart), map_add' := ⋯, map_smul' := ⋯ } g

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
f : α →C_c ℝ≥0
⊢ (toRealPositiveLinear Λ) f.toReal = ↑(Λ f)

Tactic:
by

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

Goal state before:
pp: α : Type u_2
inst✝ : TopologicalSpace α
Λ : (α →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0
⊢ toNNRealLinear (toRealPositiveLinear Λ) = Λ

Tactic:
by

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

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

Tactic:
by

Goal state after:
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✝⁷ : TopologicalSpace α
inst✝⁶ : R1Space α
inst✝⁵ : Group α
inst✝⁴ : TopologicalSpace β
inst✝³ : R1Space β
inst✝² : Group β
inst✝¹ : ContinuousMul β
inst✝ : NormedAddCommGroup γ
φ : α →* β
hφ : Topology.IsClosedEmbedding ⇑φ
f : β →C_c γ
b : β
⊢ Continuous fun a ↦ f (b * φ a)

Tactic:
by

Goal state after:
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✝⁷ : TopologicalSpace α
inst✝⁶ : R1Space α
inst✝⁵ : Group α
inst✝⁴ : TopologicalSpace β
inst✝³ : R1Space β
inst✝² : Group β
inst✝¹ : ContinuousMul β
inst✝ : NormedAddCommGroup γ
φ : α →* β
hφ : Topology.IsClosedEmbedding ⇑φ
f : β →C_c γ
b : β
⊢ HasCompactSupport fun a ↦ f (b * φ a)

Tactic:
by

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

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

Tactic:
by

Goal state after:
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
i : ι
hi : As i ⊆ B
x : α
⊢ (As i).indicator 1 x ≤ B.indicator 1 x

Tactic:
by

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

Goal state before:
pp: p b : ℝ
hp : 0 < p
hb : 0 < b
⊢ ∫ (x : ℝ) in Ioi 0, rexp (-b * x ^ p) = b ^ (-1 / p) * Gamma (1 / p + 1)

Tactic:
by

Goal state after:
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:
by

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

Goal state before:
pp: 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
A_mble : MeasurableSet A
As_mble : ∀ (i : ι), MeasurableSet (As i)
h_lim : ∀ᵐ (x : α) ∂μ, ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
⊢ μ univ ≠ ∞

Tactic:
by

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

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ ∫ (x : ℂ), ‖x‖ ^ q * rexp (-‖x‖ ^ p) = ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-|x.1| ^ p))

Tactic:
by

Goal state after:
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
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 : ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A
⊢ Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A))

Tactic:
by

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

Goal state before:
pp: 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
⊢ Tendsto (fun i ↦ μ (As i)) L (𝓝 (μ A))

Tactic:
by

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

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-|x.1| ^ p)) =
    (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)) * ∫ (x : ℝ) in Ioo (-π) π, 1

Tactic:
by

Goal state after:
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
h : ¬CompleteSpace F
ω : E → E →L[𝕜] F
γ : Path a b
⊢ ∫ᶜ (x : E) in γ, ω x = 0

Tactic:
by

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

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ -π ≤ π

Tactic:
by

Goal state after:
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
inst✝ : NormedSpace ℝ E
ω : E → E →L[𝕜] F
γ : Path a b
t : ℝ
⊢ curveIntegralFun ω γ t = (ω (γ.extend t)) (derivWithin ⇑γ.extend I t)

Tactic:
by

Goal state after:
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
inst✝ : NormedSpace ℝ F
ω : E → E →L[𝕜] F
γ : Path a b
⊢ curveIntegral ω γ = ∫ (t : ℝ) in 0..1, curveIntegralFun ω γ t

Tactic:
by

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

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)) * ∫ (x : ℝ) in Ioo (-π) π, 1 =
    2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
a b : E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
ω : E → E →L[𝕜] F
γ : Path a b
⊢ MeasurableSet (Ioo 0 1)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
a b : E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
ω : E → E →L[𝕜] F
γ : Path a b
x : ℝ
hx : x ∈ Ioo 0 1
⊢ I ∈ 𝓝 x

Tactic:
by

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

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ 2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-|x| ^ p) = 2 * π * ∫ (x : ℝ) in Ioi 0, x ^ (q + 1) * rexp (-x ^ p)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CurveIntegral.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁶ : RCLike 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
a b : E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace ℝ F
ω : E → E →L[𝕜] F
γ : Path a b
⊢ ∫ᶜ (x : E) in γ, ω x = ∫ (t : ℝ) in 0..1, (ω (γ.extend t)) (deriv (⇑γ.extend) t)

Tactic:
by

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

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ 0 < p

Tactic:
by

Goal state after:
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
ω : E → E →L[𝕜] F
a : E
⊢ curveIntegralFun ω (Path.refl a) = 0

Tactic:
by

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

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ -1 < q + 1

Tactic:
by

Goal state after:
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
ω : E → E →L[𝕜] F
a : E
⊢ ∫ᶜ (x : E) in Path.refl a, ω x = 0

Tactic:
by

Goal state after:
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
ω : E → E →L[𝕜] F
a : E
⊢ CurveIntegrable ω (Path.refl a)

