mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case hf
α : Type u_1
inst✝⁶ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SecondCountableTopology α
inst✝³ : BorelSpace α
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
a : Set α
ha : 0 < μ a
h'a : μ a < ∞
h''a : IntegrableOn f a μ
⊢ Integrable f ((μ a)⁻¹ • μ.restrict a)

pp: case hg
α : Type u_1
inst✝⁶ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SecondCountableTopology α
inst✝³ : BorelSpace α
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
a : Set α
ha : 0 < μ a
h'a : μ a < ∞
h''a : IntegrableOn f a μ
⊢ Integrable (fun x_1 ↦ f x) ((μ a)⁻¹ • μ.restrict a)

Tactic:
·

Goal state after:
case hf
α : Type u_1
inst✝⁶ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SecondCountableTopology α
inst✝³ : BorelSpace α
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
a : Set α
ha : 0 < μ a
h'a : μ a < ∞
h''a : IntegrableOn f a μ
⊢ Integrable f ((μ a)⁻¹ • μ.restrict a)
Module: Mathlib.MeasureTheory.Covering.Differentiation

Goal state before:
pp: case hg
α : Type u_1
inst✝⁶ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SecondCountableTopology α
inst✝³ : BorelSpace α
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
a : Set α
ha : 0 < μ a
h'a : μ a < ∞
h''a : IntegrableOn f a μ
⊢ Integrable (fun x_1 ↦ f x) ((μ a)⁻¹ • μ.restrict a)

Tactic:
·

Goal state after:
case hg
α : Type u_1
inst✝⁶ : PseudoMetricSpace α
m0 : MeasurableSpace α
μ : Measure α
v : VitaliFamily μ
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : SecondCountableTopology α
inst✝³ : BorelSpace α
inst✝² : IsLocallyFiniteMeasure μ
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : α → E
hf : LocallyIntegrable f μ
x : α
hx : Tendsto (fun a ↦ ⨍ (y : α) in a, ‖f y - f x‖ ∂μ) (v.filterAt x) (𝓝 0)
h'x : ∀ᶠ (a : Set α) in v.filterAt x, 0 < μ a
a : Set α
ha : 0 < μ a
h'a : μ a < ∞
h''a : IntegrableOn f a μ
⊢ Integrable (fun x_1 ↦ f x) ((μ a)⁻¹ • μ.restrict a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMul S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : ¬p a
⊢ cfc (fun x ↦ s • f x) a = s • cfc f a

Tactic:
·

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝¹³ : CommSemiring R
inst✝¹² : StarRing R
inst✝¹¹ : MetricSpace R
inst✝¹⁰ : IsTopologicalSemiring R
inst✝⁹ : ContinuousStar R
inst✝⁸ : TopologicalSpace A
inst✝⁷ : Ring A
inst✝⁶ : StarRing A
inst✝⁵ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
S : Type u_3
inst✝⁴ : SMul S R
inst✝³ : ContinuousConstSMul S R
inst✝² : SMulZeroClass S A
inst✝¹ : IsScalarTower S R A
inst✝ : IsScalarTower S R (R → R)
s : S
f : R → R
a : A
hf : ContinuousOn f (spectrum R a)
ha : ¬p a
⊢ cfc (fun x ↦ s • f x) a = s • cfc f a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : p a ∧ ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

Tactic:
·

Goal state after:
case pos
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : p a ∧ ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

Tactic:
·

Goal state after:
case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

Tactic:
obtain (ha | hf) := not_and_or.mp h

Goal state after:
case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
ha : ¬p a
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
ha : ¬p a
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

pp: case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

Tactic:
·

Goal state after:
case neg.inl
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
ha : ¬p a
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)

Tactic:
·

Goal state after:
case neg.inr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁸ : CommSemiring R
inst✝⁷ : StarRing R
inst✝⁶ : MetricSpace R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : ContinuousStar R
inst✝³ : TopologicalSpace A
inst✝² : Ring A
inst✝¹ : StarRing A
inst✝ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
f : R → R
a : A
h : ¬(p a ∧ ContinuousOn f (spectrum R a))
hf : ¬ContinuousOn f (spectrum R a)
⊢ cfc (fun x ↦ star (f x)) a = star (cfc f a)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inl
α : Type u_1
inst✝⁴ : LinearOrder α
E : Type u_2
inst✝³ : PseudoEMetricSpace E
inst✝² : CompleteSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
hs : s ∩ Iio x = ∅
⊢ ∃ l, Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)

pp: case inr
α : Type u_1
inst✝⁴ : LinearOrder α
E : Type u_2
inst✝³ : PseudoEMetricSpace E
inst✝² : CompleteSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
hs : (s ∩ Iio x).Nonempty
⊢ ∃ l, Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝⁴ : LinearOrder α
E : Type u_2
inst✝³ : PseudoEMetricSpace E
inst✝² : CompleteSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
hf : BoundedVariationOn f s
x : α
hs : s ∩ Iio x = ∅
⊢ ∃ l, Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case pos
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : CompleteSpace E
inst✝ : DenselyOrdered α
f : α → E
a : α
hf : BoundedVariationOn f univ
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
ha : IsBot a
⊢ eVariationOn f (Iic a) = eVariationOn f (Iio a) + edist (f a) (Function.leftLim f a)

pp: case neg
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : CompleteSpace E
inst✝ : DenselyOrdered α
f : α → E
a : α
hf : BoundedVariationOn f univ
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
ha : ¬IsBot a
⊢ eVariationOn f (Iic a) = eVariationOn f (Iio a) + edist (f a) (Function.leftLim f a)

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : CompleteSpace E
inst✝ : DenselyOrdered α
f : α → E
a : α
hf : BoundedVariationOn f univ
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
ha : IsBot a
⊢ eVariationOn f (Iic a) = eVariationOn f (Iio a) + edist (f a) (Function.leftLim f a)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case mp
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : StarModule R A
a : A
ha : p a
f : R → R
⊢ (fun x ↦ cfc x a) f ∈ ↑(cfcHom ha).range

pp: case mpr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : StarModule R A
a : A
ha : p a
f : C(↑(spectrum R a), R)
⊢ (↑(cfcHom ha)).toRingHom f ∈ Set.range fun x ↦ cfc x a

Tactic:
·

Goal state after:
case mp
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : StarModule R A
a : A
ha : p a
f : R → R
⊢ (fun x ↦ cfc x a) f ∈ ↑(cfcHom ha).range
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital

Goal state before:
pp: case mpr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : StarModule R A
a : A
ha : p a
f : C(↑(spectrum R a), R)
⊢ (↑(cfcHom ha)).toRingHom f ∈ Set.range fun x ↦ cfc x a

Tactic:
·

Goal state after:
case mpr
R : Type u_1
A : Type u_2
p : A → Prop
inst✝⁹ : CommSemiring R
inst✝⁸ : StarRing R
inst✝⁷ : MetricSpace R
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : ContinuousStar R
inst✝⁴ : TopologicalSpace A
inst✝³ : Ring A
inst✝² : StarRing A
inst✝¹ : Algebra R A
instCFC : ContinuousFunctionalCalculus R A p
inst✝ : StarModule R A
a : A
ha : p a
f : C(↑(spectrum R a), R)
⊢ (↑(cfcHom ha)).toRingHom f ∈ Set.range fun x ↦ cfc x a
Module: Mathlib.Topology.EMetricSpace.BoundedVariation

Goal state before:
pp: case inl
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : 𝓝[s ∩ Iio x] x = ⊥
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Icc y x)) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))

pp: case inr
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : (𝓝[s ∩ Iio x] x).NeBot
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Icc y x)) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))

Tactic:
·

Goal state after:
case inl
α : Type u_1
inst✝³ : LinearOrder α
E : Type u_2
inst✝² : PseudoEMetricSpace E
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → E
s : Set α
l : E
hf : BoundedVariationOn f s
x : α
h'f : Tendsto f (𝓝[s ∩ Iio x] x) (𝓝 l)
hx : x ∈ s
h : 𝓝[s ∩ Iio x] x = ⊥
⊢ Tendsto (fun y ↦ eVariationOn f (s ∩ Icc y x)) (𝓝[s ∩ Iio x] x) (𝓝 (edist (f x) l))
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: case hφ₁
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
⊢ Continuous[_, inst✝³] ⇑(starAlgHom (cfcHom hqa) h)

pp: case hφ₂
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
⊢ (starAlgHom (cfcHom hqa) h) (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) = a

Tactic:
·

Goal state after:
case hφ₁
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
⊢ Continuous[_, inst✝³] ⇑(starAlgHom (cfcHom hqa) h)
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: case hφ₂
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
⊢ (starAlgHom (cfcHom hqa) h) (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) = a

Tactic:
·

Goal state after:
case hφ₂
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
⊢ (starAlgHom (cfcHom hqa) h) (ContinuousMap.restrict (spectrum R a) (ContinuousMap.id R)) = a
Module: Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict

Goal state before:
pp: case pos
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
g : R → R
hg : ContinuousOn g (spectrum R a)
⊢ cfc g a = cfc (fun x ↦ (algebraMap R S) (g (f x))) a

pp: case neg
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
g : R → R
hg : ¬ContinuousOn g (spectrum R a)
⊢ cfc g a = cfc (fun x ↦ (algebraMap R S) (g (f x))) a

Tactic:
·

Goal state after:
case pos
R : Type u_1
S : Type u_2
A : Type u_3
p q : A → Prop
inst✝²¹ : Semifield R
inst✝²⁰ : StarRing R
inst✝¹⁹ : MetricSpace R
inst✝¹⁸ : IsTopologicalSemiring R
inst✝¹⁷ : ContinuousStar R
inst✝¹⁶ : Semifield S
inst✝¹⁵ : StarRing S
inst✝¹⁴ : MetricSpace S
inst✝¹³ : IsTopologicalSemiring S
inst✝¹² : ContinuousStar S
inst✝¹¹ : Ring A
inst✝¹⁰ : StarRing A
inst✝⁹ : Algebra S A
inst✝⁸ : Algebra R S
inst✝⁷ : Algebra R A
inst✝⁶ : IsScalarTower R S A
inst✝⁵ : StarModule R S
inst✝⁴ : ContinuousSMul R S
inst✝³ : TopologicalSpace A
inst✝² : ContinuousFunctionalCalculus S A q
inst✝¹ : ContinuousFunctionalCalculus R A p
inst✝ : ContinuousMap.UniqueHom R A
f : C(S, R)
halg : IsClosedEmbedding ⇑(algebraMap R S)
a : A
hpa : p a
hqa : q a
h : SpectrumRestricts a ⇑f
g : R → R
hg : ContinuousOn g (spectrum R a)
⊢ cfc g a = cfc (fun x ↦ (algebraMap R S) (g (f x))) a