mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Metrizable.Urysohn

Goal state before:
pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure[inst✝²] UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
f : X → ↑s →ᵇ ℝ
hf : IsInducing f
⊢ ∃ f, IsInducing f

pp: X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := ⋯
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⋯
this : DiscreteTopology ↑s
⊢ ∃ f, IsInducing f

Tactic:
·

Goal state after:
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure[inst✝²] UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
f : X → ↑s →ᵇ ℝ
hf : IsInducing f
⊢ ∃ f, IsInducing f
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: case he
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ μ ∅ = 0

pp: case hmono
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, s ⊆ t → μ t = 0 → μ s = 0

pp: case hunion
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, μ s = 0 → μ t = 0 → μ (s ∪ t) = 0

pp: case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, μ t = 0

Tactic:
·

Goal state after:
case he
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ μ ∅ = 0
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: case hmono
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, s ⊆ t → μ t = 0 → μ s = 0

pp: case hunion
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, μ s = 0 → μ t = 0 → μ (s ∪ t) = 0

pp: case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, μ t = 0

Tactic:
·

Goal state after:
case hmono
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, s ⊆ t → μ t = 0 → μ s = 0
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: case hunion
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, μ s = 0 → μ t = 0 → μ (s ∪ t) = 0

pp: case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, μ t = 0

Tactic:
·

Goal state after:
case hunion
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ ⦃s t : Set α⦄, μ s = 0 → μ t = 0 → μ (s ∪ t) = 0
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, μ t = 0

Tactic:
·

Goal state after:
case hnhds
α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s k : Set α
hk : IsCompact k
h'k : k ⊆ s \ μ.everywherePosSubset s
⊢ ∀ x ∈ k, ∃ t ∈ 𝓝[k] x, μ t = 0
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case e'_3
G : Type uG
E' : Type uE'
inst✝⁸ : NormedAddCommGroup E'
g : G → E'
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : BorelSpace G
inst✝⁴ : SecondCountableTopology G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : SFinite μ
inst✝¹ : NormedSpace ℝ E'
inst✝ : CompleteSpace E'
f : G → ℝ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hf : support f ⊆ ball 0 R
hnf : ∀ (x : G), 0 ≤ f x
hintf : ∫ (x : G), f x ∂μ = 1
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
hif : Integrable f μ
⊢ z₀ = ∫ (t : G), ((lsmul ℝ ℝ) (f t)) z₀ ∂μ

pp: case convert_2
G : Type uG
E' : Type uE'
inst✝⁸ : NormedAddCommGroup E'
g : G → E'
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : BorelSpace G
inst✝⁴ : SecondCountableTopology G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : SFinite μ
inst✝¹ : NormedSpace ℝ E'
inst✝ : CompleteSpace E'
f : G → ℝ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hf : support f ⊆ ball 0 R
hnf : ∀ (x : G), 0 ≤ f x
hintf : ∫ (x : G), f x ∂μ = 1
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
hif : Integrable f μ
⊢ (‖lsmul ℝ ℝ‖ * ∫ (x : G), ‖f x‖ ∂μ) * ε ≤ ε

Tactic:
·

Goal state after:
case e'_3
G : Type uG
E' : Type uE'
inst✝⁸ : NormedAddCommGroup E'
g : G → E'
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : BorelSpace G
inst✝⁴ : SecondCountableTopology G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : SFinite μ
inst✝¹ : NormedSpace ℝ E'
inst✝ : CompleteSpace E'
f : G → ℝ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hf : support f ⊆ ball 0 R
hnf : ∀ (x : G), 0 ≤ f x
hintf : ∫ (x : G), f x ∂μ = 1
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
hif : Integrable f μ
⊢ z₀ = ∫ (t : G), ((lsmul ℝ ℝ) (f t)) z₀ ∂μ
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case convert_2
G : Type uG
E' : Type uE'
inst✝⁸ : NormedAddCommGroup E'
g : G → E'
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : BorelSpace G
inst✝⁴ : SecondCountableTopology G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : SFinite μ
inst✝¹ : NormedSpace ℝ E'
inst✝ : CompleteSpace E'
f : G → ℝ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hf : support f ⊆ ball 0 R
hnf : ∀ (x : G), 0 ≤ f x
hintf : ∫ (x : G), f x ∂μ = 1
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
hif : Integrable f μ
⊢ (‖lsmul ℝ ℝ‖ * ∫ (x : G), ‖f x‖ ∂μ) * ε ≤ ε

Tactic:
·

Goal state after:
case convert_2
G : Type uG
E' : Type uE'
inst✝⁸ : NormedAddCommGroup E'
g : G → E'
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : BorelSpace G
inst✝⁴ : SecondCountableTopology G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : SFinite μ
inst✝¹ : NormedSpace ℝ E'
inst✝ : CompleteSpace E'
f : G → ℝ
x₀ : G
R ε : ℝ
z₀ : E'
hε : 0 ≤ ε
hf : support f ⊆ ball 0 R
hnf : ∀ (x : G), 0 ≤ f x
hintf : ∫ (x : G), f x ∂μ = 1
hmg : AEStronglyMeasurable g μ
hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε
hif : Integrable f μ
⊢ (‖lsmul ℝ ℝ‖ * ∫ (x : G), ‖f x‖ ∂μ) * ε ≤ ε
Module: Mathlib.Topology.Metrizable.Urysohn

Goal state before:
pp: case refine_1
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure[inst✝²] UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure[inst✝²] (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1
hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x ↦ ε UV) (↑UV).2ᶜ
hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ := fun x ↦ { toFun := fun UV ↦ (f UV) x, continuous_toFun := ⋯, map_bounded' := ⋯ }
hF : ∀ (x : X) (UV : ↑s), (F x) UV = (f UV) x
x : X
⊢ 𝓝 x ≤ comap F (𝓝 (F x))

pp: case refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure[inst✝²] UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure[inst✝²] (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1
hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x ↦ ε UV) (↑UV).2ᶜ
hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ := fun x ↦ { toFun := fun UV ↦ (f UV) x, continuous_toFun := ⋯, map_bounded' := ⋯ }
hF : ∀ (x : X) (UV : ↑s), (F x) UV = (f UV) x
x : X
⊢ comap F (𝓝 (F x)) ≤ 𝓝 x

Tactic:
·

Goal state after:
case refine_1
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure[inst✝²] UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure[inst✝²] (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1
hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x ↦ ε UV) (↑UV).2ᶜ
hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ := fun x ↦ { toFun := fun UV ↦ (f UV) x, continuous_toFun := ⋯, map_bounded' := ⋯ }
hF : ∀ (x : X) (UV : ↑s), (F x) UV = (f UV) x
x : X
⊢ 𝓝 x ≤ comap F (𝓝 (F x))
Module: Mathlib.Topology.Metrizable.Urysohn

Goal state before:
pp: case refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure[inst✝²] UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure[inst✝²] (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1
hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x ↦ ε UV) (↑UV).2ᶜ
hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ := fun x ↦ { toFun := fun UV ↦ (f UV) x, continuous_toFun := ⋯, map_bounded' := ⋯ }
hF : ∀ (x : X) (UV : ↑s), (F x) UV = (f UV) x
x : X
⊢ comap F (𝓝 (F x)) ≤ 𝓝 x

Tactic:
·

Goal state after:
case refine_2
X : Type u_1
inst✝² : TopologicalSpace X
inst✝¹ : RegularSpace X
inst✝ : SecondCountableTopology X
B : Set (Set X)
hBc : B.Countable
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure[inst✝²] UV.1 ⊆ UV.2}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure[inst✝²] (↑UV).1) (↑UV).2ᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (⇑(f UV)) 0 (↑UV).1
hfε : ∀ (UV : ↑s), EqOn (⇑(f UV)) (fun x ↦ ε UV) (↑UV).2ᶜ
hf0ε : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), (f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ := fun x ↦ { toFun := fun UV ↦ (f UV) x, continuous_toFun := ⋯, map_bounded' := ⋯ }
hF : ∀ (x : X) (UV : ↑s), (F x) UV = (f UV) x
x : X
⊢ comap F (𝓝 (F x)) ≤ 𝓝 x
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: case refine_1
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
K : Set G
K_comp : IsCompact K
K_closed : IsClosed[inst✝⁷] K
r : ℝ≥0∞
hr : r < μ K
L : Set G := μ.everywherePosSubset K
L_comp : IsCompact L
L_closed : IsClosed[inst✝⁷] L
⊢ (fun s ↦ ∃ f, Continuous[inst✝⁷, _] f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) L

pp: case refine_2
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
K : Set G
K_comp : IsCompact K
K_closed : IsClosed[inst✝⁷] K
r : ℝ≥0∞
hr : r < μ K
L : Set G := μ.everywherePosSubset K
L_comp : IsCompact L
L_closed : IsClosed[inst✝⁷] L
⊢ r < μ L

Tactic:
·

Goal state after:
case refine_1
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
K : Set G
K_comp : IsCompact K
K_closed : IsClosed[inst✝⁷] K
r : ℝ≥0∞
hr : r < μ K
L : Set G := μ.everywherePosSubset K
L_comp : IsCompact L
L_closed : IsClosed[inst✝⁷] L
⊢ (fun s ↦ ∃ f, Continuous[inst✝⁷, _] f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) L
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: case refine_2
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
K : Set G
K_comp : IsCompact K
K_closed : IsClosed[inst✝⁷] K
r : ℝ≥0∞
hr : r < μ K
L : Set G := μ.everywherePosSubset K
L_comp : IsCompact L
L_closed : IsClosed[inst✝⁷] L
⊢ r < μ L

Tactic:
·

Goal state after:
case refine_2
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
K : Set G
K_comp : IsCompact K
K_closed : IsClosed[inst✝⁷] K
r : ℝ≥0∞
hr : r < μ K
L : Set G := μ.everywherePosSubset K
L_comp : IsCompact L
L_closed : IsClosed[inst✝⁷] L
⊢ r < μ L
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case hf
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
k : Set G := ⋯
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := ⋯
k'_comp : IsCompact k'
⊢ ContinuousOn (uncurry fun x y ↦ g (y⁻¹ * x)) (t ×ˢ univ)

pp: case hfs
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
k : Set G := ⋯
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := ⋯
k'_comp : IsCompact k'
⊢ ∀ (p x : G), p ∈ t → x ∉ k' → g (x⁻¹ * p) = 0

Tactic:
·

Goal state after:
case hf
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
k : Set G := ⋯
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := ⋯
k'_comp : IsCompact k'
⊢ ContinuousOn (uncurry fun x y ↦ g (y⁻¹ * x)) (t ×ˢ univ)
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case hfs
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
k : Set G := ⋯
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := ⋯
k'_comp : IsCompact k'
⊢ ∀ (p x : G), p ∈ t → x ∉ k' → g (x⁻¹ * p) = 0

Tactic:
·

Goal state after:
case hfs
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : LocallyCompactSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : IsFiniteMeasureOnCompacts μ
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
g : G → E
hg : Continuous[inst✝⁸, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
h'g : HasCompactSupport g
k : Set G := ⋯
k_comp : IsCompact k
x₀ : G
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := ⋯
k'_comp : IsCompact k'
⊢ ∀ (p x : G), p ∈ t → x ∉ k' → g (x⁻¹ * p) = 0
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case inl
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : f = 0
⊢ ∫ (x : G), f x ∂μ = (∫ (y : G), f y * (∫ (z : G), g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ (x : G), g x ∂μ

pp: case inr
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : LocallyCompactSpace G
⊢ ∫ (x : G), f x ∂μ = (∫ (y : G), f y * (∫ (z : G), g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ (x : G), g x ∂μ

Tactic:
·

Goal state after:
case inl
G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : f = 0
⊢ ∫ (x : G), f x ∂μ = (∫ (y : G), f y * (∫ (z : G), g (z⁻¹ * y) ∂ν)⁻¹ ∂ν) * ∫ (x : G), g x ∂μ
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ HasCompactSupport f

pp: case refine_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ ContDiff ℝ (↑n) f

pp: case refine_3
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ range f ⊆ Icc 0 1

pp: case refine_4
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ f x = 1

Tactic:
·

Goal state after:
case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s : Set E
x : E
n : ℕ∞
hs : s ∈ 𝓝 x
d : ℝ
d_pos : 0 < d
hd : Euclidean.closedBall x d ⊆ s
c : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }
f : E → ℝ := ↑c ∘ ⇑toEuclidean
f_supp : support f ⊆ Euclidean.ball x d
f_tsupp : tsupport f ⊆ Euclidean.closedBall x d
⊢ HasCompactSupport f