Tactic:
by

Goal state after:
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 c d : E
ω : E → E →L[𝕜] F
γ : Path a b
hc : c = a
hd : d = b
⊢ curveIntegralFun ω (γ.cast hc hd) = curveIntegralFun ω γ

Tactic:
by

Goal state after:
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 c d : E
ω : E → E →L[𝕜] F
γ : Path a b
hc : c = a
hd : d = b
⊢ ∫ᶜ (x : E) in γ.cast hc hd, ω x = ∫ᶜ (x : E) in γ, ω x

Tactic:
by

Goal state after:
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 c d : E
ω : E → E →L[𝕜] F
γ : Path a b
hc : c = a
hd : d = b
⊢ CurveIntegrable ω (γ.cast hc hd) ↔ CurveIntegrable ω γ

Tactic:
by

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

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ 2 * π * ∫ (x : ℝ) in Ioi 0, x ^ (q + 1) * rexp (-x ^ p) = 2 * π / p * Real.Gamma ((q + 2) / p)

Tactic:
by

Goal state after:
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
t : ℝ
⊢ curveIntegralFun ω γ.symm t = -curveIntegralFun ω γ (1 - t)

Tactic:
by

Goal state after:
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
h : CurveIntegrable ω γ
⊢ CurveIntegrable ω γ.symm

Tactic:
by

Goal state after:
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
h : CurveIntegrable ω γ.symm
⊢ CurveIntegrable ω γ

Tactic:
by

Goal state after:
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
⊢ ∫ᶜ (x : E) in γ.symm, ω x = -∫ᶜ (x : E) in γ, ω x

Tactic:
by

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

Goal state before:
pp: p q : ℝ
hp : 1 ≤ p
hq : -2 < q
⊢ ∫ (x : ℂ), ‖x‖ ^ q * rexp (-‖x‖ ^ p) = 2 * π / p * Real.Gamma ((q + 2) / p)

Tactic:
by

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ ∫ (x : ℂ), ‖x‖ ^ q * rexp (-b * ‖x‖ ^ p) =
    ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-b * |x.1| ^ p))

Tactic:
by

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ ∫ (x : ℝ × ℝ) in Ioi 0 ×ˢ Ioo (-π) π, x.1 * (|x.1| ^ q * rexp (-b * |x.1| ^ p)) =
    (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)) * ∫ (x : ℝ) in Ioo (-π) π, 1

Tactic:
by

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ -π ≤ π

Tactic:
by

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ (∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)) * ∫ (x : ℝ) in Ioo (-π) π, 1 =
    2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p)

Tactic:
by

Goal state after:
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 c : E
t : ℝ
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
ht : t < 1 / 2
instE : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
H₁ : ⇑(γab.trans γbc).extend =ᶠ[𝓝 t] fun s ↦ γab.extend (2 * s)
⊢ 2 • I =ᶠ[𝓝 (2 * t)] I

Tactic:
by

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ 2 * π * ∫ (x : ℝ) in Ioi 0, x * |x| ^ q * rexp (-b * |x| ^ p) =
    2 * π * ∫ (x : ℝ) in Ioi 0, x ^ (q + 1) * rexp (-b * x ^ p)

Tactic:
by

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ 0 < p

Tactic:
by

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ -1 < q + 1

Tactic:
by

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ 2 * π * ∫ (x : ℝ) in Ioi 0, x ^ (q + 1) * rexp (-b * x ^ p) =
    2 * π / p * b ^ (-(q + 2) / p) * Real.Gamma ((q + 2) / p)

Tactic:
by

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

Goal state before:
pp: p q b : ℝ
hp : 1 ≤ p
hq : -2 < q
hb : 0 < b
⊢ ∫ (x : ℂ), ‖x‖ ^ q * rexp (-b * ‖x‖ ^ p) = 2 * π / p * b ^ (-(q + 2) / p) * Real.Gamma ((q + 2) / p)

Tactic:
by

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

Goal state before:
pp: p : ℝ
hp : 1 ≤ p
⊢ -2 < 0

Tactic:
by

Goal state after:
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 c : E
t : ℝ
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
ht : t < 1 / 2
⊢ curveIntegralFun ω (γab.trans γbc) t = 2 • curveIntegralFun ω γab (2 * t)

Tactic:
by

Goal state after:
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 c : E
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
⊢ 0 ≤ 1 / 2

Tactic:
by

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

Goal state before:
pp: p : ℝ
hp : 1 ≤ p
⊢ p ≠ 0

Tactic:
by

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

Goal state before:
pp: p : ℝ
hp : 1 ≤ p
⊢ ∫ (x : ℂ), rexp (-‖x‖ ^ p) = π * Real.Gamma (2 / p + 1)

Tactic:
by

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

Goal state before:
pp: p b : ℝ
hp : 1 ≤ p
hb : 0 < b
⊢ -2 < 0

Tactic:
by

Goal state after:
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 c : E
ω : E → E →L[𝕜] F
γab : Path a b
γbc : Path b c
⊢ curveIntegralFun ω (γab.trans γbc) =ᵐ[volume.restrict (Ι 0 (1 / 2))] fun t ↦ 2 • curveIntegralFun ω γab (2 * t)

Tactic:
by

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

Goal state before:
pp: p b : ℝ
hp : 1 ≤ p
hb : 0 < b
⊢ p ≠ 0

Tactic:
by

Goal state after: